[handbook of differential equations: ordinary differential equations] volume 3 || chapter 4...

CHAPTER 4

Bifurcation Theory of Limit Cycles of Planar Systems

Maoan Han Department of Mathematics, Shanghai Normal University, Guilin Road 100, Shanghai, 200234, China

Contents 1. Limit cycle and its perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

1.1. Basic notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

1.2. Multiplicity, stability and their property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

1.3. Perturbations of a limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

2. Focus values and Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

2.1. Poincar6 map and focus value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

2.2. Normal form and Lyapunov technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

2.3. Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

2.4. Degenerate Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

3. Perturbations of Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

3.1. General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

3.2. Existence of 2 and 3 limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

3.3. Near-Hamiltonian polynomial systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

3.4. Homoclinic bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

H A N D B O O K OF DIFFERENTIAL EQUATIONS

Ordinary Differential Equations, volume 3

Edited by A. Cafiada, E Dr~ibek and A. Fonda

�9 2006 Elsevier B.V. All rights reserved

341

Bifurcation theory of limit cycles of planar systems 343

In this chapter we present the bifurcation theory of limit cycles of planar systems with rel- atively simple dynamics. The theory studies the changes of orbital behavior in the phase space, especially the number of limit cycles as we vary the parameters of the system. This theory has been considered by many mathematicians starting with Poincar6 who first introduced the notion of limit cycles. A fundamental step towards modem bifurcation theory in differential equations occurred with the definition of structural stability and the classifica- tion of structurally stable systems in the plane in 1937 developed by Andronov, Leontovich and Pontryagin. More precisely, Andronov and Pontryagin introduced the notion of a rough system and presented the necessary and sufficient conditions of roughness for systems on the plane. Almost at the same time, Andronov and Leontovich carried out a systematic clas- sification of all principal bifurcations of limit cycles on the plane for the simplest nonrough systems. A further development of the theory had taken yet another direction, namely by selecting bifurcation sets of codimension one for primary bifurcations, and of arbitrary codimension in the general case for degenerate bifurcations. In the two-dimensional case, as was proved in Andronov et al. [2], rough systems compose an open and dense set in the space of all systems on a plane. The nonrough systems fill the boundaries between different regions of structural stability in this space.

In the following sections we concentrate on an in-depth study of limit cycles with general methods of both local and global bifurcations in high codimensional case. Many results are closely related to the second part of Hilbert's 16th problem which concems with the number and location of limit cycles of a planar polynomial vector field of degree n posed in 1901 by Hilbert [73].

1. L i m i t cyc le and its p e r t u r b a t i o n s

1.1. Basic notations

Consider a planar system defined on a region G C ~;~2 of the form

Yc = f (x), (1.1)

where f ' G --+ ]~2 is a C r function, r ~> 1. Then for any point xo 6 G (1.1) has a unique solution qg(t, xo) satisfying qg(0, xo) -- xo. Let q9 t (xo) = qg(t, xo). The family of the trans- formations qgt'G --+ ]~2 satisfy the following properties:

(i) ~o ~ -- Id; (ii) ~0 t+s = qgt o q)s.

The function ~0 is called the flow generated by (1.1) or by the vector field f . Let I(xo) denote the maximal interval of definition of ~0(t, x0) in t. If x0 6 G is such that q)(t, x0) is constant for all t 6 I (x0), then f (xo) = 0. In this case, x0 is called a singular point of (1.1). A point that is not singular is called a regular point.

For any regular point x0 6 G, the solution ~0(t, x0) defines two planar curves as follows"

V+(xo)={qg(t, xo)" t e l (xo) , t~>O}, V-(xo)={qg(t , xo)" te I (xo) , t~<O},

344 M. H a n

which are called respectively the posi t ive and the negative orbit of (1.1) through x0. The union y ( x o ) -- y + ( x o ) U y - ( x o ) is called the orbit of (1.1) through x0. The theorem about the existence and uniqueness of solutions ensures that there is one and only one orbit through any point in G. A per iodic orbit of (1.1) is an orbit which is a closed curve. The minimal positive number satisfying ~0(T, x0) = x0 is said to be the per iod of the periodic orbit y (x0). Obviously, y (x0) is a periodic orbit of period T if and only if the corresponding representation ~0(t, x0) is a periodic solution of the same period.

DEFINITION 1.1. A periodic orbit of (1.1) is called a limit cycle if it is the only periodic orbit in a neighborhood of it. In other words, a limit cycle is an isolated periodic orbit in the set of all periodic orbits.

Now let (1.1) have a limit cycle L: x = u(t), 0 ~< t <~ T. Since (1.1) is autonomous, for any given point p ~ L we may suppose p = u(0), and hence, u( t ) = ~o(t, p). Further, for definiteness, let L be oriented clockwise. Introduce a unit vector

1 Z0 = I f (p) l ( - f 2 ( p ) , f l (p))T.

Then there exists a cross section 1 of (1.1) which passes through p and is parallel to Z0. Clearly, a point x0 ~ 1 near p can be written as xo = p + aZo, a -- (xo - p)Tz0 E R.

LEMMA 1.1. There exist a constant e > 0 and C r func t ions P and r" ( - e , e) --+ I~ with

P (0) -- 0 and r (0) = T such that

q)(r(a), p + aZo) -- p + P ( a ) Z o ~ 1, [a[ < e. (1.2)

PROOF. Define Q(t , a) = [ f (p) lT( tp( t , p + aZo) -- p) . We have

Q ( T , O) O, Q t ( T , 0) ' - ' I f ( p ) [ 2 m ~ > 0 .

Note that Q is C r for (t, a) near (T, 0). The implicit function theorem implies that a C r

function r (a) = T + O(a) exists satisfying

Q( r ( a ) , a) - 0 or [ f (p)]T(qg(r(a) , p + aZo) - p ) -- O.

It follows that the vector ~0(r(a), p + aZo) - p is parallel to Z0. Hence, it can be rewritten as qg(r(a), p + aZo) - p - P ( a ) Z o , where P ( a ) = Z~(qg(r(a), p + aZo) - p) . It is obvious that P ~ C r for lal small with P (0) = 0. This ends the proof. 71

The above proof tells us that the function r is the time of the first retum to I. By Defini- tion 1.1, the periodic orbit L is a limit cycle if and only if P ( a ) :/= a for [a[ > 0 sufficiently

small.

DEFINITION 1.2. The function P" ( - e , e) --+ I~ defined by (1.2) is called a Poincar~ map

or return map of (1.1) at p 6 I. For convenience, we sometimes use the notation P" l --+ I.


Fig. 1. Behavior of a stable limit cycle.

DEFINITION 1.3. The limit cycle L is said to be outer stable (outer unstable) if for a > 0 sufficiently small,

a(P(a) - a) < 0 (> 0).

The limit cycle L is said to be inner stable (inner unstable) if the inequality above holds for - a > 0 sufficiently small. A limit cycle is called stable if it is both inner and outer stable. A limit cycle is called unstable if it is not stable.

For example, if L is stable, then the orbits near it behave like the phase portrait as shown in Fig. 1.

Let pk (a) denote the kth iterate of a under P. It is evident that { pk (a)} is monotonic in k and pk (a) > 0 (< 0) for a > 0 (< 0). Thus, it is easy to see that L is outer stable if and only if pk (a) --+ 0 as k --+ ec for all a > 0 sufficiently small. Similar conclusions hold for outer unstable, inner stable and inner unstable cases.

REMARK 1.1. If the limit cycle L is oriented anti-clockwise we can define its stability in a similar manner by using the Poincar6 map P defined by (1.2). For instance, it is said to be inner stable (inner unstable) if a(P(a) - a) < 0 (> 0) for a > 0 sufficiently small.

DEFINITION 1.4. The limit cycle L is said to be hyperbolic or of multiplicity one if P ' (0) ~ 1. It is said to have multiplicity k, 2 ~ k <, r, if P ' (0) = 1, P(J)(0) --O, j = 2 . . . . . k - 1, p(k)(0) ~ O.

By Definition 1.3, one can see that L is stable (unstable) if lP'(0)l < 1 (> 1).

1.2. Multiplicity, stability and their property

Next, we give formulas for P' (0) and P" (0). For the purpose, let

Ut(O) -- (Vl (19) 1)2(0)) T, v(O)- lu'(0)l z(o)- (-v2(o), Vl (o)) T,

346 M. Han

and introduce a transformation of coordinates of the form

x - u(O) + Z(O)b, O <<. O <<. T, I b l < e . (1.3)

LEMMA 1.2. The transformation (1.3) carries (1.1) into the system

dO db = 1 + gl(O, b), = A(O)b -k- g2(O, b),

dt dt (1.4)

where

d lnlf(u(O))[, A(O) = zT(O)fx(u(O))Z(O) - trfx(u(O)) --

gl(O,b) =h(O,b)[ f (u (O) + Z(O)b) - f (u(O))] - h(O,b)Z'(O)b,

g2(O, b ) - zT(o)[ f (u (O) + Z(O)b) - f (u(O)) - fx(u(O))Z(O)b],

h(O. b) - ([ f (u(o)) l + I)T (o)Zt (O)b)-ll)T (o),

and tr f x (u (0)) denotes the trace of the matrix f x (u (0)), which is called the divergence of the vector field f evaluated at u(O).

PROOF. By (1.3) and (1.1) we have

dO db (u' + Z'b )--d-; + Z--d- ~ - f (u + Z b ). (1.5)

dO and db from (1.5). First, multiplying (1.5) by In order to obtain (1.4) we need to solve -d-/- ?7 v T from the left-hand side and using

V T Z - - 0 , = i . , i = i f ( . ) l .

we can obtain

dO

dt = [If(u)[ + v T Z ' b ] - l v T f ( u + Z b ) = h ( O , b ) f ( u + Zb).

Note that

h(O, b) f (u) = h(O, b ) [ f (u) + Z'b] - h(O, b)Z'b = 1 - h(O, b)Z'b.

It follows that

h(O, b ) f ( u + Z b ) - h(O, b ) [ f (u + Z b ) - f (u ) ] - h ( O , b)Z'b + 1.

Then the first equation in (1.4) follows.


Now multiplying (1.5) by Z T from the left and using

Z T Z - '- 1, Z T f (u) -- O, z T z t 1 = ' - o ,

we obtain

db

dt T -- z T [ f (u -+- Zb) - f (u) - f x ( u ) Z b ] + Z f x ( u ) Z b .

It is direct to prove that

Z T f x ( u ) Z = trfx (u) - - d

~ln l f (u) l .

Then the second equation of (1.4) follows. This finishes the proof.

Set

B(O) = [fx(U + zb)];la= o, c ( o ) - ~T[L(u)Z - Z'(O)], (1.6)

and

R(O,b) -- A(O)b n t- g2(O, b)

1 + gl (0, b)

Then by Lemma 1.2, we can write

1 [ 2 A C lb2 + O(b3 ) R(O, b) -- A(O)b + -~ Z T B Z l / (u)l

1 = A ( O ) b + -~Al(O)b 2 + O(b3). (1.7)

For Ibl small we have from (1.4)

db - - R ( O , b ) (1.8)

dO

which is a T-periodic equation. From Lemma 1.2 we know that the function R is C r - 1 in (0, b) and C r in b. Let b(O, a) denote the solution of (1.8) with b(0, a) - a. We have:

LEMMA 1.3. P(a) -- b(T, a).

PROOF. Consider the equation

dO -- 1 + gl (0, b(O, a)).

dt

348 M. Han

It has a unique solution 0 = O(t, a) satisfying 0(0, a) = 0. From (1.7) it implies b(O, O) = O. 30 (T, 0) -- 1 Hence, by the implicit function theorem a unique This yields O(T, O) -- T, 37

function f'(a) = T + O(a) exists such that 0(f', a) = T. For x0 = u (0) + Z (0)a, we have by (1.3)

qg(t, x o ) - u(O(t, a)) 4- Z(O(t, a))b(O(t, a), a).

In particular,

qg(~, xo) -- u(T) 4- Z (T)b(T , a) -- u(O) 4- Z(O)b(T, a).

Thus, it follows from Lemma 1.1 t h a t r = ~" and P(a) = b(T, a). The proof is completed. V]

For lal small we can write

b(O, a) - bl (O)a + b2(O)a 2 + O(a3),

where bl (0) -- 1, b2(0) - 0 . By (1.7) and (1.8) one can obtain b] = Abl, b~2 -- Ab2 4- 1 2 A 1 b I , which give

f0 bl (0) -- exp A(s) ds, f0 1 b2(0) = bl(0) -~Al(s)bl(s)ds.

Then by Lemma 1.3 we have

P ' (0) -- bl (T) -- exp fo T A (s) ds -- exp fo T trfx (u(t)) dt,

f0 T P " ( O ) - - 2 b 2 ( T ) - - b l ( T ) Al (s )b l (s )ds .

Thus, noting (1.7) we obtain the following theorem.

THEOREM 1.1. Suppose P is a Poincard map of (1.1) at p ~ L. Then

(i) P ' (0) -- exp fL div f dt, div f - tr fx,

" fo T [ 2A( t )C( t ) ] dt" (ii) P (0) -- P'(O) efoA(s)ds Z T(t)B(t)Z(t) -- I f (u( t)) l

Hence, L is stable (unstable) if I (L) -- fL div f dt < 0 (> 0).

We remark that Theorem 1.1 remains true in the case of counter clockwise orientation of L.


EXAMPLE 1.1. Consider the quadratic system

= - - y ( 1 + c x ) - (x 2 -Jr- y2 _ 1),

~ = x ( l + c x ) , O < c < l.

This system has the circle L: X 2 + y2 = 1 as its limit cycle. We claim that the cycle is

unstable. In fact, we have

I (L) -- ( - 2 x - cy) dt - 1 + cx 2dy )

l + c x

_ f f d x d y 2+y2~1 (1 + CX) 2

> 0 .

EXAMPLE 1.2. The system

2 = - y - x ( x 2 -Jr- y 2 1)2,

= x -- y ( x 2 + y2 _ 1)2

has a unique limit cycle given by L: (x, y) = (cost, sint), 0 ~< t ~< 2re. For the system, it is easy to see that v (0) = ( - sin 0, cos 0)T, Z (0) = ( - cos 0, - sin 0)T. By Lemma 1.2 and

(1.6) we then have

8 cos 2 0 8 sin 0 cos 0 ) A (0) -- 0, B (0) -- 8 sin 0 cos 0 8 sin 2 0 "

Thus from Theorem 1.1 it follows p1 (0) = 1, pt~ (0) = 16zr. This shows that L is a limit cycle of multiplicity 2.

From (1.6)and formulas for PI (0) and P"(0) in Theorem 1.1 the derivatives P ' (0) and P"(0) are independent of the choice of the cross section 1. This fact suggests that the stability and the multiplicity of a limit cycle should have the same property. Below we will prove this in detail even if the cross section 1 is taken as a C r smooth curve.

To do this, let L be a limit cycle of (1.1) as before and let ll be a C r curve which has an intersection point pl E L with L and is not tangent to L at Pl. Then it can be represented

as

ll" X - - p l + q ( a ) , q(O)--O, d e t ( f ( p l ) , q ' ( O ) ) > 0 ,

where q : ( - e , e) -+ 1R is C r for a constant e > O. The condition d e t ( f ( p l ) , q1(O)) > 0

means that the point pl + q (a) is outside L if and only if a > O. In the same way as Lemma 1.1 we can prove that there exist two C r functions

P1, rl : ( - e , e) ~ ll~, P1 (0) = 0, rl (0) = T

350 M. Han

Fig. 2. Two Poincar6 maps.

such that

~0 ('rl (a), Pl -+- q(a)) -- Pl -+- q(P1 (a)) E ll. (1.9)

This yields another Poincar6 map P1" ( - e , e) -+ R.

LEMMA 1.4. Let P and P1 be two Poincard maps defined by (1.2) and (1.9), respectively. f Then there exists a C r function hi" ( - e , e) --~ IR with hi (0) = 0, h I (0) > 0 such that hi o

P = P l o h l .

PROOF. Since p = u(0) we can suppose Pl - -U( t l ) for some tl E [0, T). Similar to Lemma 1.1 again, there exist two C r functions hi and r*, both from ( - e , e) to R, with

h i (0) = 0 and r* (0) = tl such that

q)(r*(a), p + aZo) -- Pl + q (h l (a ) ) E l l . (1.10)

See Fig. 2. Let xo = p + aZo, Xl = q)(r*(a), xo), x2 -- q)(r (a), xo). By (1.10) and (1.2) we have Xl = pl + q(a l ) , al -- hi (a) and x2 = p + P(a )Zo . Hence, by (1.9) and (1.10) we

have

go(V1 (al ) , Xl) ---- Pl -i- q(P1 (a l ) ) ,

go(v*(P(a)),x2) - - P l -k- q(a2), a 2 - h l ( P ( a ) ) .

On the other hand, by the flow property of ~o we have

X3 -- gO(Vl (al) , Xl) -- qg(Vl (a l ) + v*(a) , xo) -- go('c*(P(a)) + v(a), xo) = ~ ( ~ * ( P ( a ) ) , ~ ) ,

which, together with the above, follows that q(P1 (al)) - q(a2) or a2 - P1 (al). Hence

hi o P = P1 o hi .


It needs only to prove h~ (0) > 0. Let a ~> 0. Introduce one more cross section below

l'" x -- u (tl) + Z( t l )a , 0 ~< a ~< e.

Let "~1 (a) -- tl -4- O(a) be such that 0('~1, a) = tl. By (1.3) we have

Xl = tp( 'rl , XO) --" U(tl) + Z(tl)b(tl , a) E l'.

Then b(tl, a) = IplXl I. By the proof of Lemma 1.3,

Ob _ fO tl Oa (tl, 0) exp A(s) ds > 0.

Consider_~_the triangle formed by points pl , Xl and x-1. There exists a point x* on the orbital arc x1~1 such that f ( x*) is parallel to the side XlX-1. Since the arc XlXl approaches Pl as a -+ 0 we have x* -~ pl , f ( x*) --~ f ( P l ) as a -+ 0. Hence, if we let Otl denote the angle between sides p l X l and X-lXl, and or2 the angle between sides plXl and X l X l , then

7r we have Oil --+ ~-, Ct2 --+ Oe0 as a -+ 0, where or0 6 (0, ~ ] is the angle between the vectors f ( p l ) and qt(0). That is, or0 is the angle between L and ll at pl . By the Sine theorem, it follows

IPlYll IplXll

sin or2 sin a 1 or

sinotlb(tl a) -- a [q(hl(a))[ = sin~-----2 ' sin~o

f0 tl ~ e x p A(s)ds(1 + O(a)) .

On the other hand, q(hl (a)) = q'(O)h~l (O)a + O(a 2) which gives

Iq(hl(a))l = Iq'(o)l . Ih](O)la + O ( a 2 ) , a > O.

Hence, we obtain

1

]h] (o)J - Iq'(0)l sinc~o f0 tl exp A (s) ds # 0.

! Noting that h l (a) > 0 for a > 0 we have h 1 (0) > 0. The proof is completed. D

COROLLARY 1.1. The stability and the multiplicity of the limit cycle L are independent of the choice of cross sections.

PROOF. By Lemma 1.4 we have

h~ (a)[P(a) - a] = Pl(hl(a)) - hi(a), (1.11)

352 M. Han

where ~ lies between a and P(a ) . By (1.11) and Definition 1.3, the stability of L under P is the same as that of it under P1. Also, if the limit cycle L has multiplicity k under P, then P ( a ) - a -- c~a ~ + O(a k+l) for some c~ # 0. It follows from (1.11) that

Pl(a) - a = ? k a k + O(ak+l), ck -- r [h t 1 ( o ) ] k - 1

Therefore, L has the same multiplicity under P1. This ends the proof. D

In the following we discuss the relation of equivalence between two planar systems. Consider (1.1) and another system of the form

y = g(y), (1.12)

where g" D --+ ]1~ 2 is C r (r ~> 1) on a region D C ]1~ 2.

DEFINITION 1.5. Let U C G, V C D be two regions. Two planar systems defined by vector fields f l u and g lv are said to be C k (1 <<, k <<, r) equivalent if there exists a C k diffeomorphism h" U --+ V which takes orbits of (1.1) on U to orbits of (1.12) on V preserving their orientation.

Let 7r(t, y) denote the flow generated by (1.12). Then under the condition of Defini- Os tion 1.5 a C ~ function s( t , x ) exists with s(O, x) -- O, ~ > 0 such that

h - ( s ( t , (1.13)

as long as qg(t, x) ~ U. If the functions f and g have the following relationship:

g ( h ( x ) ) = D h ( x ) f (x) ,

then the system (1.12) is obtained from (1.1) by making the coordinate transformation y -- h ( x ) . In this case, (1.13) becomes hocp t - 7r t oh. If f and g satisfy g ( x ) - K ( x ) f ( x ) where K" R 2 ---> R is a C r positive function, then orbits of (1.1) and (1.12) are identical, and the flows q9 and 7t satisfy

d~ d~r K ((p(t, x ) ) ~ -- g((p(t, x)), ds =

Hence, we are to sink a function s( t , x ) with s(0, x) - 0 and satisfying

qg(t, x ) - Tt(s( t , x ) , x ) .

Differentiating the equality in t gives

dgo d~ ds

dt ds dt


Substituting the equality above into the previous equations yields

d~ ds

dq9 d~p ds = K (~o(t, x ) ) ~ - K (qg(t, x)) ds d!

Thus, the function s should satisfy ds _ 1 which has the solution - ~ - - K ( q g ( t , x ) ) '

fo t dt s(t, x) -- K (~p(t, x))

With this choice of s(t , x) above one can prove qg(t, x) = ~p(s(t, x), x) by the uniqueness

of initial solutions. For C k equivalent systems we have the following lemma.

LEMMA 1.5. Suppose ( 1 . 1 ) a n d ( 1 . 1 2 ) a r e C k equivalent under a C k diffeomorphism h : U --+ V, 1 <<, k <~ r. Let L C U and L1 = h(L) C V be limit cycles o f (1.1) and (1.12)

respectively. Then: (i) The cycles L and L1 have the same multiplicity.

(ii) The inner (outer) stability o f L is the same as the inner (outer) stability o f L1 if h( ( ln t .L) A U) C In t .L1 , and the inner (outer) stability o f L is the same as the outer (inner) stability o f L1 if h((Int .L) N U) C Ext.L1.

PROOF. For the limit cycle L, take a cross section 1 at p E L as before. Then the curve

ll -- h(l) is a cross section of L1 at Pl = h(p) . Let

1 ! Z1 -- ( - g 2 ( P l ) , gl (P l ) ) T, Z 1 -- hx(P)Zo.

[g(Pl)[

! ! Note that h(p + aZo) -- Pl + Z l a + O(a2). It is easy to see that the vector Z 1 is tangent

! to ll at Pl . According to the orientation of L 1 and the direction of Z 1 there are four cases to consider as follows:

! �9 Case 1. Z1 �9 Z 1 > 0 with L 1 clockwise oriented;

! �9 Case 2. Z1 �9 Z 1 < 0 with L1 clockwise oriented;

! �9 Case 3. Z1 �9 Z 1 > 0 with L 1 counter clockwise oriented;

! �9 Case 4. Z1 �9 Z 1 < 0 with L1 counter clockwise oriented.

See Fig. 3. By (1.13) the periods T of L and T1 of L1 have the relation T1 = s(T, p). Let P be the

Poincar6 map near L defined by (1.2). The cross section 11 has a representation

y = h (p + aZo) = Pl + q (a ) ,

! where q(a) - h (p + aZo) - h (p ) - Z l a + O(a2). Introduce a function ql as follows:

q (a) for case 1 or 3, ql(a) = q ( - a ) for case 2 or 4.

zf

11

354 M. Han

C a s e 1 Case 2 C a s e 3 Case 4

! Fig. 3. Four possible cases for L 1 and Z 1 .

! ! Then 11 can be rewritten as 11" y = pl n t- ql (a), lal << 1 with ql satisfying Z 1 �9 ql (0) > 0. On 11 we can define a Poincar6 map P1 near L1 by

~(Z'l (a), Pl + ql (a)) = Pl n t- ql (P1 (a)) E l l , (1.14)

where rl (a) = T1 + O(a) is the time of the first return to ll. Since h(p + aZo) = Pl + q(a) = Pl + ql ( ( - 1 ) i - l a ) for case i, we have

h(p + P(a )Z0) = Pl + q(P(a) ) = Pl + ql ( ( - 1 ) i - 1 P ( a ) ) E ll

and hence by (1.2)

h(q)(r (a) , p + aZo)) = P l -t- q l ( ( - - 1 ) i - 1 P(a)) Ell. (1.15)

On the other hand, by (1.13) we have

h(~0(r(a), p -+- aZo)) = 7t (s ( r (a ) , p + aZo), h(p + aZo))

= ~ ( r * ( a ) , pl Jr- ql ( ( - 1 ) i - l a ) ) E ll, (1.16)

for case i, where r* (a ) = s ( r ( a ) , p + aZo) = T1 + O(a). Hence, comparing (1.14) with (1.16) we obtain

r* (a ) = rl ( ( - 1 ) i - l a ) , h (go(r(a), p + aZo)) - pl + ql (e l ( ( - 1 ) i - l a ) ) �9

Therefore, it follows from (1.15) that ( - 1 ) i - l p ( a ) -- P l ( ( - 1 ) i - l a ) for case i. Thus for case i we have

a(P(a) - a) = ( - 1 ) i - l a [ P 1 ( ( - 1 ) i - l a ) - ( - 1 ) i - l a ] .

Hence, similar to Corollary 1.1, one can prove easily that L and L 1 have the same multiplicity. Furthermore, they have the same stability for cases 1 and 4. However, for cases 2 and 3, the inner (outer) stability of L is the same as the outer (inner) stability of L1.

Then noting that h ((Int.L) n U) C Int.L 1 for cases 1 and 4, and h ((Int.L) N U) C Ext.L 1 for cases 2 and 3, the proof follows from Corollary 1.1. D


EXAMPLE 1.3. Consider the system

m y , -- --x -Jr- y (x 2 -k- y 2 1).

By Theorem 1.1, the system has a unique limit cycle L: x 2 + y2 = 1, and it is hyperbolic

and unstable.

Define four functions

hi (x, y) = (x, y)T, x y

h2(x, Y) i

X 2 -k- y 2 ' X 2 _3 r_ y2 ( )T y x h3(x, Y)

i

x 2 + y2' x 2 + y2 h4(x, y) = (y, x) T,

and a region U = {(x, y)" X 2 -+- y2 > 1 }.

Let

[t = gi l (u, V), V = gi2 (u, V)

denote the system obtained from the previous cubic system by making the coordinate transformation (u, v) v = hi(x , y), (x, y) E U. Note that the unit circle is invariant under each hi. It is evident that the case i in Fig. 3 occurs with L 1 -- L for each i E { 1, 2, 3, 4}.

This example shows that each case in Fig. 3 can happen. In particular, the orientation of a limit cycle may be changed under a coordinate transformation.

1.3. Perturbations of a limit cycle

In the rest, we return to the perturbations of a limit cycle. Consider the following system

2 = f ( x ) + F(x , #) (1.17)

where # E II~ m is a vector parameter with m ~> 1, and F : G x R m --+ lt~ 2 is a C r function with F(x , 0) = 0. Thus, system (1.1) is the unperturbed system of (1.17). As before, let (1.1) have a limit cycle L: x = u(t), 0 <<, t <~ T. Then completely similar to Lemma 1.2 we have the following lemma.

LEMMA 1.6. The transformation (1.3) carries (1.17) into the system

O -- 1 + gl (0, b) + h(O, b)F(u(O) + Z(0)b , /z) , b - A(O)b + g2(O, b) -+- zT(O)F(u(O) + Z(0)b , /z) ,

(1.18)

where the functions A, g l, g2 and h are the same as those given in Lemma 1.2.

356 M. Han

By (1.18) we have

db = R(O,b ,#) ,

dO (1.19)

where R(O, b, I~) = A(O)b 4- zT(o)F#(u(O) , 0)/z 4- O(Ib,/zl2). Let b(0, a, #) denote the solution of (1.19) with b(0, a, /z) -- a. We can write

b(O, a, lz) - bl (O)a 4- bo(O)lz 4- O([a,/z[2),

where bl (0) = 1, b0(0) = 0. Inserting the above into (1.19) yields

b] = Abl, b~o = Abo + zTF# (u(s), 0).

It follows that bl (0) -- exp fo A(s) ds as before, and

f0 T bo(O)=bl(O) b l l ( s )ZT(s )F#(u(s ) ,O)ds .

Therefore,

b(T ,a , lz) - -exp A(s)ds a + b-~l(s)ZT(s)F#(u(s) ,O)dslz

+ O(la, #la). (1.20)

On the other hand, using (1.2) we can define a Poincar6 map P(a, lz) similarly. By Lemma 1.3 it follows

P (a, lz) = b(T, a, lz). (1.21)

Obviously, for I~1 small (1.17) has a limit cycle near L1 if and only if P has a fixed point in a near a = 0.

The simplest case is that L is hyperbolic. In this case, L will persist under perturbations. In other words, we have"

THEOREM 1.2. Let L be hyperbolic. Then there exist e > 0 and a neighborhood U of L such that (1.17) has a unique limit cycle in U for Itzl < e. Moreover, the limit cycle is also hyperbolic and has the same stability as L.

PROOF. Since L is hyperbolic we have Pa~(0, 0 ) # 1, or equivalently I ( L ) - - f L div f dt # 0. By (1.20) and (1.21),

P (a, lz) - a -- (e I (L) _ a)a + Nolz + O(la,/zl2), (1.22)

where No -- e I (L) f : exp(-- fo A (s) ds) Z T (s) F u (u(s), O) ds.


For (a, /z) near zero applying the implicit function theorem to the equation P (a, #) - a -- 0 we find a unique fixed point a = a* (/z) -- O(/z) of P. This means that for [#l small (1.17) has a unique limit cycle, denoted by L u, near L. The cycle has a representation q)(t, p + a*(#)Zo , #) where q ) ( t , x , # ) denotes the flow of (1.17) with qv(0, x , /z) = x. Moreover, by (1.22) we have P'a (a* (#) , /x) = e I(L) + O([a*(/z), #1), which gives

(e ' ( L ) - 1) [Pa~ (a* (/~), /z) - 1] > 0

for all [#l small. Then the conclusion follows from Theorem 1.1. This ends the proof. V]

Further, for the nonhyperbolic case we have

THEOREM 1.3. Let L be a nonhyperbolic limit cycle o f the C r system (1.1) with r ~ 2. (i) I f L has multiplicity 2, then there exist e > O, a neighborhood U of L and a C r

function A ( # ) = 0 ( # ) such that (1.17) has no limit cycles (respectively a limit cycle o f multiplicity 2, two hyperbolic limit cycles) in U as A (#) < 0 (respectively = O, > O)for

[lzl < ~. (ii) I f L has multiplicity k with 3 <<, k <, r, then there exist s > 0 and a neighborhood U

o f L such that (1.17) has at most k limit cycles in U for ]/z[ < e. Moreover, (1.17) has at least a limit cycle in U for ]/z] < s if k is odd.

PROOF. Since r >~ 2, I (L) -- 0 and L has multiplicity 2 we can rewrite (1.22) as

P(a , / z ) - a = Q0(/z) 4- Q1 (/z)a 4- Q2(#)a2(1 4- o(1)),

where Q0(0) = Q1 (0) -- 0, Q2(0) ~ 0. Let G(a, #) = P(a, #) - a, which is called a bifurcation function of (1.17). By the im-

plicit function theorem a unique C r function q (#) = O(#) exists such that Ga (q (lZ), I z) -- 0 for ]#[ small. Then Taylor's formula yields

1 G(a, lZ) -- G(q( lz ) , lZ) + ~Gaa(q( tz ) , lz)(a - q(/z))2(1 4- o(1)).

Now it is clear that the conclusion (i) follows by taking A(# ) = - G ( q ( l z ) , / z ) Q2(0). To prove the second conclusion let us assume (1.17) has k + 1 limit cycles in an arbitrary

neighborhood of L for some sufficiently small /z 7~ 0. Then the function G has k + 1 zero ~c has k zeros in a. Using the same theorem in a for this small #. From Rolle's theorem, -b--h-

~ c has a zero ao(#) which can be arbitrarily small as # goes to repeatedly we see that -ya-ra ~ z e r o .

On the other hand, we have

OkG okp = P (a, O) - a -- qka k + o(ak), qk ~ O,

Oa k Oa k '

which implies ok G -- k!qk 4- o(1) ~: 0 for (a, #) near zero. This is a contradiction. Hence, (1.17) has at most k limit cycles near L for I#[ small.

358 M. Han

Finally, if k is odd, we have then qka[P(a, 0) - - a ] > 0 for [al = e, where s > 0 is a small constant. Thus, there exists a constant 6 > 0 such that qka[P(a,/1)--a] > 0 for [al = s, 1/11 ~< 6. Therefore, a function a*(/1) exists with la*(/1)l < s such that P(a*(/1),/1) - a* (/1) = 0, 1/11 < 3. The proof is completed. Z]

EXAMPLE 1.4. Consider

2 = - y - + - x ( x 2 4. y 2 1) 2 - - - - X [ / 1 1 (X 2 4. y2)4 . /12] , x + y(x2 + y 2 1) 2 = -- -- Y[/11 (x 2 4. y2)4-/12] .

(1.23)

F o r / 1 1 : / 1 2 : 0 , (1.23) has a unique limit cycle

L: x = c o s t , y = s i n t , 0 ~ t ~ < 2 z r .

Similar to Example 1.2, the limit cycle L has multiplicity 2 with Pa" (0, 0) = -16zr . We make a transformation of the form (x, y) = (1 - b ) ( c o s 0, sin0), 0 ~< 0 ~ 2zr, so that (1.23) becomes

0=1, b = --(1 -- b)[h 2 - / 1 1 h - (/11 4-/12)],

where h = (1 - b) 2 - 1. Hence we have

db = - ( 1 - b)[h 2 - / 1 1 h - (/11 + / 1 2 ) ] . dO

Obviously, the solution b(O, a,/11,/12) of the above equation with b(0, a, /11,/12) = a is 27r-periodic near b -- 0 if and only if the initial data a satisfies

G* (a, /11, /12) - - a 2 - / 1 1 a - (/11 4./12) m~0.

Let A ( / 1 1 , / 1 2 ) -~- / 12 -t-4(/11 4./12). Then for (/11,/12) n e a r (0, 0), (1.23) has no limit cycles (respectively a unique multiple 2 limit cycle, two hyperbolic limit cycles) if A (/11,/12) < 0 (respectively = 0, > 0). The equation A (/11,/12) = 0 defines a curve/12 - - - - /11 - - 1 /12 on the (/11,/12)-plane, which is called a bifurcation curve of saddle-node type. See Fig. 4.

Turn back to system (1.17). Note that

d 1 A(O) = trfx(u) -~ l n l f (u ) ] , Z(O) = [f(u)[ ( - f e (u ) , f l (u)) .

1 e I (L) M, where The constant No in (1.22) can be written as No -- If(u(O))l

fo T M - - e- fo tr fx(u(s))ds f (u(t)) A Fu (u(t ), 0)dt , (1.24)

with (al , a2) A (bl, b2) -- alb2 - a2bl,/1 E R.


, L . . . . . . . . . . . . . . .

o f12

Fig. 4. Bifurcation diagram (saddle-node type) for (1.23).

Thus, if P (a, 0) - a = qka k + o(ak), we can rewrite (1.22) as

G(a, #) = P (a, #) - a = qka k + el(L)

If(u(O))l M # + o([/z, akl). (1.25)

Using (1.25) we can prove

THEOREM 1.4. Let (1.17) be analytic with tx ~ R. I f M # O, then exist e > 0 and a neighborhood U of L such that for I/zl < e and in the region U, (1.17) has

(i) a unique limit cycle with multiplicity one if L is of odd multiplicity; (ii) two limit cycles with each having multiplicity one for IZ lying one side of I~ = 0 and

no limit cycles for # lying the opposite side if L is o f even multiplicity; (iii) no limit cycles if L is nonisolated.

PROOF. By Theorem 1.2 we may suppose I ( L ) = 0. By (1.25), a unique function of the form

qk (ak+l) # , # = - - - ~ l f ( u ( O ) ) l a k + 0 = (a) (1.26)

exists such that G(a, #*(a)) = O. If k is odd with qk 7 ~ 0 the function #* has a unique inverse

a _ ( _ M # ) l / k

qkl f (u(O))l (1 + o (# l /k ) ) -- a*(#) ,

OG (a* which satisfies -ya- (#), #) # 0. Then the conclusion (i) follows. The conclusion (ii) follows just similarly. In the case that L is a nonisolated periodic orbit, we have G(a, 0) = 0 for lal small. This

implies that #*(a) - 0 since G(a, #) = 0 if and only if # = #*(a) . Hence, for all I~1 > 0 we have G (a, #) # 0. This means that (1.17) has no limit cycles near L for I~1 > 0 small. This ends the proof. D

360 M. Han

EXAMPLE 1.5. Consider (1.23) again with (/z 1,/z2) varying on a straight line. That is, let /Zl =/Z,/Z2 --C/Z, where c is a constant. Then (1.23) becomes

2 = - y + x (x 2 + y2 _ 1) 2 _ #x (x 2 + y2 + c),

x + y (x 2 -+- y2 1) 2 y2 c). = -- --/xy (X 2 -+- -+- (1.27)

By (1.24) we have M -- (1 + c)2yr. Note that q2 -- 1Pa" (0, 0) - -8yr . The formula (1.26) 4 a 2 becomes/z* (a) = i -~ + O(a3), which has inverse functions

l + c ]1/2 a=aj ( t z ) - - 4 # [(-- l)J + O(1#ll/2)] ' j = 1,2.

Thus, when 1 + c 7~ 0, (1.27) has no limit cycles (2 limit cycles) for (1 + c)/z < 0 (> 0). When (1 + c) = 0, then (1.27) has always two limit cycles given by x 2 + y2 __ 1 and x 2 nt_ y2 = 1 - / x for all I#l > 0 small. This shows that the condition M ~ 0 in Theorem 1.4 is somehow necessary.

One can give the bifurcation diagram of (1.27) and compare it with Fig. 4.

REMARK 1.2. A typical result on qualitative theory of differential equations is the so- called Poincar6-Bendixson theorem which says that the positive limit set of a bounded positive semi-orbit is a connected curve consisting of singular points and orbits connecting them. One can find a proof of the theorem in Hale [35], Chicone [14] and Zhang et al. [127]. An important corollary of the theorem is that for an analytic system a positively invariant set with no singular points contains at least one limit cycle. This corollary has many applications to various planar systems to study the existence of a limit cycle. On the other hand, there are also many results on the nonexistence of a limit cycle or the existence of two or more limit cycles. The reader can consult Zhang et al. [ 127], Ye [ 116], Ye et al. [ 117] and Luo et al. [ 101] for a general theory of limit cycles. For a given planar system with or without parameters it is usually very significant and also difficult to find the number of limit cycles of it. Thousands of papers have been published on this aspect. For recent works, one can see [1,6,8,11,15,18,19,22-26,28-34,97-99].

Results in Theorems 1.1-1.4 are fundamental which can be found in Andronov [2], Chow and Hale [16] and Han [51]. The conclusions in Lemma 1.5 seem very natural, but not obvious. The proof presented here is new.

2. Focus values and Hopf bifurcation

2.1. Poincard map and focus value

In this section we consider local behavior of a planar C ~ systems near an elementary focus.


After removing the focus to the origin, the system can be written as

2 = ax + by + F(x , y) = f ( x , y) , y -- cx + dy + G ( x , y) = g (x , y) ,

(2.1)

where c # 0, F and G" ]R 2 --+ IR are C ~ functions with

F(0 , 0) -- G(0, 0) - 0, O(F, G)

a(x, y) ~ ( 0 , 0) - 0,

and the eigenpolynomial h O 0 = ) 2 _ (a + d))~ + ad - bc has a pair of conjugate complex

zero ot + ifl # 0. Let

o/ - -a (1) C - - fl C -1 c

0 /~

o / - - a ) 1

= C 0 /7 "

C

By using h (a 4- ifi) -- 0, or

012 _ f12 __ (a + d)ot + ad - bc = O, (2or - (a + d))/3 -- 0,

it is easy to see that

C_ 1 ( a c d - /3 ot '

1 o t - = (a + d), Z

1 v / _ 4 b c _ (a - d ) 2 I /~1- ~

(2.2)

Then the linear transformation (x, y)T __ C ( u , v) T carries (2.1) into the form

it --oeu + fly + F ( u, v ), iJ -- - f l u + otv + G(u , v),

where F, G - O(lu, ol2). We call the above system a first order normal form of (2.1). Let us define a Poincar6 map of (2.1) near the origin. For any given 00 E [0, 27r), let

Zo -- (cos 0o, sin 0o) T, Z~- - (sin 0o, - cos 0o) T.

The unit vector Zo determines a cross section I below

I: (x, y )T _ rZo, 0 < r < ro

3 6 2 M. Han

for ro > 0 small. For (xo, yo) T = rZo ~ l, let (x(t, xo, Yo), y(t, xo, Yo)) be the solution

of (2.1) with x(0, xo, Y0) = x0, y(0, xo, Yo) = Y0. Then the formula of constant variation

follows:

t [ ( a' (x(t, xo, Yo) = exp \ y(t, xo, yo) ct

b t ) ] ( x ~ yo,2). dt yo

(2.3)

Thus,

( ) _ ( ) [ ( a t b t ) ] x(t, xo, Yo) xo -- exp - I Zor y(t, xo, Yo) Yo ct dt

+ O ( r 2) =_rV(t,r).

By (2.2) we have

[ V ( t , r ) = C exp - / 3 t ott~t)-I] C-1Zo+O(r).

Let H (t, r) = IV(t, r)] TZ~-. Define

V (t, 0) = r~01im V (t, r) = C exp - /~ t ~ t ) _ I] C - 1 Z o . ott

Also, if we allow the variable r to be negative in the definition of 1 then H is well defined

for [r[ < ro. It is easy to see that

H(r, 0) = (e a r - 1 ) Z o Z ~ = 0, Ht (T, 0) = e ~r r r 0,

2~r. Hence, the implicit function theorem yields that a unique C ~ function where T = I-N r ( r ) = T + O(r ) exists such that H(r(r), r) = 0 for Ir] < ro. That is to say, for 0 < r < ro, r (r) is the time of the first return to 1 by a circle. Therefore, we can write

x(r, xo, Yo) ) _ P ( r ) Z o ~ 1 y( r , xo, Yo)

0 < r < ro, (2.4)

where P 6 C ~ for [rl < ro with r P (r) > 0 for r :fi 0.

DEFINITION 2.1. Let (2.1) satisfy (2.2). The function P defined by (2.4) is called a

Poincar~ map of (2.1) near the origin. The map is often written as P :l ~ 1.

! ! DEFINITION 2.2. If there exists r o > 0 such that P(r) - r < 0 ( > 0) for 0 < r < r o, we

say the origin is a stable (unstable)focus. If P(r) - r = 0 for 0 < r < r~, we say the origin

is a center.

REMARK 2.1. Similar to Corollary 1.1 we can prove that the stability of the origin is

independent of the choice of 00 which is the angle defining the section 1.


In fact, for any given 0o, 01 �9 [0, 2Jr) with 0o ~ 01, let the corresponding cross sections and Poincar6 maps be lo, ll and Po'lo ~ lo and P1 "11 ~ 11, respectively. Similarly, there exist unique C ~ functions r* (r) = tl + O(r) and h l (r) -- O(r) such that

x ( r* , rZo) ) _ hi ( r )Zl , y(r* , rZo)

tl �9 (0, T).

Differentiating the equation above in r and using formula of constant variation (2.3) we obtain

exp ( a t l Ctl

btl ) ] Zo - h~l (O)Z1, dtl

which yields h] (0) 7~ 0. It is clear that h l (r) > 0 for r > 0. Thus, h~ (0) > 0. Then in the same way as in Lemma 1.4 we have

hi o PO---- P1 o hl ,

which gives the conclusion in Remark 2.1 easily.

LEMMA 2.1. For any given Oo �9 [0, 2Jr) the origin is a stable (unstable)focus of (2.1) if and only if

r[P(r) - r] < 0 (> O) forO < Irl < ro.

Hence, if for some k ~ 1 it holds that

P(r) - r -- 2rcvkr k + O(r~+l) , Vk --fi 0, (2.5)

then k is odd, and the origin is stable (unstable) if vk < 0 (> 0).

PROOF. Let P0 denote the function P defined by (2.4). For any 0 �9 [0, 2Jr) there exists a unique 01 �9 [0, 2jr) such that

01 = O o + j r ifO~<Oo <Jr ; 00--01 +J r ifJr ~<0o <2Jr .

Let P1 denote the Poincar6 map associated to 01. We claim that

Po(r) -- - P 1 ( - r ) , Ir[ < ro. (2.6)

For definiteness, assume 0 ~< 0o < Jr and 0 < r < r0. Let Z1 = (cos 01, sin 01). Then Z1 = - Z 0 and (x0, y0) T = rZo = ( - r ) Z 1 . Hence, by (2.4) we have

x (r, xo, Yo) ) y(r , xo, Yo) -- Po(r)Zo -- Pl ( - r ) Z l ,

364 M. Han

y

X )

Fig. 5. The Poincar6 maps P0 and P1 (b > 0).

which gives (2.6). See Fig. 5. By Definition 2.2, the origin is stable if and only if

Po(r) - r < O f o r 0 < r < r 0 . (2.7)

By Remark 2.1, the above inequality is equivalent to

P1 (r) - r < 0 for 0 < r < r0.

Note that by (2.6)

Pl ( r ) -- r -- - [ P o ( - r ) - ( - r ) ] .

Hence, (2.7) is equivalent to

Po(r) - r > O f o r 0 < - r < r 0 .

Then combining (2.7) and the above together we see that (2.7) holds if and only if

r[Po(r) - r] < O f o r O < l r l < r o .

Thus, if (2.5) holds, the number k must be odd and the sign of v~ determines the stability of the origin. This ends the proof. D

DEFINITION 2.3. Let (2.5) hold with k = 2m + 1, m ~> 0. We call the origin to be a focus

o f order m, and v~ the mth Lyapunov constant or focus value. The focus at the origin is called rough or hyperbolic (weak or f ine) as m = 0 (m ~> 1).

REMARK 2.2. Similar to Remark 2.1 and Corollary 1.1, one can prove that the order and the first nonzero Lyapunov constant of the origin are independent of the choice of 00 and 1.


For the sake of convenience, we will take 00 -- 0 below. In this case we have (x0, Y0) -- (r, 0), and hence from (2.4) and (2.2)-(2.3) we obtain

P ( r ) = (x(r , xo, YO), y( r , xo, yo))Zo

= Ce~r cos/3r sin/3r C_ 1 r - sin/3r cos fl'r 0 -k- O( r 2 Z0,

which yields P ( r ) -- re ~T + O(r 2) since r (0) -- T. Comparing with (2.5) gives that

Vl -- ~-= 1 [e~r - 1 ] - - ~1 [ea+-----~ T -1].

Thus, the origin is stable (unstable) if

d iv(f , g)10 - fx(O, O) + gy(O, 0) < 0 (> 0).

Next, we suppose ot - - 0 and give a computat ion formula for 1) 3. For the purpose it is convenient to use the first order normal form of (2.1). Without loss of generality we may suppose (2.1) has been of the form. In other words, we may assume (2.2) holds with C being the identity. Introducing the polar coordinate to (2.1) we obtain

? - - c o s O F + s i n O G , O - - - b + ( c o s O G - s i n O F ) / r (2.8)

and

dr cos 0 F + sin 0 F = -- R(O, r). (2.9)

dO - b + ( c o s 0 G - s i n O F ) / r

Let r(O, ro) be the solution of (2.9) with r(0, r0) = r0. We have

LEMMA 2.2. Let a -- d = 0, b -- - c ~ O. Then

(i) P(ro) -- r (2=, ro) i f b < O, and P ( r ( 2 = , ro)) -- ro i f b > O,

_ _ 1 s g n ( - b ) f g = (ii) v3 ~-y R3 (0) dO, where R3 (0) -- ~.-gUr 3 ~ R (0, O)

PROOF. The conclusion (i) can be proved in a similar manner to L e m m a 1.3. Let

R(O, r) = R2(O)r 2 + R3(O)r 3 + . . . (2.10)

and

r(O, ro) = rl (O)ro + r2(O)r 2 + r3(O)r 3 + . . . ,

where rl (0) - 1, r2(0) - r3(0) - 0. Inserting the above solution and (2.10) into (2.9), and then comparing the like powers of r0 we obtain

t t __ R2r2, r l (0) = 0 , r2(O ) r~(O) -- R3 r2 + 2R2rlr2 ,

366 M. Han

or

f0 rl (0) = 1, r2(0) -- R2(0) dO,

f0 r3(0) = [R3(0) q- 2R2(O)r2(O)] dO.

Hence, if we let

P(ro) = r(2zr, r o ) = ro -I- 2zr v2r 2 + 2zr v3ro 3 + O(r4),

then

lfo2 V2 -= ~ R2 (0) dO,

1 f0 2zr v3 = ~ - [R3(0) + 2R2(O)r2(O)] dO

[fo fo 1 = 2---~- R3(0) dO + 2r2(O)r;(O) dO

1 f0 2zr = 2re R3(0) dO + 27r(~2) 2.

If b < 0, we have P (ro) = P (ro), and by Lemma 2.1, v2 = 0. Thus

lfo2~ V3 -- V3 -" ~ R3(0) dO.

If b > 0, we have P(ro) -- P - 1 (ro). Note that if /~(ro) = ro + 2z r~ rg + O(r k+l) then

j~- i (ro) -- ro -- 2zr ~/r + O(ro~+l).

Hence, using Lemma 2.1 we have fi2 = 0 and

V3 ----- --V3 lfo2 27r R3 (0) dO.

This ends the proof. D

Based on the above lemma we can prove:

Bifurcation theory of limit cycles of planar systems

T H E O R E M 2 . 1 . L e t a = d = O, b = - c 7 ~ O, and

3 -- X i yj O(IX, 13 F(x, y) Z aij -Jr- y ),

i + j = 2

3

G(x, y) = Z bi jx i yj + O(IX, yl3).

i + j = 2

Then

v3 = 8 - ~ 3(a30 + b03) + a12 + b21 - ~ [ a l l (a20 + a02) - b l l (b20 + b02)

+ 2(ao2b02 - a20b20)] }

1 { Fxxx + Fxyy + G yyy

16lbl

- FxxGxx + FyyGyy]} . (o,o)

PROOF. By (2.9) and (2.10),

where

R 2 ( 0 ) -- - b - I p 2 ( o ) ,

1 -- -s + Fyy) - a x y ( a x x "1-ayy)

U

R 3 ( 0 ) - - -b- l [p3(o) + b -I P2(0)$2(0)],

Pk(O) = cosOFk(cosO, s in0) + sinOGk(cosO, s in0) ,

S~(O) = cosOGk(cosO, s in0) - sinO Fk(cosO, s in0) ,

y) = aij i y j , Ck( , y ) - i y j .

i+j--k iWj=k

It follows directly that

P3(0) -- (a12 -+- b21) sin 2 0 COS 2 0 -+- a30 cos 4 0 -k- b03 sin 4 0 -+- Ko(O),

P2(0)$2(0) -- -aozbo2 sin 6 0 -k- azob20 cos 6 0 - N1 sin 4 0 cos 2 0

- N2 sin 2 0 cos 4 0 + K1 (0),

367

N1 -- 2ao2all - 2bo2bll -k- ao2b20 n t- a l l b l l -q- a20b02 - ao2b02,

N2 -- 2azoal l - 2b20bl 1 - ao2b20 - a l l b l l -k- a20b20 - a20b02.

where Ko and K1 are 2re-periodic functions with zero mean value over the interval [0, 2re], and

368 M. Han

Then we have

1 f0 2zr Jr 2Jr R3 (0) dO - - 2Jr-----b

3 1 (a30 d- b03)~ -[- (a12 + b21)~

b (ao2b02 -a20b20)~ + (N1 + N2) �9

Thus, the conclusion follows from Lemma 2.2(ii). D

Along the above line one can find formulas for computing v5, l)7, etc. See [31 ] for more detail.

2.2. Normal form and Lyapunov technique

Next, we introduce another method (called normal form method) to find focus values.

LEMMA 2.3. Let a = d = O, b = - c 7~ O. Then for any integer m ~ 1 there exists a polynomial change of variables of the form

(x, y)T _ (U, V) T + O(lu, vl 2) = Q(u, v)

which transforms the system (2.1) into

m

it -- by + E ( a j u + bjv)(u 2 + v2) j + O([u, 1)12m+2), j=l

m i) = - b u + E ( - b j u + ajv)(u 2 + v2) j + O([u, v[2m+2).

j=l

(2.11)

PROOF. For convenience we introduce complex variables z -- x + iy, ~ = x - iy, or

z 1 i x ( ~ , ) -- ( 1 - i ) ( y ) (2.12)

so that (2.1) becomes

f z + z z - z ' z + ~ ' z - 2 ' - - = h ( z , z) 2 ' 2i + ig 2 ' 2i

z f z + z z - z ig z + z z z - - - - - = - - h ( z , z).

2 ' 2i 2 ' 2i

(2.13)

It is easy to verify that h is the conjugate of h. Hence, we may neglect the second equation in (2.13), since it can be obtained from the first one by taking complex conjugation.


By (2.3) we can write

h(z, #) - - i b z + Z AjkZJ#k + O([Zl2m+2)" (2.14) 2 <~j +k <~ 2m+ l

We would like to make a change of variables of the form

g -- 09 4- Z Cjko)J-~ k -- (_o + p(co,-~), (2.15) 2<~j+k<~2m+l

and expect that the resulting equation which has the form in general

d) - - iboo + Z BJkc~ + 0(10912m+2) (2.16) 2<~j+k<~2m+l

is as simple as possible. For the purpose, differentiating both sides of (2.15) in t and using (2.13), (2.14) and

(2.16) we obtain

[ 2m+1 ][ 2m+1 ] I + ~ Cjk jcoJ- l -~ k - i b c o + ~ BjkcoJ-~ k

j+k=2 j+k=2

-+- Cjkko)J-~ k-1 ib-~ + ~ --Bjkcok-~ j

j+k=2 j+k=2

= -ib[co q- Z Cjk~

2<~j+k<~2m+l

2m+l + Z Ajk(co -+- p(co,-~))J (-~ -4- fi(co, _~))k + O(]co]2m+l).

j+k=2

By considering terms of o)J-~ k for j + k = 2 we obtain

- i b C j k j r.o J -~ k -4- B j k w J -~ k -+- i b C j k k w J -~ k -- - i b C j k oo J -~ k -4- A j k w J -~ k,

which yields

Bjk -- A jk + i b C j k ( j - k - 1). (2.17)

Hence, to nullify the coefficients B jk we can choose

Ajk Cjk -- ib(1 + k - j)

370 M. Han

if

j r k + 1, (2.18)

which always holds for j + k -- 2. Thus it appears that Bjk "- 0 for the chosen Cjk for j + k - 2. In other words, all quadratic terms will be nullified in the new equation (2.16) so far if we choose the coefficient of quadratic terms in (2.15) proper.

By equating the coefficients of coj-~k for j + k = 3, we obtain

Bjk - Ajk + ibCjk(j - k - 1) + Zjk, (2.19)

where Zjk depends o n Cj,k, with j~ + k ~ - 2. Thus, in this case, for j and k satisfying (2.18) we can also choose Cjk by

1 Cjk = ib(1 + k - j ) [Ajk -4- Zjk]

so that Bjk -- 0 for the new equation. The only monomial left up to now is B21 co2~ which is called a resonant term, and by (2.19) the coefficient C21 can be chosen freely, say C21 = 0.

For higher values of j + k, the expression (2.19) remains valid, where Zjk depends only on C j, k, with 2 ~ j t + k t < j + k. Hence, by continuing the way above, an appropriate

change of variables can be found that eliminates all those monomials BjkcoJ-~ k for which (2.18) are satisfied. The only monomials which survive, called resonant terms, have the form Bk+l,kcok+l-~ k. Therefore, eventually, Eq. (2.16) takes the form

m

go = -ibco + ~ Bk+l,kcok+l-~ k --[- O([col 2m+2) = R(co, N).

k = l

More precisely, there exists a change of variables

z = co + p(o~, N), g = ~ + t3(~o, ~ )

which carries (2.13) into the form of

-- R (co, ~), ~ = R(co, ~).

Substituting c o - u + iv, ~ = u - iv into the above gives the desired system (2.11) with

aj = R e B j + I , j , bj - - - I m B j + l , j .

The coordinate change from (2.1) to (2.11) is given by

(x) (1 i)l( ) ( ) ( ) ( ) co~ p(co, ~) co 1 i u y 1 - i ~ + / 3 ( o ) , ~ ) ' ~ = 1 - i v

Bifurcation theory of limit cycles of planar systems 3 7 1

where (x, y) and (u, v) are both real variables, and

( 1 i ) - I ( ! 2 �89 = i i �9 1 - i - ~

The proof is completed. D

We call the system (2.11) the normal form of (2.1) of order 2m + 1. It is easy to see that in polar coordinates the normal form equation takes the form

i" - al r3 + . . . + am r2m+l + O(r2m+2), 0 -- - ( b + bl r2 + ' " + bm r2m) + O(r2m+2). (2.20)

For a relationship between focus values appeared in (2.5) and the constants aj in (2.20) we have

LEMMA 2.4. Let a = d = O, b = - c =7/= O. Then (2.5) holds with k = 2m + 1 ~ 3 i f

and only i f aj - -O, j = 1 . . . . . m - 1, am =7/= O. Moreover, V2m+l = am/lb[ as aj = O, j = l . . . . . m - 1 .

PROOF. Let P and P* denote respectively the Poincar6 maps of (2.1) and (2.11) both associated to 00 -- 0. For r0 > 0 small, let

l: ( x , y ) = ( r , 0), 0 < r < r 0 ,

1 1 - Q - l ( / ) = { ( q x ( r ) , q 2 ( r ) ) , O < r <r0} ,

where Q-1 is the inverse of the transformation Q appeared in Lemma 2.3, and q l (r) = r + O(r2), q2(r) = O(r2). Let P1 :ll ~ 11 denote the Poincar6 map of (2.11) defined by a similar way to (1.14). Then by the proof of Lemma 1.5 with the case 1 we have

P1 = P. (2.21)

Let l' = {(u, v): u = r, v = 0, 0 < r < r0} which is tangent to 11 at the origin. Denote by ( u ( t , r ) , v ( t , r ) ) the solution of (2.11) with (u(O,r) , v (O,r ) ) = (r, 0). It is easy to see from (2.11) that

u(t , r) -- r cosbt + O(r2), v(t , r) - - r sin bt + O(r2).

Note that 11 can be represented as

ll = {(r', c ( r ' ) ) , 0 < r ' < ql (r0)}

where c is a C ~ function satisfying c(O) = c' (0) = O. Consider the function

K ( t , r) = c (u( t , r)) - v( t , r) -- rK1 (t, r) ,

372 M. Han

where K1 (t, r) = sin bt + O ( r ) . By the implicit function theorem, there exists a C e~ function r*( r ) = O(r) such that K l ( r * ( r ) , r) = 0. Then similar to (1.10), there exists a unique function hi (r) such that

(u ( r*(r) , r) , v ( r*( r ) , r)) -- (ql (hi (r)), q2(hl ( r ) ) ) E 11. (2.22)

Moreover, the function satisfies

hi o P* = P1 o h 1. (2.23)

By (2.22) it follows that

ql (hi) -- u(r*, r) - r cos br* + O(r 2) - r + O(r2),

which gives that

hi -- q l l ( r + O(r2)) - r + O(r2).

Therefore, by (2.21) and (2.23) we obtain

hl o P * = P oh l , h l ( r ) - r + O(r2). (2.24)

Then, it is easy to verify that (2.5) holds if and only if

P* - r -- 2rr vl~r I~ + O ( r 1~+ 1), Ok ~k O. (2.25)

On the other hand, let aj = 0, j = 1 . . . . . n - 1, an =/: O. We then have from (2.20)

dr _ _ _ a n r2n+ 1 -q- O(r2n+2) dO b \ !

which has the solution

2n+l r(O, ro) -- ro Or 0 + O(r2n+2).

Thus,

r (27r, r0) = r 0 - 2:rranr2n+l + O(r2n+2). b

Hence, by Lemma 2.2 we have

2Jran r2n+ 1 + O( r2n+2)" P*(r ) -- r + Ibl

Obviously, for n ~> 1 the above holds if and only if (2.25) holds with k = 2m + 1 -- 2n + 1. This ends the proof. E]


We remark that some algorithms have been established for calculating the coefficients al . . . . . am in (2.20). An efficient one can be found in Yu [119].

We finally give a third method to determine the stability and the order of a focus origi- nated by Lyapunov.

LEMMA 2.5. For any given integer N > 1 and expansions of f and g of the form

f -- by + f l + O([x, yI2N+2), g -- - b x -+- gl -+-O([x, y[2N+2),

where

f l (x, y) -- Z aijxi yj ' gl (x, y) -- ~ bijx i y J, 2<~i+j ~<2N+l 2~<i+j ~<2N+l

there exist constants L 2 . . . . . L N+ 1 and a polynomial

2N+2

V(x, y) -- ~ Vk(x, y), k=2

where

V2(x, y) - - x 2 -+- y2, Vk(x, y) -- Z r yj, iWj--k

3 ~ < k ~ < 2 N + 2

such that

N+I Vxf + V y g - ~ Lk(x 2 + y2)k + O(Ix, yI2N+3).

k=2

(2.26)

Moreover, for 2 <<. k <<. N + 1, Lk+l depends only on ai j and bij with i + j <<. 2k + 1.

PROOF. We want to find a polynomial V with the supposed form and constants L2 . . . . . LN+I satisfying (2.26). Note that for the given form of f, g and V we have

Vx f -+- Vyg -- b(y Vx - x V y ) -+- (Vx f l -k- Vygl) -+-O([x, y[ 2N+3)

2N+2 2N+2

k=3 k=3

where

2N+2

Gk -- - (Vx f l + Vygl) + O(]x, y[ 2N+3)

k=3

(2.27)

374 M. Han

with G~ being a homogeneous polynomial of degree k depending only on the coefficients aij, bij and cij with 2 <~ i + j < k.

Then neglecting terms of degree more than 2N + 2 Eq. (2.26) is equivalent to

2N+2 2N+2 N+I

b ~ (yVtcx - X W k y ) - Z ak -t- ~ t k ( x 2 -+- y2)/~. k=3 k=3 k=2

In order to find V~ and Lk satisfying this equation it suffices to solve the following equation for 3 ~< k <~ 2N +2 :

b ( y Vlc x -- X Vk y ) -- G lc

b(yV~x - XVky) = G~ + tk /2 (x 2 --F y2)k/2

for k odd, (2.28)

for k even. (2.29)

The above equations can be rewritten as

d b~--~ V~(cos 0, sin0) = Gk(cos0, sin0)

d b~--~ Vk(cos 0, sin0) - Lk/2 + G~(cos0, sin0)

for k odd, (2.30)

for k even (2.31)

since

b(yVl~x - xVky)(r cosO, r sinO) = b r lcd Vk (cos 0, sin 0).

Let us solve (2.30) and (2.31) by induction in k. First, for k = 3, we have

f0 2rr G3(cos0, sin0) dO = 0.

Hence there is a function G3 of the form

G3(0) "- ~ gij c o s / 0 sin j 0 i+j=3

such that G3(0) - G3(cos0, sin0). Denote G3 by

f G3(0) -- G3(cos 0, sin0) dO.

We can write

f G3(cosO, sin0) dO = ~ ff, ij COS/0 sin j 0. i-k-j=3


Then by (2.30) we have a solution for V3 below

V3(x, y) = -~ ~ gijx t y j . i+j=3

That is, we have cij __ ~.1 g i j for i + j - 3. For k = 4, G4 will be known when V3 is definite. Thus, we can choose L2 to be such

that

'f0 L2 -k- ~ - G4(cos0, sin0) dO = 0.

It then follows that

f [L2-+-G4(cosO, s i n O ) ] d O - ~ gi jcos i Osin jO

i+j=4

as before. By (2.31), V4 will be determined by taking cij - - 1 g, i j for i + j = 4. For higher values of k, Vk can be determined in the same procedure. This ends the

proof. D

The following lemma says that the first nonzero constants of L2 . . . . . LN+I will determine the stability and the order of the focus of (2.11) at the origin if a = 0.

LEMMA 2.6. Let a = d = 0, b = - c ~: O. Then (2.5) holds with k = 2m + 1 ~ 3 if and only if

Lj ----0, j = 2 . . . . . m, Lm+l 5~0.

Lm+l - -0 , j - - 2 , . m. Moreover, V2m+l m ~ as L j .. ,

PROOF. Let r > 0 be small and L denote the orbit arc of (2.1) from (r, 0) to (P(r), 0). Note that for any # 6 (0, 1),

r + / z [ P ( r ) - r ] - - r + O(r3).

The mean value theorem implies that

V(P(r), O) - V(r , O) -- Vx(r + ~,[P(r)- r], O)(P(r)- r)

= 2 r [ P ( r ) - r](1 + O(r21),

where V is the function in Lemma 2.5. On the other hand, by the formula of constant variation we have x 2 + y2 = r2(1 + O(r))

along L. Thus, by (2.26)

376 M. Han

V(P(r),O) - V(r,O) = fL dV = fL (Vx f + Vyg)dt

-- fo r(r) N~I Lkr2k(l + O(r)) + O(r2N+3) 1 dt

k=2

N+I

2Zrlb] E L~r 2/~ (1 + O(r)) + O(r2N+3), k=2

where r ( r ) - 27r ]E + O(r) is the time along L. It follows that

N+I 7r 1 O(r2N+2) P ( r ) - r = [b~l E Llcr21r (1 + O ( r ) ) +

k=2

Then the conclusion follows easily and the proof is completed. []

The proof of Lemma 2.6 presents an algorithm to compute constants L2, L3 . . . . . It includes the following three steps in a loop:

(i) Find G2m+l by (2.27), (ii) Find Vzm+l by (2.30),

(iii) Find G2m+2, Lm+a and g2m+2 by (2.31). To begin with m = 1 we can get L2 by executing the 3 steps. We can get L3 further by

doing the same procedure for m = 2. See Chicone [ 14] for more detail. By (2.3) and (2.5) we can prove the following corollary in a similar way to Lemma 2.6.

COROLLARY 2.1. Let a = d = 0, b -- - c # O. Suppose there exists a function V (x, y) - X 2 _q_ y2 + O(]x, y[ 3) such that

= Vxf + Vyg = H2~(x, y) + O(Ix, yl2k+l), k >~ 2,

where H2~ is a homogeneous polynomial of order 2k satisfying

Ibl f 27r/lb[ H2~(cosbt, - sinbt) dt < 0 (> 0). Lk = ~ dO

Then the origin is a stable (unstable)focus of order k - 1 of Eq. (2.1).

Now let us recall some facts obtained so far in this section. We will outline them for analytic systems. Suppose f and g in (2.1) are analytic functions near the origin and (2.3) holds with a + d = 0. Then by Lemma 2.1, the proof of Lemma 2.6 and the definition of the Poincar6 map in (2.4) we have

N+I oo 7r 1 O(r2N+2 P ( r ) - r = -~l E Llcr21r (1 + O ( r ) ) + ) = 2zr E vlcrk'

k=2 k=3

(2.32)


where

l)2m --O(Iv3, v5 . . . . . U2m-ll), m >~ 2.

The implicit function theorem ensures the convergence of the series in the right-hand side of (2.32).

If a = d -- 0, b = - c 5~ 0, by Lemma 2.3, there exists a formal change of variables given by the formal series

(x, y)T _ (u, I))T + ~ qijui l)J -- Q(u, v)

i+j>/2

which carries (2.1) formally into

fi -- by + Z ( a j u + b jv ) (u 2 + v2) j,

j>~l ~0 -- - b u -+- Z ( - b j u -Jr-ajv)(u 2 -Jr- v2) j .

j>~ l

(2.33)

By Lemma 2.5, there exist a formal series

V (x, y) - x 2 -+- y 2_+_ ~ Cij X i yj

i+j>/3

and constants L2, L3 . . . . . such that

Vx f + V y g - ~ Lk(x 2 + yZ)k. (2.34)

k >/2

Lemmas 2.4 and 2.6 show that the following three statements are equivalent to each other:

(i) vZj+l -- 0, j = 1 . . . . . m - 1, V2m+l ~ 0; (ii) a j = O, j - - 1 . . . . . m - 1, am 5~ 0;

(iii) Lj = O, j = 1 . . . . . m , L m + l ~: O.

Moreover, when one of the conditions (i)-(iii) holds, we have

l)2m+l am Lm+l

Ibl 21bl

Based on this relationship, we call either V2m+l, am or Lm+l the mth Lyapunov quantity or focus value.

Note that the series in (2.32) is always convergent for Ire small if (2.1) is analytic. We therefore obtain

THEOREM 2.2. Suppose (2.1) is an analytic system satisfying a = d = O, b - - c 5/= O. Then (2.1) has a center at the origin if and only if one of the following holds:

378 M. Han

(a) V 2 k + l ~" 0 for all k >~ 1; (b) ak = 0 for all k >~ 1; (c) Lk = O for all k >>, 2.

Lyapunov proved that the formal series for V (x, y) is convergent near the origin if Lk = 0 for all k ~> 2. Hence, by (2.34) in this case the function V represents a first integral whose level sets are orbits of the system (2.1).

Bryuno [9] proved that the formal series for Q(u, v) is convergent near the origin if and only if ak - 0 for all k ~> 1. In this case, system (2.33) becomes

j / > l

j ) l

which has a first integral of the form U 2 -+- 1) 2 . This implies that the original system (2.1) has a first integral of the form WZ(x, y) + WZ(x, y), where W(x, y) = (WI (x, y), Wz(x, y)) is the inverse of the coordinate change (x, y)T = Q(u, v).

Thus, by Theorem 2.2 and the next theorem we know that under the condition of Theo- rem 2.2 system (2.1) has a center at the origin if and only if it has an analytic first integral of the form X 2 q - y2 _+_ O(Ix, y[3).

The following theorem gives a sufficient condition for (2.1) to have a center at the origin.

THEOREM 2.3. Consider the C c~ system (2.1). Let (2.2) hold. If one of the conditions below is satisfied:

(i) f ( - x , y) = f (x, y), g ( - x , y) = - g ( x , y); (ii) there exists a C ~ function H (x, y) for (x, y) near the origin satisfying H (0, 0) --

0, H (x, y) ~ 0 for 0 < x 2 -+- y2 << 1 such that Hx f + Hy g = 0, then (2.1) has a center at the origin.

PROOF. Let (i) hold first. Without loss of generality we assume that the orbits of (2.1) near the origin are oriented clockwise. For y0 > 0 small let (x(t), y(t)) be the solution of (2.1)

with (x(0), y(0)) = (0, Y0). Set

L1 -- { (x(t) , y(t)) [ 0 tl }, L2 -- { (x(t) , y(t))[tl t2 }, = { ( - x ( - t ) , y( - t ) l [ - t l <.t 01,

where

tl -- min{t > 0 1 x(t) -- O, y(t) < 0}, t2 = min{t > 0 I x(t) = O, y(t) > 0}.

! By our assumption, L 1 is an orbit arc of (2.1) starting at (0, y(tl)) . This implies that L2 =

l L 1 and hence y(t2) = yo. In other words, (x(t), y(t)) is a periodic solution.


Now let (ii) hold. For definiteness, we suppose H (x, y) > 0 for 0 < x 2 -+- y2 ~ 1. Then the function z = H(x , y) takes a minimal value at (x, y) = (0, 0). This implies that for sufficiently small h > 0, the equation H(x , y) = h defines a closed curve near the origin. Our assumption ensures that the curve is a periodic orbit which approaches the origin as h --+ 0. The proof is completed. IS]

EXAMPLE 2.1. Consider a C ~ system of the form

Yc- y - x h ( x , y), y = - x - yh(x , y),

where

0, for (x, y) -- (0, 0), h(x, y) = 1

e x2+y2 otherwise.

Let V (x, y) = x 2 + y2. Then for the system, (2.34) becomes

V x f + Vyg -- - 2 ( x 2 + y2)h(x , y) < 0

for x 2 -k- y2 > 0. By the proof of Lemma 2.6, the origin is a stable focus. However, in this case we have L k --- 0 for all k ~> 2.

This example shows that Theorem 2.2 is no longer true for C ~ systems.

EXAMPLE 2.2. Consider the following cubic Lirnard system:

-- y - (a3 x3 + a2x 2 + a lx ) , (2.35) - - m X ,

where al , a2 and a3 are real constants. The divergence of (2.35) takes value - a l at the origin. Hence, the origin is a stable (unstable) focus if al > 0 (< 0). Further, by Theorem 2.1 we have V3 -- - 3 a 3 as al -- 0. Thus, in this case, the origin is stable (unstable) if a3 > 0 (< 0).

Let al -- a3 -- 0. Theorem 2.3(i) implies that (2.35) has a center at the origin now.

EXAMPLE 2 .3 . For a given C ~ function H:II~ 2 ~ 11~, it induces a planar system of the form

which is called a Hamiltonian system with the Hamiltonian function H. The level sets of the function give orbits of the system. By Theorem 2.3(ii), if there is a point (x0, Y0) 6 IR2

380 M. H a n

Fig. 6. The phase portrait of (2.36) with g ( x ) = x 2 - x .

satisfying

Hx (xo, Yo) -- Hy(xo, Yo) --O,

- [nxy(XO, y0)] 2 -k- nxx(XO, yo)nyy(XO, Yo) > 0

then the above Hamiltonian system has a center at the point. For example, let

ly2 H ( x , y) -- -~ + G(x) .

Then the induced system is

.;c -- y, ~ - - g ( x ) , (2.36)

where g(x) = G I (x). In mechanics, the function H is called the total energy of the system (2.36), while the term �89 y2 is called the kinetic energy and the function G is the potential energy. It is evident that (2.36) has a center at point (x0, 0) if g(xo) - O , g~(xo) > O. A singular point (x0, 0) satisfying g~(xo) < 0 is a saddle point. Taking g(x) = x 2 - x,

(2.34) becomes

k - - y , y - " X - - X 2

which has two singular points: center (1,0) and saddle (0,0). Note that H (0, 0) = 0. There is a nontrivial orbit y defined by the equation H (x, y) = 0. The orbit approaches the saddle point both positively and negatively. An orbit with this property is called a homoclinic orbit. The phase portrait of the above quadratic Hamiltonian system has been drawn in Fig. 6.


2.3. H o p f bifurcation

In the next part, we consider a planar C ~ system with a vector parameter of the form

2 -- f (x, y, lz), y -- g(x , y, lz), (2.37)

where/z e I[~ m , m ~> 1. Suppose (2.37) has an elementary focus for # = 0. Without loss of generality we can assume that for all small 0#1 (2.37) has a focus at the origin. Then we can write near the origin

f (x, y, lz) -- a( lz)x + b(tx)y + O([x, y[2), g(x , y, # ) -- c( lz)x + d( l z )y + O([x, y[2),

(2.38)

where a(0) = d(0), b(0) -- - c (0 ) # 0. Let P(r, # ) denote the Poincar6 map of (2.37) near the origin. By Lemma 2.1, similar

to (2.32) for any integer N > 0, P has the following expansion:

N

P(r, #) -- r + 2re Z Vk(lz)rk + O(rN+l) ' k=l

(2.39)

where

V2m --O(Iv l , v3 . . . . . V2m-11), N

1 ~< m ~< -~-. (2.40)

Sometimes, for convenience, we call I)2m+l (/Z) in (2.39) the mth Lyapunov constant of (2.37) at the origin, m >~ 1. Introduce

d(r, lz) = P(r, lz) - r,

which is called a succession function or bifurcation function of (2.37). This function is also called the displacement function.

By using an analogous formula to (2.6) it is easy to obtain

LEMMA 2.7. For Igzl small, Eq. (2.37) has a limit cycle near the origin i f and only i f there exist rl (#) > 0, r2(#) < 0 near r = 0 such that

d (rj (lz), Ix) - O, j - 1, 2.

By (2.38) and the discussion after Remark 2.2 we know that Vl (/Z) has the same 1 [eZ:r~/Ir 1] with c ~ - �89 + d ) 1/31- sign as a ( # ) + d(/z). In fact, Vl ( /Z) -~-y

l v / - 4 b c - (a - d) 2. Hence, if a(0) + d(0) # 0 then Vl (0) # 0 and d(r, lZ) has no pos- 2 itive zero near r = 0 for all I#1 small. Thus, if a(0) + d(0) # 0, there is no limit cycle near the origin for all I#l small.

382 M. Han

Let a (0) + d (0) = 0, and let v30 denote the first order focus value of (2.37) for IX = 0 at the origin. Then by (2.39) and (2.40) we have

d(r, Ix) -- 2 r r r [ v l (IX) + v2(ix)r q- v3(ix)r 2 + O ( r 3 ) ] ,

where Vl(0) = V2(0) = 0, V3(0) = V30. Similar to Theorem 1.3(i) we can prove that if v30 ~ 0, d(r, IX) has two zeros in r with one positive and the other negative for Vl (ix)v30 < 0 and has no nontrivial zero for Vl (ix)v30 > 0. Thus, by Lemma 2.7 and noting

Vl a(ix) + d(ix) (1 + O(IX)) 21b(O)l

we have proved the following

THEOREM 2.4. I f for IX = 0 (2.37) has a first order focus at the origin, then it has at most one limit cycle near the origin for all I~1 small. Moreover, the limit cycle exists if and only if (a(ix) -q- d(ix))v3o < O.

In general, similar to Theorem 1.3(ii), we have:

THEOREM 2.5. I f for IX = 0 (2.37) has a kth order focus at the origin (k >1 2), then it has at most k limit cycles near the origin for Itzl small. Moreover, k limit cycles can appear by suitable perturbations.

EXAMPLE 2.4. Consider a cubic Li6nard system

2 = y - (x 5 + IxlX 3 + Ix2x), (2.41)

where IX 1 and IX2 are small parameters.

First, by (2.39) and (2.40), we have

1 Vl -- --~IX2 -[- O(IX2), V2 = O(IX2),

3 V3 -- --gixX if- 0( / / ,2) , V4 ---- O(IVX, V31).

It is easy to check that for

V (x, y) - x 2 + y2 5 ~ 3 _ 11 - -~xy 5 - x3y --~x5y

it holds along the orbits of (2.41)

dV

dt #1 =#2--0

5 8 (x2 + y2)3 + O(Ix, Nil~


Hence, by Lemma 2.6, it follows that

5 vg -- 16 -+-O([Ixl' Ix21)-

This shows that for Ix 1 "-- Ix2 = 0 the origin is a stable focus of order 2 for (2.41). We claim that there is a function

9 2 5/2) ~ ( I x l ) - - 4--oIxl "+" O(lIxll

such that for IIxll -q- IIx21 small Eq. (2.41) has (i) no limit cycle if either Ix1 >~ 0 and//,2/> 0 or IXl < 0 and//,2 > •(IXl);

(ii) a unique simple limit cycle if either IX1 ~> 0 and IX2 < 0 or IX l < 0 and Ix2 ~ 0;

(iii) a unique double limit cycle if Ixl < 0 and Ix2 -- S(Ixl) ;

(iv) two simple limit cycles if Ix a < 0 and Ix2 < S(Ix 1).

In fact, by v2 -- O(vl) , v3 -- --~IxI + O(va) and v4 - O(Ivll + [Ixa l) we can write

[ 3 ] d(r, ix) - 2rCrPl(r ,u) Vl - -~IXlr2p2(r, IX) + v~r4p3(r, ix)

= 2rcr P1 (r, u)da (r, Ix),

: # m where Pi = l + O(r) 6 C ~ , i = 1 ,2 ,3 , v 5 - written further as

5 16 + O(IIXll). The function dl can be

3 p2 p4 dl(r, i x ) = V l - - ~ i x l -k-v~ P~(p , IX)=~d2(p, IX),

where p = r~/Pz(r , IX) - r + O(r 2) E C ~ P3* - 1 + O(p) 6 Cc~ Od2 Clearly, for IX1 ~> 0 we have -by < 0 for 0 < p << 1. This implies that for IX l /> 0, dl

has a unique zero if and only if Vl > 0. Note that Eq. (2.41) has a double limit cycle near the origin if and only if the function dz(p , IX) has a double zero in p near p -- 0. Let us consider the equations below

Od2 d2(p , IX) - 0, ~ ( p , IX) - 0.

0p

We can solve form the above equations

3

9 l ) 1 - 256v~ #2(1-t - 6 2 ( ~ - ~ ) 1

384 M. Han

~ d

f p -'-~ p

(a) fll >- O, fl~ < 0 (b) Pa > 0, /2 2 > 0

~ d

p f

(~ < 0 , / 4 < 8 0 ~ )

d

->

p

(~)/4 <0, & - 3 0 4 )

~ d

/q f

Fig. 7. Curves determined by d2(p, #).

Q)'

_. l /

P2

o

Fig. 8. The bifurcation diagram of (2.41).

for/~1 < 0, where ~i (U) = O(tt) E C w, i - 1, 2. The second equation above determines a

unique function

9 ~,~ = ~o,1) - ~, ,2 + oOd<,).


Now it is easy to draw curves determined by the function d = d2(p, /z) on (d, p) plane, see Fig. 7.

Then the claim follows easily. For the bifurcation diagram of (2.41) see Fig. 8. On the (#1,/z2) plane in Fig. 8, the line /Z2 = 0 (near the origin) is Hopf bifurcation

curve, and the curve /Z2 = 3 ( / Z l ) is the double limit cycle bifurcation curve, or in other words, the saddle-node bifurcation curve for limit cycles.

If (2.37) has a center at the origin for/z = 0, we can also study the bifurcation of limit cycles for ]/zl small. We give an example to illustrate this phenomenon.

EXAMPLE 2.5. Consider the cubic system

.f -- y + X 2 -q- 8 ( X 3 - - 8 X ) ,

- - - - ( X - - X 3 ) .

We prove that the system has a unique limit cycle near the origin for 0 < e << 1. For the purpose, let us consider

- y + x 2 + e (x 3 - 3x) , (2.42) - _

and prove that the system has a unique limit cycle near the origin for 0 < lel < e0, 0 < ~ < e0 for a small constant e0 > 0.

First, by Theorem 2.3(i), (2.42) has a center at the origin for e -- 0. Hence, the succession function d(r, e, 3) of (2.42) can be written as

d(r, e, 8) - 27rer[v~(e, 8) + v~(e, 8)r + v~(e, 6)r 2 + O(r3)]

= 2re erd* (r, e, 3),

where

, 1 [e_e~ ~ _ 1 ] _ e3(l+O(e3)) 81; 1 - - 131 - - ~ - - - - ~ - ,

3 - - - +

Therefore,

- -~(1 -+-O(e3)) + O(3)r -+- + 0(3) r 2 + O(r 3)

which has a positive zero r - ~3 ~/6(1 + O(e6)) near r - 0 if 0 < 8 << 1. Then the conclu-

sion for (2.42) follows directly.

386 M. Han

2.4. Degenerate Hopf bifurcation

We now turn to the degenerate Hopf bifurcation near a center and establish a general theory by using the first order Melnikov function.

Consider a C ~ planar system of the form

2 = f (x, y) + ep(x, y, e, 6), ~ = g(x, y) + eq(x, y ,e , 6),

(2.43)

where e 6 IR, 6 e D C I~ m (m >~ 1) with D compact, and

( f (x, y), g(x, y)) -" lz(Hy, - H x ) ,

H(x , y ) = K ( x 2 n t- y2) at_ O(Ix, y[3),

p(O,O,e, 6 )=q(O,O,e , 6) = 0 .

# =-t-1, /

K > 0 , (2.44)

Then for 0 < h << 1, the equation H(x , y) = h defines a periodic orbit Lh of (2.43) (e = 0) which intersects the positive x-axis at A(h) = (a(h), 0). Let B(h, e, 6) denote the first intersection point of the positive orbit of (2.43) starting at A(h) with the positive x-axis. Then by (2.44) we have

H ( B ) - H ( A ) = fAB dH -- s[M(h , 6) + O(s)], (2.45)

where

M(h, 6) = -- ( f q - gp)le=odt = - (q dx - p dy)le=0 lz h tX h

= <<.h(PX -I- qy)le=0 dx dy. (2.46)

We call M the first order Melnikov function. We will see that the function plays an important role in the study of the number of limit cycles.

Let u(t, c) be the solution of (2.43) (e = 0) with initial value (c, 0). Then we have by (2.44)

H (u (t, c)) -- H (c, 0) -- r 2 (c), t E I~,

r(c) - - c v / K + S(c), S(c) - -O(c) ~ C ~ .

r Denote by c = c(r) -- ~ + O(r 2) the inverse of r = r(c). Then

H ( v ( t , r ) ) - - r 2, r) = (c(r), 0),


where v(t, r) = u(t, c(r)), and T(r) is the period of the function v in t. Clearly, the func- Jr tions v(t, r) and T(r) are both C ~ with T(0) = KIuI > 0. Introduce a C ~ function as

follows:

G ( O, r ) = v ( T2(-~~ O, r ) .

Obviously, G = O(r) is 2zr-periodic in 0, and

H (G(O, r)) - r 2, 0 6 R. (2.47)

LEMMA 2.8. The change of variables

(x, y)T = G(O, r) (2.48)

transforms the system (2.43) into the C cc system

2re I1 8 Grm(p(G s, 6 )q(G, 8 6))1 0 - T(r) - 2#----~ . . . .

8 ? = 2---~DH(G). (p(G,s, 6),q(G,e, 6)) T.

PROOF. Differentiating (2.48) in t yields

GoO + Gri" - ( f (G) + 8p(G, s, ~), g(G) + eq(G, s, ~))T.

By (2.47) and the definition of G, we have

DH(G)Go = 0 , DH(G)Gr = 2 r , T(r)

Go -- --~ ( f (G), g(G)).

Multiplying (2.50) by DH(G) from the left-hand side gives

2ri" -- 8DH(G) . (p(G, s, ~), q(G, e, 6)),

which gives the second equation in (2.49). Further, noting that

Gr A GO = T(r) 2rr

~ G r A ( f ( G ) , g ( G ) ) - #T(r )DH(G)G r _ 2re

#T(r) 7r

it follows from (2.50) that

(ar /k ao)O - ar /k ( f (G), g(G)) + Bar/k (p, q),

(2.49)

(2.50)

o r

#T(r) - ~ r O = -21zr + eGr A (p, q)

388 M. Han

which gives the first equation in (2.49). The proof is completed. D

By (2.49) we obtain

d r = s R ( O , r , s , 8), (2.51)

dO

where R is C ~ and 2Jr-periodic in 0 with R = O(r) and

T(r ) R(O r, 0 8) -- D H ( G ) �9 ( p ( G 0 8), q (G , O, ~))T. (2.52)

' ' 4zrr ' '

Let r(O, p, s, 8) denote the solution of (2.51) with r(0, p, s, 8) = p, and (x(t, c, s, 8), y( t , c, s, 8)) the solution of (2.43) with (x(0, c, e, 8), y(0, c, s, 8)) - (c, 0). Then the Poincar6 map P (c, s, 8) of (2.43) is given by

P(c , s, 8) = x ( r , c, s, 8) - P(c)

where

r -- min{t > O lcx ( t , c, s, 8) > O, y( t , c, s, 8) -- 0}.

By the definition of G we have G(0, r) - (c(r), 0) which yields

G(0, P I ( P ) ) = (c(P1 (p)), 0),

where

P1 (P) -- r (2Jr, p, s, 8) = P1 (P, s, 8)

which is called the Poincar6 map of (2.51). On the other hand, by (2.48) for c = c(p) we have

(P(c , s, 8), O) -- G(2Jr, r(2zr, p, s, 8)) = G(0, PI(P)).

Hence

(P(c(p), o) = (c(P, (p)), o),

or

P o c -- c o P1. (2.53)

THEOREM 2.6. Let (2.43) satisfy (2.44). Then:

(i) The funct ion M is C c~ at h - O. It is analytic at h = 0 i f (2.43) is analytic.


(ii) I f there exist a compact subset Do of D and an integer k >~ 0 such that for 0 < h << 1

M(h, 6) = Bk(6)h k+l + o(hk+2) , Bk(6) 5/= O, 6 ~ Do, (2.54)

then there exist eo > 0, an open set U(Do) D Do and a neighborhood V of the origin such that (2.43) has a tmos t k limit cycles in V forO < [e] < eo, 6 ~ U(Do).

PROOF. It is easy to see that the Poincar6 map P1 of (2.51) can be written as

Pl(r ,e , 6 ) - r + e r F ( r , e , 6), (2.55)

where

fo zr rF(r ,O, 6 ) = R(O,r,O, 6 )dO--Ro(r , 6).

By (2.44) and (2.52) we have

1 fH ( f ' g) A (p, q)[e=odt Ro(r, 6) = ~ - - r 2

_-- 1 fi4 (q dx - pdy)l~=o (2.56) 2 r ~ =r 2

which immediately follows R o ( - r , 6) -- -Ro ( r , 6). Hence, the function M*(r) defined by

M* (r) -- 2r Ro(r, 6)

is even. Since M* 6 C ~ , for any integer j > 1, we have

J M* (r) = Z Air 2i -+- N (r),

i=1

where N 6 C ~ is even and N (i) (0) -- 0, i = 0, 1, . . . , 2 j . Let N ( h ) -- N(~/-h). We claim that

~ ( i ) ( 0 ) _ 0 , i - 0 , 1 . . . . . j .

In fact, we can prove

-~(i) (h) -- h j - i ~[i (~/-h ), i - O, 1 . . . . . j , (2.57)

N e . ~

by induction in i, where Ni (r) is C ~ in r and Ni (0) - O. First, it is not hard to see that

N (i) (r) -- r 2j-i Ni (r), Ni E C o~ , Ni (0) -- O, i = O, 1 . . . . . 2j .

390 M . H a n

Hence, (2.57) holds for i = 0. That is,

= hJNo(Vfi) = hJgo(Vfi).

It then follows that

[ 1 1 = hJ -1 jNo(e )+ VfiN (e ).

Let

Nl (r) -- jNo(r ) + ~rN~(r) . Z ~

Then N1 E C ~ and N1 (0) - -0 . Thus, (2.57) holds for i = 1 . Let (2.57) hold for i = k. Then we have

[ 1 /-~)~ (~/-~)] ~(k+l) (h) -- [hJ-kNk(~/-h )]t = h j - k -1 (j - k)Nk(~/-h ) + -~

Set

~ 1 ,~ l Nk+l (r) -- (j -- k)Nk(r) + -~rN~:(r).

It follows that (2.57) holds for i - k + 1. Hence, (2.57) has been proved. By (2.57) it is immediate that N E cJ for 0 ~< h << 1. Let

J M(h) = E Aihi + -N(h).

i=1

Then M E C j . Therefore M E C ~ since j is arbitrarily large. Note that by (2.46) and (2.56) we have

M(h) -- M* (4'-h) - 2v/-hRo (4'-h , 8) -- M(h, 6).

Thus, M E C ~ in h at h = 0, and if (2.54) holds, then

1 (r2k+3) Ro(r, 8) = -~ Bk(6)r 2k+l -+- 0

By (2.53) and (2.55) and Lemma 2.7, Eq. (2.43) has a limit cycle near the origin if and only if F has two zeros correspondingly with one positive and the other negative. Hence, by Rolle theorem, (2.54) ensures that at most k limit cycles can appear near the origin for Is l small and 8 in a neighborhood of Do.


Finally, if (2.43) is analytic, then (2.51) is analytic. Hence the functions R0 and M* are

also analytic. This implies that the series

M* (r) -- E Ai r 2j

i>/1

is convergent for Ir[ small. Therefore M is analytic since M ( h , 3) = M*(~/-h). The proof

is completed. D

By Theorem 2.6, for any k >~ 1 we have the following expansion for M"

M ( h , 3) -- h[bo(3) + bl (3)h + . . . 4- bk(3)h k + O(hk+l)] , 0 < h < < l . (2.58)

We can use the coefficients b0, bl . . . . . bk to study the Hopf bifurcation of limit cycles.

COROLLARY 2.2. Suppose that there exist k >~ 1, 60 ~ D such that bk (30) 7 ~ 0 and

O(bo . . . . . bk -1) , (30) 5 ~ 0, (2.59) bj(3o) -- 0 j = 0, 1 . . . . . k - 1, det 0 (31 , . . . , 3k)

where 3 = (31 . . . . . 3 m ) , m >~ k. Then f o r any eo > 0 and any ne ighborhood V o f the origin

there exist 0 < e < eo and 13 - 30[ < e0 such that (2.43) has precisely k limit cycles in V.

PROOF. Fix 3j --- 3 j0 for j -- k + 1 , . . . , m. By (2.59) the change of parameters

bj - b j (3 ) , j - O, . . . , k - 1

has the inverse 3 j - - 3j(bo . . . . . bk -1) , j = 1 . . . . . k. Then (2.58) becomes

, - - . . . h k-1 (hk+ 1 M ( h 3) h[bo + b l h -% + bk-1 + bk hk + 0 )]

where bk -- bk (6o) 7/= 0 as b0 . . . . . bk-1 = 0. By changing the sign of bk-1 , bk-2 . . . . . bo

in turn such that

b j - l b j < O, j -- k, k - 1 . . . . . 1, 0 < Ib0[ << Ibll << "'" << Ibk-ll << 1,

we can find k simple positive zeros hi, h2 . . . . . hk with 0 < hk < hk-1 < - " < hi << 1. Let rj -- ~ / ~ , j -- 1 . . . . . k. Then r l , . . . , rk are simple positive zeros of Ro(r, 6). By (2.55) and the implicit function theorem, the function F has k zeros rj -% O(e) , j - 1 . . . . . k. This

ends the proof. D

The following is evident from Theorem 2.6(ii).

392 M. Han

COROLLARY 2.3. I f for any eo > 0 and any neighborhood V of the origin (2.43) has k limit cycles in V for some (e, 3) satisfying 0 < lel < eo, 16 - 6ol < eo, then M(h , 6o) -- O(hk+l) .

In the multiple parameter case it often occurs that M(h, 60) ---- 0, M(h, 6) ~ 0 for 3 ~ 30. In this case (2.54) fails to hold, and we have further the following result.

THEOREM 2.7. Let (2.44) hold. Suppose

(i) the coefficients bo . . . . . bk-1 in (2.58) satisfy (2.59); (ii) there exists a k-dimensional vector function 99(e, 3 k + l . . . . . 3m) such that (2.43) has

a centerat the origin when (61 . . . . . 3k) - - q9(6, 3 k + l . . . . . 3m)fOr [e[ + [3 -3o [ small. Then there exist eo > 0 and a neighborhood V of the origin such that (2.43) has at most k - 1 limit cycles in V for 0 < [el < eo, [3 - 30[ < eo. Moreover, k - 1 limit cycles can appear in an arbitrary neighborhood of the origin for some (e, 3) sufficiently near (0, 6o).

PROOF. By (2.58) and noting that M(h, 3) -- 2rRo(r, 6), r = x / h , we have

j=O

Then the function F in (2.55) has the following expansion:

2 k - 1

F(r, s, 3) -- Z Cj(S, 3)r j -+- r j=0

2kQ(r,e, 3), (2.60)

where Q 6 C ~176 and

C 2 j ( 0 , 3) = bj(3), c2j+l (0, 3) = 0, j = 0 . . . . . k - 1.

By (2.59), the equations

bj -- r 3), j = 0 . . . . . k - 1

have the solution

(31 . . . . . 3k) = qS(e, bo . . . . . bk-1,3k+l . . . . . 3 m ) . (2.61)

By condition (ii) we have F(r, e, 6) = 0 and hence r 3) = 0 , j = 0 . . . . . k - 1 as long

as (31 . . . . . 3k) - qg(e, 3~+1 . . . . . 3m). The uniqueness of the solution q3 implies that

q3 = q9 if and only if b0 . . . . . bk-1 - - 0. (2.62)


Inserting (2.61) into (2.60) we obtain

k-1 F = E [ b j r 2 J -+- c2j+l (~, q~, 8k+l . . . . . 3m)r 2j+l ] -+- r 2k Q(r, t~, ~, 8k+l . . . . . 8m)

j=o

= / 7 ( r , s, ~), ~- - (b0 . . . . . b k - 1 , 8 k + l , - . . , 8m). (2.63)

It follows from (2.61) and (2.62) t h a t / 7 - 0 as bj - -0 , j = 0 , . . . , k - 1. Hence, by the mean value theorem we can write

k-1

c 2 j + l (8, q3, 8k+l . . . . . 8m) -- 6 E bi Aij(8, ~), i=0

k-1

Q(r, s, ~p, 3k+, . . . . . 8m) -- E b ia i ( r ' ~' ~) ' i=0

where Qi E C c~, i = 0 . . . . . k - 1. Note that sr2F -- r(P1 - r) keeps sign fo r0 < Irl << 1

by Lemma 2.1 and (2.53). By (2.63), it follows that

c2j+l (S, q~, 8k+l . . . . . 8m) = 0, if b0 . . . . . bj -- 0, j -- 0 , . . . , k - 1.

This yields that

Aij --0, j + 1 <~ i <~ k - 1, j - O . . . . . k - 2.

Therefore, the function F in (2.63) can be written as

k-1

F(r,s , 6 ) - - E b j r 2 j P j ( r , s , 3 ), j=o

(2.64)

where

k-1 Pj = 1 + e E Aji (s, 3) r 2(i-j)+l + r 2(k-j) Q j,

i=j

O < ~ j < ~ k - 1 .

We particularly have

k-1

F(r, s, 3) -- E bjr2J + O([r]2k + is[).

j=O

By using this form we can prove, in a similar manner to Corollary 2.2, that k - 1 limit cycles can appear in an arbitrary neighborhood of the origin for some (s, 8) sufficiently

close to (0, 80).

394 M. Han

It needs only to prove that F has at most k - 1 positive zeros near r = 0 for [bol +

�9 "" + Ibk-11 > 0 small. We do this by induction in k. For convenience, denote by Fk-1 the right-hand side function of (2.64).

First, for k -- 1, we have

Fo -- boPo -- b0(1 + O(lel + Irl)) ~ 0 for Ib01 > 0.

Suppose for k = n the function

n-1

Fn-1 = ~ bj r 2j Pj j=O

has at most n - 1 positive zeros in r near r = 0 for lel and Ib0l + " " + Ibn-ll > 0 small, where Pj -- 1 + O(lel + Irl) ~ C ~ , j = 0 . . . . . n - 1. Consider the function

H

Fn = Z bJ r2j PJ' j=O

where Pj = 1 + O(lel + Ir[) E C a . We have

Fn = POFn, Fn -- ~-~bjr2J ff j, j=0

where P0 = 1, Pj = Pj/Po = 1 -+-O(lel + [r[) 6 C a , j = 1 . . . . . n. Then

"~ ~ ( f f j r d f f j ) n-1 dEndr ---- 2jbjr2j-1 -Jr- -~ ~ = r Z bjr2J-fiJ -- r-fin-l,

j = l j=O

where

/ ~ j - 2 ( j + 1)bj+l , P j -- ej+l -]- r dPj+l 2 dr

= l + O ( l e l + l r l ) ~ C ~

By the induction assumption, the function Fn-1 has at most n - 1 positive zeros in r near

r - - 0 for lel and I/~01 + " " + Ibn-ll > 0 small. Hence, by Rolle 's theorem it follows that

Fn has at most n positive zeros near r - 0 in r for lel and Ib01 + " " + [bn I > 0 small. This finishes the proof. D

Observe that in both Theorems 2.6 and 2.7 the parameter 6 is required to vary near 80.

The results are local in this sense. In many cases the functions p and q in (2.43) depend

on 8 linearly. This enables us to obtain a global result based on Theorems 2.6 and 2.7 as follows.


THEOREM 2.8. Let (2.44) hold, and let the functions p and q in (2.43) be linear in 6.

Suppose further that for an integer k >~ 1 O(bo . . . . . bk-1)

(i) rank = k, m >~ k; 0(61 . . . . . 6m)

(ii) Eq. (2.43) has a center at the origin as bj (3) = O, j = O, 1 . . . . . k - 1. Then for any given N > O, there exist eo > 0 and a neighborhood V o f the origin such that Eq. (2.43) has at most k - 1 limit cycles in V for 0 < le[ < e0, [31 <~ N. Moreover, k - 1

limit cycles can appear in an arbitrary neighborhood o f the origin for some (e, 6).

PROOF. Since p and q are linear in 6, each coefficient bj in (2.58) is also linear in 6.

Hence, by condition (i) we can suppose

O(bo . . . . . bk-1) det :/:: O.

0(61 . . . . . 6k)

It follows that the equations bj(3) = 0, j = 0 . . . . . k - 1 have a solution (61 . . . . . 6k) =

qg(6k+a . . . . . 6m). , , ~ m For 60 - (qg(0) 0) E , by Theorem 2.7, Eq. (2.43) can have k 1 limit cycles near

the origin for some (e, 6) near (0, 6~). Then we need to prove that k - 1 is the maximal

number of limit cycles. If it is not the case, then there exist N > 0 and sequences en ~ 0, 6 (n) E R m with 16(n)[ ~ N such that for (e, 6) = (en, 6 (n)) Eq. (2.43) has k limit cycles which approach the origin as n --+ cx~. We may suppose 3 (n) --+ 30 for some 60 E ~m as

n -+ Cx~. First, by Theorem 2.6 we must have bj (30) = 0, j = 0 . . . . . k - 1. By our assump-

tion (ii), Eq. (2.43) has a center at the origin for (31 . . . . . 3k) = q9(6~+1 . . . . . 6m). Hence, it follows from Theorem 2.7 that (2.43) has at most k - 1 limit cycles near the origin for all (e, 6) sufficiently close to (0, 60). This is a contradiction since Eq. (2.43) has k limit cycles

approaching the origin as (e, 6) = (en, 6 (n)) ~ (0, 60). This ends the proof. [B

Theorems 2.5 and 2.6 tell us that the coefficients in the expansion of M act like focus values. The following lemma gives a relation between the two groups of the values.

LEMMA 2.9. Let (2.44) and (2.58) hold. Then

* b j - - 4 7 r [ * (1 * * * I b o = 4 r c v 1, - ~ Vej+l + O v 1, v 3 . . . . . Vej_ 1)] , j - - 1 . . . . . k - l ,

where

V2j+I 0V2j+I

0e 6=0 j = 0 . . . . . k - l ,

and l)2j+l is the j th Lyapunov constant o f (2.43) at the origin.

PROOF. Let P(r, e, 6) denote the Poincar6 map of (2.43). Then the point B in (2.45) is

given by (P(a , e, 6), 0), where a - ~ / h ( 1 + O(~/-h)). Note that P - a -- O(leal) . By

/ , _ . _ _ _ _

the

396 M. Han

mean value theorem we have

H ( B ) - H ( A ) = Hx(a(1 + O(s)), 0 ) (P - a)

= 2Ka[1 + O(l~l + la l ) ] (P - a ) .

By (2.39) and (2.40) we can write formally

la2i+l P (a, s, 6) - a = 2re V2i+ Pi , i >>.o

where Pi = 1 + O(a), i ~ O. Hence

H ( B ) - H(A) = 4Jrh E V2i+.l hi--ff i ' K l

i >~o

where f f i = 1 + O(Isl + ~/~), i ~> 0. Inserting the above into (2.45) and noting V 2 i + l - -

sv~i+l + O(s 2) we obtain

4rch Z v~i-k-1 hi g i (1 + O(~/-h)) = M(h, ($). i>~0

Since M is C ~ at h = 0 it follows from the above that

[ 1 1 ] M(h, ($ ) - 4rch v~ + -~(v~ + O(v~))h + -K-~(v~ + O(v~, vj) )h 2 + . . . .

Thus the conclusion follows by comparing with (2.58). This completes the proof. D


2 - y - (/zox 5 + / Z l x3 -+- ]z2x), (2.65) --- mX.

We claim that there exist so > 0 and a neighborhood V of the origin such that (2.65) has at most 2 limit cycles in V for l/z0[ + I/Zll + ]/ZZ[ < 60, and 2 limit cycles can appear.

In fact, by the discussion to (2.41) we know that

1 Vl -- --~/Z2 + O(/z2), /33 -- - -~/Zl + O(/z2),

5 v5 - - ~ / z 0 + O(1~11 + 1~21).

Let . i - 6($i, G -- V//z 2 -Jr- . 2 _it_ . 2 , ($2 _+_ ($2 + ($2 ~_. 1. Then by L e m m a 2.9 (taking/or -- 1

and K = 1/2 ) w e have

b0 = -2zr($2, bl = -3zr($1 + 0(($2), 8 2 - -5JrSo 4- O(ISll + IS21).


For the sake of convenience, we take 30, r r as free parameters with 16il ~ 1. Then the conditions of Theorem 2.8 are satisfied with m - k -- 3. Therefore, the claim follows.

Note that for e = 0 the origin is a linear center of (2.65). Hence, in this case the function

M can be obtained directly by using (2.46).

DEFINITION 2.4. We say that Eq. (2.43) has Hopf cyclicity k - 1 at the origin if the

conclusion of Theorem 2.8 holds.

Thus, both systems (2.41) and (2.65) have Hopf cyclicity 2 at the origin.

EXAMPLE 2.7. Consider a Li6nard system of the form

2n+1

Yc -- y - e E ai X i , ~ = --X (2.66)

i=1

where n >~ 1, Jail <~ 1, and e > 0 is small. We claim that the system has Hopf cyclicity n at

the origin. In fact, we have H ( x , y ) - l (x2 + y2) with # = 1, K - 1. The curve Lh given by

H ( x , y ) -- h has the representation (x, y) -- x/~h(cos t , - sin t). Hence, by (2.46) we have

F/

M ( h ) -- - E 2 J + l N j a 2 j + l h J + l

j=O

where

f0 7r Nj = COS 2(j +1) t d t > 0.

Set 6 = ( a l , a2 . . . . . a 2 n + l ) , bj -- - 2 j + l Nja2j+l , j -- 0 . . . . . n and k = n 4- 1. Note that by Theorem 2.3(i) Eq. (2.66) has a center at the origin as bj --O, j = 0 . . . . . n. Thus, the

claim follows from Theorem 2.8.

In the proof of Corollary 2.2 we have given a way to find limit cycles near the ori-

gin. In the case of (2.66), M ( h ) can have n zeros hi > h2 > . . . > hn > 0 for some ( a l , a3, . . . , aZn+l ) . Then the corresponding function F in (2.55) has n zeros x / ~ + O(e),

j = 1 . . . . . n. It follows that the n limit cycles of (2.66) approach the curves 1 (x 2 + y2) _

h j , j - - 1 , . . . , n, respectively, as e ~ 0. In the following we give another way to obtain limit cycles near the origin. For conve-

nience, take a2j = O, j = 1 . . . . . n.

First, from Lemma 2.9, we have

* ( ) , j - o . . . . . n , l )2j+l -- eV2j+I 4- O 6. 2

398 M. Han

where

1 No v~ = ~--~-b0 = - 2--~- a l ,

V2j+l : 2j+2rcbj + O(Ib0 . . . . . b j - l l ) - - - - a2 j+ l + O(la l ,a3 . . . . . a2 j - l l ) .

On the other hand, for V = x 2 -k- y2 we have along the orbit of (2.66)

"~ ~ X 2j+2. V = - 2 e a2 j+ l

j=0

By Corollary 2.1, it holds that

Lk+l = - - - - a 2 k + l N k

if a2j+l - 0 for j -- 0 . . . . . k - 1. It then follows that

v 2 j + l - - - - e l ~-~a2i+l+zrc _ O(la l ,a3 . . . . . a 2 j - l l ) ] ( l + O(e)), j ~ 0 ~ . ~ 1 7 6

For e > 0, let us vary al , a3 . . . . . a2n+l such that

a2j - la2 j+l < O, 0 < l a 2 j _ l l < < l a 2 j + l l < < e , j = l . . . . . n

which yields

V2j-1132j+1 < 0, 0 < Iw2j-ll << Iw2j+ll << 8 2 j = 1 n , ~ , o o , �9

Thus, by (2.32), the corresponding function P (r) - r will have n zeros which give n limit cycles of (2.66). Geometrically, the n limit cycles are obtained by changing the stability of the origin n times. Moreover, unlike above they approach the origin as e ~ 0.

REMARK 2.3. There are different ways to prove Lemma 2.3 for the normal form system (2.11). Here the proof is given by following [111]. The conclusions in Lemmas 2.4 and 2.6 are well known. However, the definite relationship between constants am, V2m+l and Lm given in the lemmas appears for the first time here. For more discussions on the level set of a Hamiltonian system, the reader can consult [36]. On the bifurcation of limit cycles, Theorems 2.4 and 2.5 are basic tools, while Theorems 2.6-2.8 were recently obtained by [47]. More discussions similar to Lemma 2.9 can be found in [80]. Under condition (2.44), it was proved in [64] that the right-hand side function in (2.45) is C c~ in ~/h at h = 0, but not C 2 in h at h = 0 generally.


As we know, in 1901, Hilbert [73] posed 23 mathematical problems of which the second part of the 16th one is to find the maximal number and relative position of limit cycles of planar polynomial systems. Many works have been done on the study of the problem, especially for quadratic and cubic systems, see [3,4,8,10-13,15,18-34,38-51,54-72,74-110, 113-126,128,129]. A detailed introduction and related literatures can be found in Li [85], Schlomiuk [109] and Ilyashenko [78]. As was showed in Examples 2.4 and 2.7, a typical way to find limit cycles in Hopf bifurcation is to compute focal values and change the stability of the focus by using the values. Certain nice results have been obtained for quadratic and cubic systems. Bautin [5] proved that a focus of a quadratic system has order at most three and that for this system a focus or center can generate at most three limit cycles under perturbations of its coefficients. Then Chen and Wang [ 12] (by bifurcation method) and Shi [ 110] (by using Poincar6-Bendixson theorem) separately found a quadratic system having four limit cycles. More and more mathematicians believe that quadratic system have at most four limit cycles. However, up to now the problem is still open. Li and Li [89], Li and Huang [88], Li and Liu [91-93], and Liu, Yang and Jiang [96] found different cubic systems having eleven limit cycles by using Melnikov function method. James and Lloyd [79] gave a cubic system having eight limit cycles in a neighborhood of a focus. Recently, Han, Lin and Yu [60] and Yu and Han [120,121] obtained sufficient conditions for a cubic system to have 10 or 12 limit cycles, respectively (all limit cycles having small amplitude). It seems that the maximal number of limit cycle for cubic system is 12.

For Hopf bifurcation in higher dimension, the reader can see Hale [35], Chow and Hale [16], Chow, Li and Wang [17] and Han [51] etc. For bifurcation of periodic solutions of delay-differential equations, see [7,37,52,53] et al.

3. Perturbations of Hamiltonian systems

3.1. General theory

In this section we will study a C ~ system of the form

Yc = Hy + ep(x, y, e, 6), -- - H x + eq(x, y, e, ~), (3.1)

where H(x , y), p(x , y, e, 6), q(x, y, e, 6) are C ~ functions, e >~ 0 is small and 6 E D C ]~m is a vector parameter with D compact. For e = 0 (3.1) becomes

= Hy, ~ = - H x (3.2)

which is Hamiltonian. Hence, Eq. (3.1) is called a near-Hamiltonian system. For Eq. (3.2) we suppose there exist a family of periodic orbits given by

Lh: H(x , y) = h, h E (~, ~)

such that L h approaches an elementary center point, denoted by L~, as h --+ c~, and an invariant curve, denoted by L~, as h --+ 13. Without loss of generality, we can assume that each L h is oriented clockwise.

400 M. Han

Fig. 9. The phase portrait of (3.2) with a L/~ homoclinic loop.

L~

If L~ is bounded, it usually is a homoclinic loop consisting of a saddle and a connection or a heteroclinic loop consisting of at least two saddles and connections between them. In the homoclinic case, the phase portrait of the family {Lh" c~ ~< h ~</3 } is given in Fig. 9.

Introduce an open set G as follows:

G - U Lh. or<h</3

Our main purpose is to study the number of limit cycles of Eq. (3.1) in a neighborhood of the closure G of G for e > 0 small and 8 6 D.

Note that ifEq. (3.1) has a limit cycle L(e, 3) for e > 0 small and 3 6 Do C D, then the limit of the cycle as e --+ 0 is either the center L,~, a periodic orbit Lh with h 6 (or,/3) or the boundary L~. That is,

lim L(e, 3) - - L h , h ~ [c~,/3]. e-+0

In this case, we say that the limit cycle L(e, 3) is generated from Lh. Thus, in order to study the number of limit cycles, we first need to study the number of limit cycles generated from

each Lb. For the purpose, similar to Definition 2.4 we first introduce a notation below.

DEFINITION 3.1. We say that Eq. (3.1) has cyclicity k at a given Lh, h 6 [or,/3], if there exist e0 > 0 and a neighborhood V of Lh such that Eq. (3.1) has at most k limit cycles in V for 0 < e < e0, 6 6 D and if k limit cycles can appear in an arbitrary neighborhood of Lh for some (e, 3) with e > 0 sufficiently small. More specifically, k is said to be Hopf (Poincard or homoclinic) cyclicity when h = ot (h ~ (or,/3) or h = /3 with Lr being a homoclinic loop).

In the last section we gave some method to find Hopf cyclicity. This section concems with global bifurcations of limit cycles and presents further methods to find Hopf, Poincar6 and homoclinic cyclicity.

Take h = h0 a (or,/3) and A(ho) E Lho. Let l be a cross section of Eq. (3.2) passing through A(ho). Then for h near h0 the periodic orbit Lh has a unique intersection point with l, denoted by A(h). That is, A(h) = Lh (q 1. Consider the positive orbit v(h, e, 3)


of Eq. (3.1) starting at A(h). Let B(h, e, 6) denote the first intersection point of the orbit with I. Then similar to (2.45) we have

H(B) - H(A) -- e[M(h, 6) + 0@)] = eF(h, e, 6), (3.3)

where

M(h, 6) -- fL (Hyq + HxP)le=odt h

---fL (qdx -pdy ) l~=o- - f f . h <.h

(Px + qy)le=o dx dy. (3.4)

The function F(h, e, 6) in (3.3) is called a bifurcation function of Eq. (3.1). It has the following property.

LEMMA 3.1. For e > 0 small and 6 E D, Eq. (3.1) has a limit cycle near Lho, ho E (c~,/3), if and only if the equation F (h, e, 6) = 0 has a zero in h near ho.

PROOF. Since the orbit y(h, e, 6) starting at A(h) is closed if and only if A = B, we need only to prove that A = B if and only if H(A) = H(B) by (3.3). It is easy to see that B -- A + O(e). Hence, by Taylor formula for e > 0 small we have

H(B) - H(A) -- (Hx (A), Hy(A)) . (B - A) + O(]B - Ale).

Note that the cross section I can be taken to be parallel to the gradient (Hx (A), Hy (A)). It follows that

H ( B ) - H ( A ) - [ + v / H 2 ( A ) + H 2 ( A ) + O(IB - AI) ] . [B - AI,

which gives the desired conclusion. The proof is completed. V]

As in the situation of Hopf bifurcation, the Melnikov function M(h, 6) can also be used to determine the cyclicity at a periodic orbit. First, by (3.3) we have

THEOREM 3.1. Let ho ~ (or, ~), 6o ~ D. (i) There is no limit cycle near Lho for e + 16 -- 601 small if M(ho, 60) =/= O.

(ii) There is exactly one (at least one, respectively) limit cycle L(e, 6) for e + 16 - 6ol small which approaches Lho as (e, 6) --+ (0, 60) if M(ho, 60) = O, Mh (ho, 60) 5/= 0 (ho is a zero of M(h, 6o) with odd multiplicity, respectively).

(iii) The cyclicity of Eq. (3.1) at Lho is at most k if for any 6 ~ D there exists 0 ~ j <, k such that

g~ j) (ho, ~) r O.

402 M. Han

PROOF. By (3.3) and Lemma 3.1, the first conclusion is clear. For the second one, the conclusion follows from the implicit function theorem if h0 is a simple zero of M(h, 60). Let h0 be a multiple zero of M(h, 60) with odd multiplicity. Then for e0 > 0 small we have

M(ho - eo, 60). M(ho 4- eo, 60) < O.

Hence, by (3.3) we have

F(ho - e0, e, 6). F(ho + eo, e, 6) < 0

for 0 < e < eo, 16 - 6ol < eo as long as eo is sufficiently small. Thus the function F(h, e, 6) has a zero h* 6 (ho - eo, ho + eo).

For the third conclusion, if Eq. (3.1) has cyclicity at least k 4- 1 at Lho, then there exist some en --+ O, 6 ~n) ~ D such that for (e, 6) = (en, 6 ~n)) Eq. (3.1) has at least k 4- 1 limit cycles which approach Lho as n ~ c~. We can suppose 6 (n) ~ 60 as n --+ cxz. On the other hand, by our assumption M(h, 6o) has ho as a zero of multiplicity at most k. By (3.3) and Rolle's theorem for e 4- 16 - 6ol small the function F(h, e, 6) has at most k zeros in h near ho, which follows that at most k limit cycles exist near Lho for e 4- 16 -- 6ol small. Thus a contradiction appears if (e, 6) = (en, 6 ~n)) with n sufficiently large.

The proof is completed. [3

By the above proof we have immediately

COROLLARY 3.1. Suppose for some 60 ~ D, M(h, 60) has k zeros in h ~ (c~, fl) with each having odd multiplicity. Then for e + 16- 6ol small Eq. (3.1) has k limit cycles in a compact subset of the open set G.

COROLLARY 3.2. If there exist ho ~ (c~, fl), 60 ~ D such that for an arbitrary neighborhood of Lho, Eq. (3.1) has k limit cycles in the neighborhood for some (e, 6) with e + 16 - 6ol sufficiently small, then M(h, 60) = O(Ih - holk).

Let L(e, 6) be the limit cycle appeared in Theorem 3.1. To determine its stability we need to consider the sigh of the integral

e fL(e,~) ( p x + q y ) d t = e l f L (px+qy) le=odt+O(e)] . ho

Obviously, if

f a(ho, 60) = ~p (Px + qy)lE=0,~=~0 dt r 0

,J L ho

then the stability will be determined easily. The following lemma gives a relation between cr (h, 6) and Mh (h, 6).


LEMMA 3.2. Suppose Lh is oriented clockwise. Then

Mh(h, 6) = -+- fc (Px + qy)le=odt = +or(h, 6), h

where "+" (respectively " - " ) is taken when Lh expands (respectively shrinks) with h increasing.

PROOF. Fix h0 6 (or,/5). For definiteness, suppose that Lh expands with h increasing. Then for h > h0, applying Green's formula we have

f f M(h, 6) - M(ho, 6) -- ] ] (Px + qy)le=0 dx dy,

JJA (h) (3.5)

where A(h) denotes the annulus bounded by Lh and Lho. Let u(t, h) denote a representation of L h satisfying

H(u(t,h))-h, O<<.t<~r(h), hE(u,~).

Here T (h) denotes the period of Lh. Consider the integral transformation of variables given by

(x, y) = u(t, r), O <, t <, T(r), ho < r < h.

Note that

Ou(t,r) DH(u) �9 Dru = det ~ = 1.

O(t,r)

We obtain from (3.5)

f h foT(r) M(h, 6) - M(ho, 6) = dr o

(Px q- qy)(U(t, r), O, ~) dt.

Then differentiating the above in h yields

Mh(h, 6) = fo r(h) (px -k- qy)(U(t, h), O, 6) dr.

This ends the proof. [3

EXAMPLE 3.1. Consider van der Pol equation

404 M. H a n

The equation is equivalent to

.it = y, ~ - - - X - - 8 ( X 2 - - 1)y. (3.6)

For e = 0, Eq. (3.6) has periodic orbits L h " 1 (x 2 _+. y2 ) _ h, h > 0. By (3.4) we have

M(h) - - ffx2+y2<~2 h (1 - x2) dx dy - rch(2 - h).

The function M has a unique positive zero h -- 2. By Lemma 3.2 we have

cro = fL (1 - X 2) dt - - M ' ( 2 ) - - -2zr. 2

Hence, by Theorem 3.1, Eq. (3.6) has a unique limit cycle L(e) for e > 0 which is stable, simple and approaches the circle x 2 + y2 _ 4 as e --+ 0.

In many cases (polynomial systems for example), the first order Melnikov function has

the form

k

M(h, 8) -- Z bi(a) l i (h) , i=1

k~>2.

Let

I~ (or) -J: O, I i ( h ) -~0 , o r < h < / 5 . (3.7)

Then we can write

k

M(h, 6) -- I i ( h ) ~ bi Ji(h) - - ll (h )N(h , b), i=1

(3.8)

where b - (bl . . . . . bn), bi - bi(6), Ji(h) - l i ( h ) / I I (h ) , i - 1,2 . . . . . k. Introduce the Wronskian of the functions J1, J2 . . . . . Jk

W ( h ) =

Jl(h) J2(h) . . . Jk(h) g~ (h ) J~ (h ) . . . J[ (h )

.. l (k -1) (h) j k-l h) �9 ok

(3.9)


Clearly, by a property of determinant we have

Wt(h) =

J l (h) J2(h) . . . Jk(h) J[(h) J~(h) ... J~(h)

.. /(k-2) (h) j ~ - 2 ) ( h ) J2 (~-2~(h) . ,,~

j(k) (h) J2 (k) (h) . . . j(k) (h) 1

We have

(3.10)

THEOREM 3.2. Suppose that (3.7) and (3.8) hold. Let Ibl > 0 for all 6 ~ D and ho [~, ~).

(i) If W(ho) ~= 0, then Eq. (3.1) has cyclicity at most k - 1 at Lho. (ii) If W(ho) = 0 , Wt(ho) ~ 0, then Eq. (3.1) has cyclicity atmost k at Lho.

PROOF. For the first conclusion, let us suppose Eq. (3.1) has cyclicity at least k at Lho. Then there exist Sn --+ c~, 3n ~ D such that for (s, 3) = (Sn, 3n) Eq. (3.1) has k limit

(n) with (n) cycles L j L j --+ L ho as n --+ c~, j -- 1, 2 . . . . . k. Without loss of generality, we can suppose

6n-+ 60, b(6n)-+ bo=(blo . . . . . bko) a s n - + r

Near h0 we have the following expansion for N (h, b):

N(h, b) = hi +/92(h - ho) + . . . + bk(h - ho) k-1 -l- bk+l (h - ho) k + . . . , (3.11)

where, by (3.8),

bj+l - [ ~ M(h' 6) ] (j) l [bl J(J) (ho) + + bk~k , . . . . . " (J) (ho)] I i (h) Jh=ho J!

j = 0 , 1 . . . . . k - l , k . (3.12)

It follows that

a({~ . . . . . ~ ) W(ho) det O(bl, . . . , bk) 1!2!.. . (k - 1)! -~ 0. (3.13)

Note that Ib0l > 0 by our assumption. By (3.12) and (3.13) we obtain

(/91 . . . . . /gk) 15=80-- ( ~910 . . . . . lgk0) ~ O.

Thus, an integer I satisfying 1 ~< l ~< k exists such that

/~j0--0, j - - 1 . . . . . 1 - 1 , b l 0 r

406 M. Han

Hence, we have by (3.11)

N(h , bo) - fglo(h - ho)/-1 -I- O(Ih - hol/), /91o -~ O. (3.14)

Therefore, by (3.7) and Corollary 2.3 (if h0 = or) or Corollary 3.2 (if h0 > ct), there exist e0 > 0 and a neighborhood U of Lho such that for 0 < e < e0, 16 - 601 < e0 Eq. (3.1) has at most 1 - 1 limit cycles in U. This contradicts that Eq. (3.1) has k limit cycles approaching Lho for (e, 6) = (en, 6n) and n --+ oe. This finishes the proof of conclusion (i).

Let W(ho) = O, W'(ho) # O. If the conclusion (ii) is not true, then, as before, there exists a sequence (en, 6n) approaching (0, 60) such that for (e, 6) = (en, 6n), Eq. (3.1) has k + 1 limit cycles approaching Lho as n --+ ~ . In this case, formulas (3.11) and (3.12) remain true with

0(/91 . . . . . bk-1, bk+l) W'(ho) det O(bl . . . . . bk) -- 1!2!. . . (k - 2)!k! ~ 0, (3.15)

and hence

(b~ . . . . . E'~-,, bk+l)I~-_~o--(/~10 . . . . . bk-l,O,/gk+l,O) ~=0.

It follows that there exists 1 ~< l ~< k + 1 and I # k such that (3.14) holds. Thus, a contradiction appears in the same way as above. This ends the proof. D

It often happens that b -- 0 for some 6 6 D. In this case, the condition of Theorem 3.2 fails and the following one can apply further.

THEOREM 3.3. Suppose the following conditions are satisfied. ~b _ k and b(6o) = 0 for some 60 ~ D (a) The vector b is linear in 6 with rank ~

(b) Eq. (3.1) is analytic on the closure G and has a center near L~ when b = O. (c) There exists ho ~ [or, ~) such that IW(h0)l + IW'(ho)l # 0.

Then (i) When W(ho) # O, Eq. (3.1) has cyclicity k - 1 at Lho.

(ii) When W(ho) - -0 , W'(ho) # O, Eq. (3.1) has cyclicity k (respectively k - 1 or k) at

Lho if ho > ot (respectively ho = ~). (iii) For each h ~ [t~, ~), Eq. (3.1) has cyclicity at least k - 1 at Lb.

PROOF. Let W(ho) :/: 0 first. By Definition 3.1, it suffices to prove the following two points:

(1) There are at most k - 1 limit cycles near L ho for e > 0 small and 6 ~ D. (2) There can appear k - 1 limit cycles in any neighborhood of L ho for some arbitrarily

small e + 16 - 601. We proceed the proof by contradiction. If the conclusion (1) is not true, then there exists

a sequence (en, 6n) --~ (0, 6*) with 6" 6 D such that for (e, 6) = (en, 6n) Eq. (3.1) has k


limit cycles which approach Lho as n --> e~. The proof of Theorem 3.2 implies b(3*) -- 0. By condition (a) we may assume

det O(bl, . . . . bk) 0(31, 7 3k) 5;~ O.

Then for 3 near 3" the linear equation b = b(3) h a s a unique set of solutions 3 j - -

6j(b, 3k+l . . . . . 3m), j = 1 . . . . . k. By (3.12) we have

0 (/91,- �9 �9 O(/)l . . . . . /gk) det = det det

0(31 . . . . . 3k) O(bl . . . . . bk)

O (bl , . . . . bk) 0(31, 7 3k) 5;~ O.

Further, by condition (b) Eq. (3.1) has a center near L~ when (hi . . . . . /)k) = 0. Thus, in the case of h0 -- or, by (3.7), (3.8), (3.11) and Theorem 2.7 we know that Eq. (3.1) has Hopf cyclicity k - 1 at L~ for e + 13 - 3"1 small. This contradicts to the existence of k limit cycles near L~ for (e, 3) = (en, 3n). The conclusion (1) above is proved for h0 = oe. Since b(30) = 0, using 30 instead of 3", the above discussion implies that k - 1 limit cycles can appear in any neighborhood of Lc~ for arbitrarily small e + 13 - 601. Then the conclusion (2) follows for h0 = ~.

For the case of h0 > oe, by (3.3), (3.7), (3.8) and (3.11) we have

j= l

= II(h) I~-~bj(h-ho)J-1 + O(,h- h0,k)], j= l

(3.16)

where {~j - /~ j + O(e), j - 1 . . . . . k. Also, by (3.13) and condition (b) Eq. (3.1) has a center near Lu for (/~1 . . . . . /~k) -- 0 and

hence for h near h0,

F(h,e , 3)--O if (/~1 . . . . . /~) - -0 .

Thus (3.16) can be rewritten as

k F(h,e , 3 ) - I i ( h ) E b j ( h - h o ) J - l [ 1 + Pj(h,e , 3)],

j= l (3.17)

where bj - / ~ j + O(E[bl . . . . . bkl), Pj(h,e , 6 ) - O ( I h - holk-J+l), j = 1 . . . . . k. Then using the form of (3.17) and similar to the proof of Theorem 2.7 we can prove that F has at most k - 1 zeros near h = h0 for e + 13 - 3"l small. A contradiction occurs too as before. Also, as before, by using (3.17) F must have k - 1 zeros in h near h - h0 for (/~1 . . . . . b~)

408 M. Han

satisfying 0 < [bj[ << [ffgj+l[ << Ib~l, bjbj+l < 0, j = 1 . . . . , k - 1. Hence, the proof of the first conclusion of the theorem is completed.

To prove the second one, suppose W(ho) = O, W'(ho) =/= 0 with h0 > or. Then by (3.9) and (3.10) there exist constants or0 . . . . . a~-2 such that

k-2

( ( ( ) . j ~ - l ) h o _ . . . . , ~ J ( / C - 1 ) ( h o ) ) = E o l J ( J [ J ) ( h o ) , " ' , ~ k "(j)(ho)). j=O

Hence, by (3.12) we have

k-1

j = l (3.~8)

for some constants &l . . . . . ~ k - 1 - Therefore, by (3.15), (3.18), similar to (3.17) we obtain

F(h, e, 6) -- ll(h) I ~ bj(h - ho) j -1 (1 + O([h - holk-j+2)) I_j=l,j#k

+ Sjbj + O(E]/~l . . . . . b k - 1 , bk+l [) " (h - h0) k-1 ,

x j=l

where

b j - b j -t- o ( E l b l . . . . . b k - 1 , bk+l 1), j = 1 . . . . . k - 1, k + 1.

Note that

b j = bj -q- O(E[ff91 . . . . . bk-1, bk+l I), j = 1 . . . . . k -- 1, k + 1.

We have further

k+l

F = I1 (h) E bj(h - ho) j -1 (1 + Pj) , j = l

(3.19)

where

Pj -- (~j + O(e))(h -- ho) k-j,

=O(,h-hol), j - - 1 . . . . . k - l ,

/gk -- O(eff)k+l) .

Using (3.19), similar to Theorem 2.7, one can prove that Eq. (3.1) has at most k limit cycles near Lh0 for e small and 6 6 D.


Furthermore, we can prove that k limit cycles can appear in any neighborhood of L ho.

In fact, let us take

h - h0 -- )~e, {?j - - B j , E k + l - j j -- 1 k - 1 k + 1

where B j a r e such constants that the polynomial in ~,

g(~) - - B1 -+- B2)~ -+-... + Bk-1)~ k-2 -+- B k + l ~ , k

has k simple zeros ~,j ~;& 0, j -- 1 . . . . . k. Then (3.19) becomes

F -- e k I1 (h) [g()O + O(e)],

which has k simple zeros ~j = ~,j + O(6) in )~, j -- 1 . . . . . k. Thus F has k simple zeros

hj -- ho + ~,je in h, j - 1 . . . . . k. For h0 -- ol, we may assume ol -- 0. Then noting (3.18), similar to (2.64) we have

F - - h i (1 + O(IE, r l ) ) +/~2r2(1 + O(le, r l ) ) + . . . + / ) k - l r 2(k-2)

• (1 + O(le, r [ ) )+ [gk+l r2 (k -1 )O(e )+{gk+l r2k(1 + O(le, r l ) ) .

It is easy to see that F has at most k zeros in r > 0 and k - 1 zeros can appear. For conclusion (iii), let h 6 [or,/3) and U be any neighborhood of Lh. Since W is ana-

lytic on [or,/3) there exists/~ > h such that W(f~) ~= 0 and L~ C U. By the conclusion (i) Eq. (3.1) has k - 1 limit cycles in U for some (e, 6). Since U is arbitrary it follows that Lh

has cyclicity at least k - 1. The proof is completed. D

By (3.17) and (3.19) one can prove easily

COROLLARY 3.3. Suppose the condit ions (a)-(c) hold with ho > or. I f

(i) W(ho) 5/=O, l ~< l ~< k - l, or (i i) W(ho) = O, W'(ho) ~ O, 1 ~< 1 ~< k, l ~: k - 1,

then there exists a func t ion 6 = 6(e) with 6(0) E D such that f o r 6 = 6(e) and e > 0 small

Eq. (3.1) has a limit cycle o f mult iplici ty I which approaches Lho as e ~ O.

REMARK 3.1. Under conditions of Theorem 3.3, the function W is analytic and has only isolated zeros on [or,/3). Thus for any/ t e (or, fl), there exist ol ~< hi < h2 < . . . < hi < [t,

1 ~> 0 such that Eq. (3.1) has cyclicity k - 1 at Lh for h e [or, h) - {hi . . . . . hi}.

EXAMPLE 3.2. Consider the Li6nard system (2.66) discussed in Example 2.7

2 n + l

Z 2 = y - e a ix t ,

i=1

m reX.

As before, suppose that e > 0 is small and lail ~ 1 for i - 1 . . . . . 2n + 1 with n/> 1. We claim that for each h ~> 0 the above system has cyclicity n at the circle x 2 + y2 _ 2h.

410 M. Han

In fact, by the discussion in Example 2.7 we have

M ( h ) -- ~ b j + l h j + l ,

j=O bj+l = - 2 j + l N j a 2 j + l , Nj > O, j = 0 . . . . . n.

Comparing with (3.8) we may take

I1 (h) = h, J j (h) = h j -1 j = 1 n + 1 ~ o o . ~ .

Hence, by (3.9) we have W (h) = 1 for all h ~> 0. Thus the claim follows from Theorem 3.3 and Theorem 2.3(i).

EXAMPLE 3.3. For the system

we have

lX2i+1 .;c -- y - e a2i + ,

i=0

~ - - - x ( 1 - x ) ,

H

M(h) -- Z bj+lhJ+l (1 -k- O(h)) j=O

for 0 < h << 1. It follows W(0) = 1. Thus, by Remark 3.1 there may exist finitely many values 0 < hi < . . . < hi in the interval (0, ~) such that for any h 6 [0, ~) - {hi . . . . , hi} (respectively h E {hi . . . . . hi}) the above system has cyclicity n (respectively at least n) at Lh .

DEFINITION 3.2. Let U 6 R 2 be a bounded set and No a positive integer. We call No the cyclicity o f U for Eq. (3.1) with 6 ~ D and e > 0 small if the following are satisfied:

(i) For any given compact set V ~ U Eq. (3.1) has at most No limit cycles in V for all 6 6 D and 0 < e < e0, where e0 = eo(V) > O.

(ii) Eq. (3.1) has No limit cycles for some (e, 6) with 6 ~ D whose limits as e --+ 0 are in U.

The cyclicity of the set G - ~h~(a ,~ Lh for Eq. (3.1) is also called the cyclicity of the period annulus {Lh : ot < h < fl }.

The following theorem gives a sufficient condition for finding the cyclicity of the open sets G and G tO Lc~.

THEOREM 3.4. Suppose (3.7) and (3.8) hold. Let the conditions (a) and (b) of Theorem 3.3 are satisfied. Let further, for each b ~ 0 the function N (h, b) has at most No zeros (taking multiplicity into account) in h ~ [~, fl) (respectively h ~ (~, fl)) and for some b ~ 0 it has No simple zeros in h ~ (or, fl). Then the cyclicity o f the set G U L~ (respect&ely G ) f o r Eq. (3.1) is No as G is bounded.


PROOF. For the sake of simplicity, we suppose that the singular point of Eq. (3.1) near Lc~ is at the origin with oe -- 0. Then we can take a cross section 1 which is on the positive x-axis with an endpoint at the origin. In this case we have

A ( h ) = L h A l - - ( a ( h ) , O ) , B(h,e , 6 ) - ( b * ( h , e , 6),O)

for h > 0 small. By (2.4), the function b* is analytic in a. Thus, b * - O ( a ) and H(B) - H(A) -----O(a 2) - - O ( h ) . Then it follows from (3.3) that F(h, e, 3) - -O(h) . As before, suppose

O(bl . . . . . bk) det 7~ 0.

0(31 . . . . . 6k)

Then we can solve 6 i - ~i(b, 6k+l . . . . . 6m), i - - 1, . . . , k from b = b(6). Hence, for h E [0,/~) we have

F(h, e, 3) -- 11(h)[N(h, b) + O(e)] = F*(h, e, b, 6k+l . . . . . 6m). (3.20)

By our assumption and Corollary 3.1 Eq. (3.1) can have No limit cycles for some (e, 3). What we need to do is to prove that for any given constants 0 < )~ </~ (respectively 0 < # < )~ < /3 ) there exists e0 > 0 such that for 0 < e < e0, 6 6 D the function F has at most No zeros in h 6 (0,)~] (respectively [/z, X]). If the conclusion is not true, then there exists a sequence {(e j , 6j)} with ej ---> O, 6j ---> 3 0 E D as j ----> oc such that F(h, e j , 6j)

has No + 1 z e r o s hij, i = 1 . . . . . NO + 1 in h E (0,)~] (respectively [#, )~]). We can suppose hij ---> hio E [0, )~] (respectively [#, )~]) as j --+ oc. Then by (3.20)

N(hio, bo) - O, i - 1 . . . . . N o + l ,

where b0 - -b (60 ) . Hence, by Corollaries 2.3 and 3.2 the function N(h, bo) has at least No + 1 zeros (multiplicity taken into account) in [0, ~.] (respectively [#, ~]). This implies b o = O .

On the other hand, we have F* [b=O = 0 on [0, ~]. This yields

k

F* = ~ hi Fi (h, e, b, 6k+l . . . . . 6m)

i=1

= I1 (h)[N(h, b) + O(eIb[)]

= [b[I1 (h)[N(h, c) + O(e)]

= IblI1 (h)G(h, e, c, 6k+1,. . . , 6m),

where c = ~ -- (cl . . . . . ck). Let

(3.21)

c(j) - b (6 j ) j >~ 1. [ b ( S j ) ] '

412 M. Han

Then b(S j ) ~ b(60) -- O, j --+ oc. We can a s s u m e C (j) ~ CO with Icol - 1. Note that

F ( h i j , s j , S j ) = O , j >~ 1, i = 1 , . . . , N 0 + 1.

It follows from (3.20) and (3.21) that N(hio, co) = 0, i = 1 . . . . . No + 1, which contradicts to our assumption. The proof is completed. [5

REMARK 3.2. Let the conditions of Theorem 3.4 be satisfied. If G is unbounded, then there exists a compact set V0 C G U Lc~ (respectively C G) such that for any compact set V satisfying V0 C V C G U Lc~ (respectively V0 C V C G) Eq. (3.1) has cyclicity No on V.

EXAMPLE 3.4. Consider Eq. (2.66) again. Since

N ( h , b ) - ~ b j + l h j j=o

in this case, by the discussion in Example 3.2 for any compact set V containing the origin its cyclicity is n.

An open problem for Eq. (3.1) is: What is the maximal number of limit cycles on the plane? The above example suggests that the answer be n. The most difficult part is to study the number of limit cycles which disappear into infinity as e --+ 0.

3.2. Existence of 2 and 3 limit cycles

By (3.8), the function N can be written in the form

N(h, b) = bl - P(h, b2 . . . . . bk). (3.22)

The function P here is called a detection function and its graph on the (h, bl) plane a detection curve. For a given system, an interesting problem is to find a point (bl0, b20 . . . . . bko) such that the line bl = bl0 and the curve bl = P (h, b20 . . . . . bko) have as many intersection points as possible.

The simplest case is that P is monotonic in h 6 (or, fl). Some sufficient conditions for some special systems on the monotonicity of P were obtained by Li and Zhang [84], and Han [49]. Obviously, if P (or, b2 . . . . . bk) ~- P (fl, b2 . . . . . b~) for some (b2 . . . . . bk) then Eq. (3.1) can have a limit cycle.

Next, we give some conditions for Eq. (3.1) to have 2 or 3 limit cycles. We will suppose 13 < cx~ and L/~ is a homoclinic loop with a hyperbolic saddle S on it. First, we prove

LEMMA 3.3. Let 4)(x, y) be a C ~176 function with dp(S) = O. Then along the orbit Lh of Eq. (3.2) the limit limh~t~ fLh 4)(X' Y)dt = fL~ ~b(x, y)dt exists finitely.


t- 0

" " - - . , ~ _ . .

0

,y

,% \

f

.x

Fig. 10.

PROOF. Without loss of generality, we may suppose the saddle S is at the origin and the Eq. (3.2) has the form

2 -- k.x + f (x, y), y = -1 .y + g(x, y) (3.23)

where f , g = O(/x, y[2), which implies that H(x , y) = )vxy + O([x, y13), X ~ 0 with fl - -0 . For definiteness, assume )v < 0 and L h locates in the first quadrant. See Fig. 10.

For to > 0 small take points A1 E Lh N{x = eo}, A2 E Lh n{x = y}, A3 E Lh n{y = eo}.

(a, a) of A2 satisfies a = a(h) = ~/h + O(h). Let y -- xu. Then r - - - -

Then the coordinate

H ( x , y) --- x2[Xu q-- xqbo(x, u) ] ,

where, by the integral mean value theorem, ~P0 6 C ~ . The orbit arc A1 A2 satisfies the equation

H(x , y ) = h , a(h) ~ x <~ eo,

which is equivalent to

V ( x , u , v ) = O , a(h) <~ x <, Eo, (3.24)

where V (x, u, v) -- )~u + x*0(x, u) - v , v = xh-7 E [ ~ , ah-7 ]. Note that

v =o, v. - x .

Taking v as a parameter we can solve uniquely from (3.24)

v 1 u = u(x, v) - -~ + O(x) with uv - ~ + O(x).

414 M. Han

It is clear that along A 1 A2

dp(x, y) dp(x, xu(x, v)) ~(S) )~x + f (x, y) )~x + f (x, xu(x, v)) )~x

-k RO(X) + R1 (x, v)v, (3.25)

where R0, R1 E C ~. Hence, by (3.23) and 4~(S) = 0

fA~2 f a f a fea 49(x, y) dt - (Ro + Rl v) dx = Ro dx + Rl v dx.

o o o

Since

a I tf a L I 1,2, o d x = h ~ - a ~o O(Ihl , o o

it follows that

f o lim [ . _ 4)(x, y) d t = Ro dx = ~b(x, y) dt 6 N, h--+0 JA1A2 o

where Aio = limh~0 Ai, i -- 1, 2, 3. Similarly,

lim f A ~ 4)(x, y) dt = fA2-~3o ~(x' y) dt 6 IR" h--+O

Also, it is obvious that

lim fA~ ~(x, y)dt = fA~o4)(x' y)dt 6 IR" h-+O

Thus

l imfL ~ ( x , y ) d t - l i m fA_.~2 h-+O ~ h-~0 U A ~ U A ~ l

dp(x, y) dt

= f A ~ o d p d t + f A ~ o d p d t + f A ~ o dpdt

= f dp(x,y) dtER.

This finishes the proof. [5

By Lemmas 3.2 and 3.3 we have immediately


COROLLARY 3.4. Let Lh be oriented clockwise and expand with h increasing. Then for any C c~ functions fi(x, y) and ~(x, y) the Abelian

l (h) -- fL ~ dx - ti dy h

has the derivative

I !(h) -- f (fix + qy) dt, h

and

I ! (or) = (fix -+- {ly)(L~) T~, I' (h) = coTh + Cl ~- c2(h),

where Th denotes the period of Lh, T~ -- l i m h ~ Th, co -- (Px + qy)(S), Cl -- fL~[Px -+-

qy -- cO] dt, limh~t~ c2(h) -- 0. Further, by (3.25) it is easy to see that

Th lim = Po < 0. (3.26) h ~ in Ih -/31

The following lemma gives formulas for computing the value of the function P and its derivative at h - or.

LEMMA 3.4. Let (3.7), (3.8) and (3.22) hold. Suppose

k

(Px + qy)(L~, O, 6) -- ~ djobj, j=l

dlo ~ 0, (3.27)

and

k k , 1

v3 -- Z v3jbj as bl - -dl----~ Z dj~ j=2 j=2

(3.28)

, Or3 where v 3 -- ~ Is=o and v3 is the first focus value of Eq. (3.1) at the focus near L~ obtained by using Theorem 2.1, and djo (1 ~ j <~ k) and v3j (2 ~ j <~ k) are constants. Then

P (or, b2 . . . . . bk) -- --

P~ (or, b2 . . . . , bk ) --

k 1 y~, djobj,

dlo j=2

k 4Jr

Kblo j~2 v3jbj,

! where K > 0, and blo - 11 (c~) which can be obtained by Corollary 3.4.

416 M. Han

PROOF. Let

Ij (h) - bjo(h - or) + bj l (h - or) 2 + O(Ih - oil3), j -- 1 . . . . . k.

Then

M(h, 8) -- b~)(h -or ) + b~ (h --Or) 2 -t- O(Ih - oel3),

where b* = Y ~ = I bj ib j , i = 0, 1. By Lemma 2.9 and Corollary 3.4 we have

b~ - 0 if and only if (px + qy)(La, O, 8) -- O,

and

K b , v ~ - - ~ 1 whenb~)=O.

Therefore, by (3.27) b~ = 0 implies

bl - - - - -

k 1

blo ~ bjobj -- j=2

k 1

dlo ~ djobj , j=2

and hence

v~ -- ~ bl l bl ff- bj l bj -- j=2

k

K j~2(blobj l - b l lb jo )b j , 4zrblo .

when b~ = 0. Hence, by (3.22) and (3.8)

k b j l j ( h ) . . . . . bk) - - Z_~V" I](h) P(h, b2

j=2

k bj[bjo + bjl (h -- or) + O(Ih - ot12)]

�9 b lo -Jr- b l l (h - or) + O ( I h - c~l 2)

k

1 ~ bj[blobjo + (blobjl - b l lb jo) (h - or) b2o j--2

k 1 ~ bjobj 47c

blo j=2 Kb]o - ~ v~ (h - ~ ) + O ( I h - or


k k

l j~2 d j o b j 4 7r j ~ 2 dl0 . Kblo . v3jbj(h - or) + O(Ih m 0ll2).

Then the conclusion follows from (3.28) and the above easily. The proof is ended. D

Now we can give a condition for the existence of 2 limit cycles.

THEOREM 3.5. Let Lh be oriented clockwise and expand with h increasing. Suppose (3.7), (3.8), (3.22), (3,27) and (3.28) are satisfied. I f there exists 6o ~ D such that

(i)

(ii)

(iii)

k

(Px + qy)(S, O, So) - E djbj(6o), j= l

k ( d l l j ( f l ) ) bj(60) (fl) a (6o) -- ~ dj - j=2

k

o'(60) E v3jbj(60) < O; j=2

b~ (6o) # 0 and

min{ P(ot, b2(60) . . . . . bk(60)), P(fi, b2(60) . . . . . bk(60)) }

bl (60) -- as a (60) I1 (fl) < O, max{ P(ot, b2(6o) . . . . . bk (60)), P(f l , b2(6o) . . . . . bk(60)) }

as a (60) 11 (fl) > 0

then Eq. (3.1) has at least two limit cycles for some (s, 6) near (0, 60).

PROOF. By (3.8) and (3.22) we have M - I1 (bl - P) . Hence M' = I l'(bl - P) - I1P ' and

p!

Th

I M t I 1 bl - P 1 �9 ~ o

rh I1 Th I1

Take bl - P (fl, b2 . . . . . bk) and apply Corollary 3.4 to function I1 and M so that

p1 1 M t lim = lim

h ---~ fi Th 11 ( fl ) h ~ fl Th b! =P(fi,b2 ..... bk)

II(fl) - ~ ( P x + qy)(S , 0, 6)]bl=P(fi,b2 ..... bk )

1

Ii(f l) a(~). (3.29)

418 M. Han

! Then noting I 1 (c~)Ii(fl) > 0, by Lemma 3.4 we know that for h near 13 the product

k P~(ot, b2 . . . . . b k )Ps b2 . . . . . bk) has the same sign as a(6) ~ v 3 j b j . Then it follows

j=2 from (ii) that for 6 = 6o the function P ( h , b2 . . . . . bk) has a minimum or maximum in the interval (or, 13). Therefore, by (3.29) and (iii) for some 6 near 6o the function bl - P ( h , b2 . . . . . bk) has two zeros with odd multiplicity in the interval (ct,/3). Then the conclusion follows from Theorem 3.1. This ends the proof. D

EXAMPLE 3.5. Consider the system

2 -- y(1 - y) - e ( x 3 - 6x ) , - - x , (3.30)

where e > 0 is small and 6 6 R is bounded. For e -- 0, Eq. (3.30) has a family of periodic orbits giving by

1 ly 3 --h, L h" -~ (x 2 + y2) _ 1

0 < h < 6, y < l .

By (3.4) we have

M ( h , 3) = 611 (h) - 312(h) = 11(h)(6 - P ( h ) )

where

312(h) P ( h ) --

I i(h) '

1 -1 _ff,. -2 X 2j dy x 2j dx dy , I j (h) = 2 j - 1 h t.Lh

j - - 1,2.

By Corollary 3.4, I1 (h) = 2zrh + O(h2), 12(h) = O(h2). By Theorem 2.1, we have v~ = 3 8 as 6 - -0 . Let

a -- = ~o. I1(~)

Since L 1/6 can be represented as x 2 - -~ (y - 1) 2 (y + 1), _ 1 ~< y ~< 1, it is easy to see that

( ~ ) f 1 I1 = 2 1/2

V/ 2 6 ( l - y ) 1 - ~ ( 1 - y ) d y = - ~ ,

2fl 12 -- 3 1/2 (1 -- y)3 1 -- ~ (1 -- y)

108

d y - 385


5

54 _ _ 7 7

/// ol

I I I

. . . . . . . . ~ I

/ ',,', - 4 . . . . . 4

/ / i

/ i I I

I i

I I

I I i

I I

I i . . . . . . )

1__ h 6

Fig. 11. The graph of P(h) for Eq. (3.27).

419

54 Thus, 60 -- 77" By Theorem 3.5 for Eq. (3.30) the system has two limit cycles for (e, 6) near (0, 54 54 77) with ~ > 77" The graph ofthe function 6 = P(h) on the (h, 6) plan is as shown in Fig. 11.

For the existence of three limit cycles, we have:

THEOREM 3.6. Let Lh be oriented clockwise and expand with h increasing. Suppose (3.7), (3.8), (3.22), (3.27) and (3.28) are satisfied. Assume further

(i) there exists 8o ~ D such that

k

(Px -}- qy)(S, O, (~o) ~_~ v3jbj(8o) > O, btl ((~o) =/= O, j=2

blQiO) -- P ( a , b2(8o) . . . . . bk(80)) -- P(fl, b2Q;o) . . . . . bk(8O));

(ii) there exists 6" near 8o such that

I1 (fl)(Px n t- qy)(S, O, So)

x [ P ( a , bl (S*) . . . . . bk (S*)) -- P(fl , bl (S*) . . . . . bk(S*))] > O.

Then for e > 0 small and 6 = 6" Eq. (3.1) has 3 limits cycles.

We will give an example to show the way to prove the above theorem.

420 M. Han

EXAMPLE 3.6. Consider a system of the form

{ :~ - y(1 - y) - - 8 ( x 3 - - 3 1 x -+- 32xy), f~ ~ r e X .

(3.31)

We claim that for e > 0, 31 < 0, 32 < - 5 4 / 1 1 and e + I~l+ 132 + 54/111 small Eq. (3.31) has 3 limit cycles.

For this system we have

M(h) = b111 (h) -k- b212(h) -k- b313(h),

where bl = 31, b2 = - 3 , b3 - 32 and I1 (h), I2(h) are the same as in Example 3.5, and I3(h) -- fLh xy dy with I3(-~) = _ 6 . Let

P (h, b2, b3) -- - 1 [b212(h) 4- b313(h)]. I i (h)

54 Then Also, let blo = 310 = 0, b20 -- - 3 , b30 = 320 = 312(~)/I3(~) = - - f t .

(1 ) blo -- P(0, b20, b30) - P g, b20, b30 -- 0.

Further, by Lemma 3.4 and (3.29) we have

1 P~ (0, b20, b30) -- ~ (3 + 320) < 0,

lim P~ (h b20, b30) 320 --- ~ 0 ,

h ~ Th I1(~)

1 ) _ 13 (1____2) (32 - 320). P -~,b2, b3 = i1(1 )

Denote by 1 -'~ and I'* respectively the graph of the function P(h, b2, b3) on the (bl, h) plane for 62 = 620 and 62 - 6~, where 3~ < 320 with 13~ - 3201 small. See Fig. 12.

It is clear that for 3~ < 0 and I~'1 sufficiently small, the line bl = 3~ and F*" b l ---

P (h, - 3 , 3~) have at least 3 intersection points. Then the conclusion follows easily.

3.3. Near-Hamiltonian polynomial systems

We now give an important application of Theorem 3.3 to polynomial systems. Consider a near-Hamiltonian polynomial system of the form

~c = Hy (x, y, lZ) + ep(x, y, a), -- - H x ( x , y, lZ) + eq(x, y, a),

(3.32)


.#

bl

0

I I I

I I I

"" / / I F * 6 \ " . . . . . . / 1 I

I

Fig. 12. Curves F 0 and F*.

421

where

m

1 (x 2 H(x, y , # ) - - -~ + y2) + E hijxi yj i + j = 3

m>~3,

n

' E x i y j p(x y, a) -- aij , i+j=l

n

q(x, y, a) = E bijxi yj ' i + j = l

a n d / z = {hij} r r -- l (m + 1)(m + 2 ) - 6 , a - - {aij ,bi j} ERn2+3n, n >/2. For each/z,

there exists fl = fl(#) �9 (0, + e e l such that for h �9 (0, fl) the equation H(x, y, #) = h defines a smooth closed curve L h which surrounds a unique singular point (the origin) of the system (3.32) (s - 0). By (3.4) we can write

n - 1 k

M(h ,a ) - E cijlij(h) - E b j l j ( h ) , i + j = 0 j = l

where

Iij -- f fH xi yj dx dy, <~h

{11 . . . . . I~}--{I i j , O<<.i+j<<.n--1},

{bl, . . . , bk } -- {cij, 0 ~ i + j <. n -- 1},

I1 - - I 0 0 ,

1 k = = n ( n + 1).

2

Denote by W(h, #) the Wronskian defined by (3.9). Then the equation W(0, #) -- 0 de-

fines an (r - 1)-dimensional surface E}I__) 1 in IR r and the equations W(h, #) - W~ (h, #) -

0 with 0 ~< h < fl(#) define another (r - 1)-dimensional surface Er 1. Let

Br R r (1) ~(2) "-" - - ~ ] r - 1 m * - ' r - l "

422 M. Han

Then Br is an open set in ]~r with boundary ~ (1) (.j ~ (2) For each/z a Br, w e have " ' r - 1 "- 'r-l"

w(0, u) r 0, ] W (h, #)14- ] Ws (h, #) ] =/= O, h 6 [0, fl).

Note that for each /z e Br and h e (0, fl), the analytic function W(h, Ix) has only finitely many zeros h i , . . . , ht on (0, hi. Then by Theorem 3.3 we have immediately"

THEOREM 3.7. There exists an open set Br in ]~r whose boundary consists of r - 1 dimensional surfaces such that for each lZ E Br, and ft ~ (0, fl) there may exist constants 0 < hi < "'" < h l with I >~ 0 such that

(i) for h ~ [0,/t] - {hi, . . . ,hi}, Eq. (3.32) has cyclicity ln(n 4- 1) - 1 at Lh;

(ii) for h ~ {hi . . . . . hi}, Eq. (3.32) has cyclicity ln(n + 1) at Lh.

In particular, for each lz ~ Br, Eq. (3.32) has Hopf cyclicity ln(n + 1) - 1 at the origin.

Roughly speaking, for almost a l l / z 6 ]l~ r , Eq. (3.32) has cyclicity �89 + 1) - 1 or 1 gn(n + 1) at each Lh with h 6 [0, fl).

REMARK 3.3. It is easy to see that in some cases, Theorem 3.7 is still valid if tt is a vector parameter of dimension r with r < �89 + 1)(m + 2) - 6.

As an application, let us consider quadratic systems. By Ye et al. [117], a quadratic system having a focus or center can be changed into the form

2 = - y 4-6x + l x 2 + m x y + ny 2, = x(1 4- ax 4- by).

The system is Hamiltonian if and only if 6 = m -- b + 21 = 0. Then taking 6 = 661, m ----- em 1, b = -210 + ebl, 1 = lo 4- ell, a = ao 4- sal, it becomes

JC = Hy 4- 6(61x 4-11 x2 4- m l x y 4- n l y 2 ) ,

= - H x 4- e(al x2 4- blxy) , (3.33)

where

1 1 1 3 H(x, y) = - ~ ( x 2 4- y2) 4- lox2y 4- _~noy3 _ -~aox .

For (3.33), we have

M(h) - 6111(h) 4-/~112(h) 4- ml I3 (h) , /~1 = bl 4- 2ll,

where

Ii (h) = f f _ dx dy, I2(h) - f f _ x dx dy, H <~h H <~h

I 3 ( h ) - f l - y d x d y , H<.h

h e [0,/~).


By [ 117], up to a positive constant, the first four focus values of the origin can be taken as

Wo m 8 Vo , Wi -8Vi -+-0(8 2) i - - 1 2 3

where

V0 - - 81, V1 - - m l (no + Io) - a o b l ,

V2 - 5mla2[(lo + n o ) 2 ( n o - 2 / 0 ) - a2no].

Thus,

det a(VO, Vl, V2)

O ( 8 1 , b l , m l ) = -5ao3 [(/o + n o ) 2 ( n o - 2 / o ) - a2no].

Hence, by Lemma 2.9 and (3.12), it is easy to see that

W(O, ao, lo, no) -- Noa3[(lo + no)2(no - 2/o) - a2no], No 7 ~ O.

Then from the discussion before Theorem 3.7 it follows that for any (ao, Io, no) E ~x~ 3 sat-

isfying

ao[(/o + no)2(no - 2/o) - a2no] 7 L O,

the Hopf cyclicity of (3.33) at the origin is 2.

3.4. Homoclinic bifurcation

In the rest of the chapter, we introduce a way to find limit cycles in a neighborhood of the

homoclinic loop L/3. As a preliminary, we first discuss the stability of an isolated homoclinic loop. Consider

a C ~ planar system of the form

2 -- f ( x , y), ~ = g(x, y). (3.34)

Suppose that (3.34) has a homoclinic loop L with a hyperbolic saddle. Assume that the

saddle is at the origin. Then on the stability of L we have

LEMMA 3.5. Let

el = ( f x -+- gy)(O), C2 -- ~L ( f x -Jr- gy -- Cl) dt. (3.35)

Then L is stable (unstable) i f C1 < O, or Cl - - 0 , c2 < 0 (c1 > 0, or Cl --O, c2 > 0).

424 M. Han

For a proof of the lemma, see Han [47,51], Han and Chen [57], and Feng and Qian [27]. Further, let Cl = c2 -- 0. We can assume the functions f and g have the following form:

f ( x , y) = kx + O(Ix, y]2), g(x , y) = -)~y + O(Ix, y]2), k > 0.

In this case, let

1 E 1 11 C3 = - ~ f x xy n t- gxyy -- - s f x y -- gxygyy ) x = y = 0

which is called the first saddle value of (3.34) at the origin. If f and g satisfy

f (x, y ) = )~y + O([x, yle), g(x , y) = )~x + O([x, y[2),

(3.36)

then, instead of (3.36), let

1[ 1 C3 = - - - ~ f x x x -- f xyy + gxxy -- gyyy -~" ~ ( f x y ( f y y -- f x x )

+ gxy (gyy -- gxx) - f x x g x x + fyygyy)] d x = y = 0

Then we have the following lemma obtained by Han, Hu and Liu [58].

(3.37)

LEMMA 3.6. Let Cl = c2 = 0, and c3 =7/= O. Then L is stable i f and only i f one o f the following occurs:

(i) c3 < O, a Poincar~ map is well-defined near inside L and L is oriented anti- clockwise;

(ii) c3 < O, a Poincar~ map is well-defined near outside L and L is oriented clockwise; (iii) c3 > O, a Poincar~ map is well-defined near outside L and L is oriented anti-

clockwise; (iv) c3 > O, a Poincar~ map is well-defined near inside L and L is oriented clockwise.

We remark that the formula (3.37) can be obtained from (3.36) by introducing a liner transformation x = ~ (u - v) and y -- ~ (u + v) where u and v are new variables.

Going back to (3.1) we introduce the following four functions:

do(6) -- f c (q dx - p d y ) - M(~ , 6), t~ e=O

dl (8) = (Px -+- q y ) ( S , O, 6),

d2(6) = fL [Px + qy -- dl (8)] dt, t~

0c3 d3 (~) -- -~e (e, 6) ,

e--O

(3.38)


where c3 is obtained by (3.36) or (3.37). The following theorem was obtained by Han [43].

THEOREM 3.8. (i) Let there exist ~o = ((]1o . . . . . (]m0) 6 D, m/> 2 such that

O(do, dl) do((]o) - dl ((]o) = O, d2((]o) -7 6 O, det ~ ((]o) :# O.

0((]1, (]2)

Then for any eo > 0 and neighborhood U of ~o there exists an open subset Vs C U for 0 < s < so such that Eq. (3.1) has 2 limit cycles near L~ for a E Vs.

(ii) Let there exist (]0 = ((]10 . . . . , (]mO) ~ D, m ~ 3, such that

O(do, dl,d2) di ((]0) = 0, i = 0, 1, 2, d3 ((]o) ~: 0, det ((]0) :fi 0.

O((]l, (]2, (]3)

Then for any so > 0 and neighborhood U o f (]o there exists an open subset Vs C U for 0 < s < eo such that Eq. (3.1) has 3 limit cycles near L~ for a ~ V~.

We briefly outline the proof of the first conclusion. For definiteness, suppose L~ is oriented clockwise and the Poincar6 map is well defined inside it as before. For s > 0 small a unique saddle Ss near S and two separatrices L~ and L s near Lr exist. The directed distance between L s and L u on a cross section is given by (see [14,51,112])

d(s, (]) = sN[do((]) + O(s)] , N > O.

~d0 ((]0) :~ 0. The implicit function theorem By the assumption, we can suppose d~0 = implies that a unique function (]1 - - (/91 (8, (]2 . . . . . (]m) = (fl10((]2 . . . . . (]m) + O(s) exists such that for s > 0 and I(] - (]ol small d(e, (]) >~ 0 if and only if doo[(]l - qgl] >~ 0. Hence, a homoclinic loop L* appears near Lt~ if (]1 -- q91.

Let (]1 -- gOl and define

Cl (8, (]2 . . . . . (]m) -- 8 ( p x --~ q y ) ( g s , O, (]) -- 8[c10((]2 . . . . . (]m) -+- 0 ( 8 ) ] ,

where

clo = (px + qy)(S, O, (])l~=~o.

Then our assumption implies that

c10((]20 . . . . . (]m0) "-- 0, , OClO

d l o = --L--j--~ ((]20, . . . , (]mO) ~ 0. 0 o 2

T h u s a unique f u n c t i o n (]2 -" q92(8, (]3 . . . . . (]m) = (fl20((]3 . . . . . (]m) + O(s) e x i s t s s u c h t h a t

!

Cl (t3, (]2 . . . . . (]m) ~> 0 i f a n d only i f 610[(] 2 - 992] ~ 0.

426 M. Han

Let 61 = r 62 = r and define

f r 63 . . . . . 6m) -~- ~ (1) (Px 4- qy) dt -- ~[c20(63 . . . . . 6m) 4- O(6)],

JL

where

r . . . . . 6m) = fL (Px 4- qy)i~=~010,~2=~o20 dt.

It is easy to see that

r . . . . . 6mO) = 62(60) 5; & 0.

Without loss of generality, suppose d2(60) > 0. Then for e > 0, 61 = qgl, 62 = ~2 and e 4- 16 - 601 small L~ is unstable by Lemma 3.6. Fix e > 0 and 6j near 6jo for j -- 3 . . . . . m, and vary 61 and 62 such that

61 = r 0 < 162 - 9921 << 1, el (t3, 62 . . . . . 6m) < 0.

Then, L* has changed its stability from unstable into stable and therefore an unstable limit cycle has appeared near it at the same time. Next, noting that we have assumed that L~ is oriented clockwise and the Poincar6 map is well-defined inside it, we then change 61 such that 0 < 161 - q911 << 162 - q921, d(e, 6) < 0. Clearly, L* has broken and a stable limit cycle has appeared. Therefore, 2 limit cycles can appear near L~.

We can summarize the method used above into 3 steps: 1. For e > 0 fixed, a homoclinic loop L* appears for 6 on a codimension 1 surface in

]~m when d(e , 6) = O. 2. Keep L~ to appear and study its stability and then change the stability of L* in turn

to produce limit cycles. 3. Find a final limit cycle by making the homoclinic loop broken. This method was first used by Han [43] and then developed by Han and Chen [57] and

Han, Hu and Liu [58] to study the number of limit cycles near a double homoclinic loop. General theorems like Theorem 3.8 on double homoclinic bifurcations can be found in

Han and Zhang [71]. Interesting applications of the method to quadratic and cubic systems et al. for the existence limit cycles are given in [65-68,70,72,122-126].


Jc - y - e ( a l x 4- a 2 x 2 4- x4), ~ X - -X 2.

(3.39)

For e = 0, (3.39) has a first integral of the form

H ( x , y ) = ~ ( y 2 - x 2 ) + x 3,


which gives a family of periodic orbits

1 Lh: H ( x , y) = h,

6 < h < 0 .

The limit of Lh as h --+ 0 is a homoclinic loop L0. We can prove that (3.39) has 2 limit cycles near L0 for some (al, a2).

In fact, for (3.39) we have

d0(al, a2) = - a l I00 - 2a2101 - 4103,

where

f03/2 lo j = x J y dx = 2 o

2x X j+ l 1 - ~ dx, j - - 0 , 1 , 3 .

It easy to get that

6 36 72 I 0 0 - ~, I01 = 35' 103 = ~I00.

Thus,

12 288 ] do(al, a 2 ) - - - I o o al + --7-a2 + - ~ .

By (3.38), we have further

d l ( a l , a 2 ) = - a l ,

d2 am, - - - ( 2 a 2 x + 4x 3) dt ,/L 0

- - - 2 d x - - 6 2 a 2 + ~ .

v/1- x 24 Hence, i f a l = 0, a2 = - i T , then

d 0 = d l = 0 , d2 :/:0.

It follows from Theorem 2.8 that Eq. (3.39) has 2 limit cycles for e > 0 small and some ( a l , a2) near (0, -- ]-]-).24

From Han [51 ] we know Eq. (3.39) has at most 2 limit cycles on the plane for all e > 0 small and (al, a2) bounded.

428 M. Han

REMARK 3.4. Study of the number of limit cycles for near-Hamiltonian systems has been a significant and important part of bifurcation theory for decades. Most results can be di- vided into two aspects. One is to give a lower bound of the number for a given system with parameters. In other words, this aspect mainly concentrates on finding limit cycles as many as possible by choosing suitable parameters. The other is to study the maximal number of limit cycles and give an upper bound of it. In this section we mainly concern with the theory and methods in the first aspect. Theorem 3.1 is well known as Poincar6- Pontryagin-Andronov theorem. The formula in Lemma 3.2 was first obtained by Han [39]. Theorems 3.2, 3.3 and 3.7 are just recently obtained by Han, Chen and Sun [56]. Theo- rem 3.4 is from Han [51 ]. Results in Lemma 3.3 were given by Luo, Han and Zhu [ 100]. Here we present a new and simple proof. Conclusions of Lemma 3.4 were first established in Han and Ye [69]. The homoclinic bifurcation under the condition of Lemma 3.6 was studied in Zhu [ 130] in detail. Theorems 3.5 and 3.6 are new and obtained by the author. There have been many interesting results in the second aspect on the estimate of an upper bound for the number of limit cycles planar systems. For the theory and methods on this aspect, the reader can consult [13,15-18,23-26,42-44,49,74,76,83,107,109,115-118,127, 128]. For the bifurcation of periodic solutions of higher dimensional systems, see [14,16, 35,37,39,41,51-53,111,112,131].

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[3] V.I. Arnold, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl. 11 (1977), 85-92.

[4] V.I. Arnold, Geometric Methods in Theory of Ordinary Differential Equations, Springer, New York (1983). [5] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equi-

librium position offocus or center type, Mat. Sb. (N.S.) 30 (1952), 181-196 (in Russian); Transl. Amer. Math. Soc. 100 (1954), 397-413.

[6] J.D. Bejarano and J. Garcia-Margallo, The greatest number of limit cycles of the generalized Rayleigh- Linard oscillator, J. Sound Vibration 221 (1999), 133-142.

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[8] T.R. Blows and C. Rousseau, Bifurcation at infinity in polynomial vector fields, J. Differential Equations 104 (1993), 215-242.

[9] A.D. Bryuno, Analytical form of differential equations, Trudy Moskov. Mat. Obsc. 25 (1971), 120-262. [10] S. Cai and G. Guo, Saddle values and limit cycles generated by separatrix of quadratic systems, Proceed-

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