existence criteria for cycles in two-dimensional quadratic systems

ISSN 1063-4541, Vestnik St. Petersburg University: Mathematics, 2007, Vol. 40, No. 3, pp. 172–181. © Allerton Press, Inc., 2007.Original Russian Text © G.A. Leonov, 2007, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2007, No. 3,pp. 31–41.

172

Existence Criteria for Cyclesin Two-Dimensional Quadratic Systems

G. A. Leonov

Received March 15, 2007

Abstract

—The existence of a bifurcation of simultaneously arising cycles in a neighborhood of twoequilibrium states of two-dimensional quadratic systems under a variation of one scalar parameter isproved.

DOI:

10.3103/S1063454107030028

1. INTRODUCTION

Is a bifurcation of the simultaneous birth of cycles in a neighborhood of two equilibrium states of two-dimensional quadratic systems under a variation of one scalar “control” parameter possible? A positiveanswer to this question is given in this paper.

During the past century, the study of cycles in two-dimensional quadratic systems was stimulated by Hil-bert’s 16th problem and its various versions [1–6].

The discovery of quadratic-fractional transformations reducing an arbitrary two-dimensional quadraticsystem to a special Lienard-type equation with discontinuous nonlinear functions [7–9] has opened newpossibilities for developing asymptotic methods for studying cycles of quadratic systems. Some of them aredemonstrated in this paper.

The combination of elements of Lyapunov’s direct method with methods of averaging and bifurcationtheory has made it possible to obtain existence criteria for small cycles in neighborhoods of stationarypoints. We suggest a new approach to calculating the first Lyapunov value [10], which does not employ theapparatus of normal forms and reduces the smoothness requirements to the system.

The use of an artificial discontinuity and the construction of families of transversals and specialLyapunov-type functions has also led to criteria for the existence of cycles.

The passage from a quadratic system to a Lienard-type equation has revealed interesting symmetries,which has made it possible to “almost automatically” prove the existence of certain “additional” cycles.

2. EXISTENCE CRITERIA FOR SMALL CYCLESIN NEIGHBORHOODS OF STATIONARY POINTS

Consider the equation

(1)

Lemma 1

[11]. A solution to Eq. (1) with initial data

z

(0) and (0) is determined by

(2)

It follows from (2) that if (0) = 0, then

(3)

Consider the equation

z z+ u t( ).=

z

z t( ) z 0( ) u τ( ) τsin τd

0

t

∫– t z 0( ) u τ( ) τcos τd

0

t

∫+ t.sin+cos=

z

z t( ) z 0( ) tsin– u τ( ) τsin τ tsind

0

t

∫ u τ( ) τcos τ t.cosd

0

t

∫+ +=

x F x ε,( ) x G x ε,( )+ + 0=

VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS

Vol. 40

No. 3

2007

EXISTENCE CRITERIA FOR CYCLES IN TWO-DIMENSIONAL QUADRATIC SYSTEMS 173

or the equivalent system

(4)

Here,

ε

is a positive number,

F

(

x

, 0) =

f

(

x

),

G

(

x

, 0) =

g

(

x

),

f

(

x

0

) =

g

(

x

0

) = 0, and, in some neighborhood ofthe point (

x

0

, 0), the functions

F

(

x

,

ε

) and

G

(

x

,

ε

) are smooth.

Theorem 1.

If

(5)

where

x

ε

is a zero of the function

G

(

x

,

ε

) in a neighborhood of

x

=

x

0

, then there exists a number

ε

0

> 0 suchthat system (4) has a cycle for any

ε

∈

(0,

ε

0

).

Proof.

We set

z

(

t

) =

x

(

t

) –

x

0

and

Without loss of generality, we can assume that

g

1

= 1.

First, consider the case where

ε

= 0 and

z

(0) is small.In this case, using the smoothness of the functions

F

and

G

, we construct approximations to solutions ofsystem (4) in a finite interval by Lyapunov’s first method.

The first approximation of a solution

z

(

t

) of system (4) is the function

The equation determining the second approximation

z

2

(

t

) to the solution

z

(

t

) is

where

By Lemma 1,

z

2

(

t

) has the form

(6)

This implies that if (0) = 0, then

It follows from the last expression that the moment

T

> 0 at which the solution

x

(

t

),

y(t) of system (4) withinitial data x(0), y(0) = 0 intersects the line y = 0 for the second time (see Fig. 1) satisfies the relation

(7)

Next, consider the function

x y=

y F x ε,( )y– G x ε,( ).–=

f '' x0( )g ' x0( ) f ' x0( )g '' x0( ), g ' x0( ) 0, F xε ε,( ) 0,> ><

f x( ) f 1 x x0–( ) f 2 x x0–( )2 O x x0–( )3( ),+ +=

g x( ) g1 x x0–( ) g2 x x0–( )2 O x x0–( )3( ).+ +=

z1 t( ) z 0( ) t.cos=

z2 z2+ u t( ),=

u t( ) z 0( )2 f 1 t tsincos g2 tcos( )2–( ).=

z2 t( ) z 0( ) tcos z 0( )2 f 1

3----- tsin( )3 tcos–⎝

⎛+=

+g2

3----- 1 tcos( )3–( ) tcos

f 1

3----- 1 tcos( )3–( ) tsin g2 tsin

13--- tsin( )3–⎝ ⎠

⎛ ⎞ tsin ⎠⎞ .–+

z

z2 t( ) z 0( ) tsin– z 0( )2 f 1

3----- tsin( )4

⎝⎛+=

–g2

3----- 1 tcos( )3–( ) tsin

f 1

3----- 1 tcos( )3–( ) tcos g2 tsin

13--- tsin( )3–⎝ ⎠

⎛ ⎞ tcos ⎠⎞ .–+

T 2π O x 0( ) x0–( )2( ).+=

V x y,( ) y f z( ) zd

x0

x

∫+⎝ ⎠⎜ ⎟⎛ ⎞ 2

2 g z( ) z.d

x0

x

∫+=

174

VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS Vol. 40 No. 3 2007

LEONOV

It is easy to see that the derivative of V along the solutions to system (4) satisfies the equality

This equality, (6), and (7) imply the relation

It follows from this relation and the continuous dependence of the solutions to system (4) on the param-eter that, if the parameter ε > 0 is sufficiently small in comparison with |z(0)|, then

(see Fig. 1). On the other hand, the equilibrium state x = xε, y = 0 of system (4) at small ε > 0 is a stablefocus. These two facts imply the existence of a cycle of system (4) in a small neighborhood of the point x =xε, y = 0 (see Fig. 2).

Corollary 1. If

(8)

then there exists a number ε0 < 0 such that, for all ε ∈ (ε0, 0), system (4) has a cycle.

3. ASYMPTOTIC ESTIMATES USING THE DISCONTINUITY POINTSOF THE COEFFICIENTS IN THE LIENARD EQUATION

Consider system (4) with ε = 0 and

(9)

(10)

V x y,( ) 2g x( ) f z( ) zd

x0

x

∫–=

= f 1 x x0–( )3–23--- f 2 f 1g2+⎝ ⎠

⎛ ⎞ x x0–( )4– O x x0–( )5( ).+

V x T( ) y T( ),( ) V x 0( ) 0,( )– f 1z2 t( )3 23--- f 2 f 1g2+⎝ ⎠

⎛ ⎞ z2 t( )4+⎝ ⎠⎛ ⎞ td

0

T

∫– O z 0( )5( )+=

= z 0( )4 23--- f 2 f 1g2+⎝ ⎠

⎛ ⎞ tcos( )4 3 f 1 tcos( )2 f 1

3----- tsin( )3 tcos–

g2

3----- 1 tcos( )3–( ) tcos+⎝

⎛+

0

2π

∫–

+f 1

3----- 1 tcos( )3–( ) tsin g2 tsin

13--- tsin( )3–⎝ ⎠

⎛ ⎞ tsin ⎠⎞ dt O z 0( )5( )+–

= f 2 f 1g2–( )π

2------------------------------z 0( )4– O z 0( )5( ).+

V x T( ) y T( ),( ) V x 0( ) 0,( )>

f '' x0( )g ' x0( ) f ' x0( )g '' x0( ), g ' x0( ) 0, F xε ε,( ) 0,<>>

f x( ) Ax B+( )x β bx+ q 2– ,=

g x( ) λ Ax B+( )xβ bx+ 2q

β bx+( )3----------------------,=

y

x

x(T)

x(0) x0

Fig. 1.



where A, B, β, b, and λ are positive numbers and

(11)

It is easy to see that, at the points x1 = 0 and x2 = –B/A, we have

Thus, for conditions (8) to hold, it suffices to require that

(12)

Here, xε denotes the zeros of the function G(x, ε) in neighborhoods of x1 and x2.

For system (4) with ε = 0 and functions (9) and (10), the following simple lemmas are valid.

Lemma 1. On the sets

where C ∈ [0, λβ(q – 1)], and

where C ≥ λβ(q – 1), the inequality > 0 holds.

On the sets

B Aβb---, 1– q 1.<≤>

f x1( ) g x1( ) f x2( ) g x2( ) 0,= = = =

g ' x1( ) λBβ2q 3– , g '' x1( ) 2λβ 2q 4–( ) Aβ Bb 2q 3–( )+( ),==

g ' x2( ) λBbB βA–

A--------------------⎝ ⎠

⎛ ⎞2q 3–

, g '' x2( ) 2λ Bb Aβ–A

--------------------⎝ ⎠⎛ ⎞

2q 4–( )Aβ Bb 2q 2–( )–( ),==

f ' x1( ) = Bβq 2– , f '' x1( ) = 2βq 3– Aβ Bb q 2–( )+( ),

f ' z2( ) BbB βA–

A--------------------⎝ ⎠

⎛ ⎞q 2–

, f '' x2( ) 2Bb Aβ–

A--------------------⎝ ⎠

⎛ ⎞q 3–

q 1–( )Bb Aβ–( ).=–=

F xε ε,( ) 0.<

xβb---– 0,⎝ ⎠

⎛ ⎞∈ y, C–=⎩ ⎭⎨ ⎬⎧ ⎫

,

x 0 y,> C–={ },

y

xBA---– β

b---–,⎝ ⎠

⎛ ⎞∈ y, C=⎩ ⎭⎨ ⎬⎧ ⎫

,

y

xx(T)

x(0) x0

Fig. 2.

176


LEONOV

where

and

where

the inequality < 0 holds.To prove the first assertion of the lemma, it is sufficient to note that the second equation of system (4)

with ε = 0 and functions (9) and (10) on the set {x ∈ (–β/b, 0), y = –C} implies

provided that C ∈ [0, λβ(q – 1)].The proofs of the remaining assertions are similar. Lemma 2. Any solution of system (4) with initial data x(0) = 0 and y(0) > 0 intersects the set {x > 0,

y = 0} with increasing t. Any solution of system (4) with initial data x(0) = –B/A and y(0) < 0 intersects theset {x < –B/A, y = 0} with increasing t.

Let us prove the first assertion of Lemma 2. For this purpose, consider the Lyapunov function

Clearly, along the solutions of system (4), we have

On the set {x > 0, y > 0}, as x2 + y2 ∞, we have

(13)

the last relation follows from the condition –1 ≤ q < 1.According to the Barbashin–Krasovskii theorem [12, 13], it follows from conditions (13) that any solu-

tion x(t), y(t) satisfying the inequalities x(t) > 0 and y(t) > 0 for all t ≥ 0 tends to zero, which is impossiblefor trajectories from the first quadrant. This contradiction proves the first assertion of the lemma. The proofof the second assertion is similar.

Lemmas 1 and 2 imply that, for system (4) with ε = 0 and functions (9) and (10), the trajectories behaveas shown in Fig. 3.

Now, consider a solution to system (4) with initial data

(14)

and

(15)

Lemma 1 (see Fig. 3) implies the existence of numbers T1 > 0 and T2 > 0 such that

C 0 λ bB βA–A

--------------------⎝ ⎠⎛ ⎞

q 1–( ), ,∈

xBA--- y,–< C=

⎩ ⎭⎨ ⎬⎧ ⎫

,

C λ bB βA–A

--------------------⎝ ⎠⎛ ⎞

q 1–( ),≥

y

y Ax B+( )x β bx+ q 2– C λ β bx+( ) q 1–( )–( ) 0,>=

V x y,( ) y f z( ) zd

0

x

∫+⎝ ⎠⎜ ⎟⎛ ⎞

2

2 g z( ) z.d

0

x

∫+=

V 2g x( ) f z( ) z.d

0

x

∫–=

V 0, V x y,( ) 0, limV x y,( )>< +∞;=

x 0( ) 0, y 0( ) λβ q 1–( )–= =

x 0( ) BA---, y 0( )– λ bB βA–

A--------------------⎝ ⎠

⎛ ⎞q 1–( )

.= =

x T1( ) 0, y T1( ) 0, x t( ) 0 t∀ 0 T1,( )∈≠>=



in case (14) and

in case (15) (see Fig. 4).The above considerations imply the following theorem.Theorem 2. For solutions to system (4) with initial data

and

there exist numbers τ1 > 0 and τ2 > 0 such that

and

(see Fig. 5).It is well known [7–9] that the two-dimensional quadratic system

(16)

where c1 is set to zero without loss of generality, reduces to a system of the form (4) with

(17)

(18)

x T2( ) BA---, y T2( )– 0, x t( ) B

A--- t– 0 T2,( )∈≠<=

x 0( ) 0, y 0( ) y T1( )>=

x 0( ) BA---, y 0( )– y T2( ),<=

x τ1( ) 0, y τ1( ) y T1( ) y 0( ), x t( ) 0 t∀ 0 τ1,( )∈≠< <=

x τ2( ) 0, y τ2( ) y T2( ) y 0( ), x t( ) 0 t∀ 0 τ2,( )∈≠> >=

x a1x2 b1xy c1y2 α1x β1y,+ + + +=

y a2x2 b2xy c2y2 α2x β2y,+ + + +=

F x ε,( ) = – b1b2 2a1c2– a1b1+( )x2 b2β1 b1β2+(–[

– 2α1c2 2a1β1)x α1β1 β1β2+( ) ] β1 b1x+ q 2– ,–+

G x ε,( ) = a2x2 α2x+β1 b1x+

-------------------------b2x β2+( ) a1x2 α1x+( )

β1 b1x+( )--------------------------------------------------------–

c2 a1x2 α1x+( )2

β1 b1x+( )3--------------------------------------+

⎝ ⎠⎜ ⎟⎛ ⎞

β1 b1x+ 2q.–

y

x

BA---–

βb---–

Fig. 3.

178


LEONOV

Here, q = –c2/b1, and it is assumed that b1 ≠ 0.

Suppose that, in system (16), a1 = a2 = c1 = 0, b1 = β1 = c2 = 1, b2 = –1, α1 = 1/3 – ε, and α2 = β2 = –1/3.It is easy to see that, in this case,

In a neighborhood of x = 0, we have xε = 0, and in a neighborhood of x = –2, we have xε = –2 – 3ε + o(ε).

Clearly, if xε = 0, then

F x ε,( ) x2 2 1 ε–( )x ε+ +( ) 1 x+ 3– ,=

G x ε,( ) 19--- x2 2x+( ) εx x2 2x

13--- εx–+ +⎝ ⎠

⎛ ⎞+⎝ ⎠⎛ ⎞ 1 x+( ) 5– ,=

f '' 0( ) f '' 2–( ) 10, g ' 0( )– g ' 2–( ) 29---.= = = =

F xε ε,( ) ε,=

y

x

BA---–

βb---–

Fig. 4.

y

xBA---– β

b---– 0

Fig. 5.



and if xε = –2 – 3ε + o(ε), then

Thus, it follows from Corollary 1 that, for sufficiently small ε < 0, the system

(19)

has at least two cycles. Moreover, each of the two equilibrium states is enclosed by a cycle.

The same result follows from Theorem 2.

4. AN INVERSE TRANSFORMATION OF THE QUADRATIC SYSTEMTO THE LIENARD EQUATION

As mentioned in the preceding section, the quadratic system (16) can be transformed into system (4) withspecial functions f and g.

Now, consider the question of the existence of an inverse transformation. In other words, the question is:When does the equation

(20)

where

admit the existence of a quadratic system of the form (16) satisfying conditions (17) and (18)? We assumethat c1 = 0 and α1 = –β2.

Proposition 1. If b1 ≠ 0, β1 ≠ 0, and α1 ≠ 0, then, without loss of generality, it can be assumed that b1 =α1 = β1 = 1 in system (16).

To prove this fact, we make the change

In the new variables, the first equation of system (16) takes the form

Proposition 2. For the coefficients A, B, Cj ( j = 1, …, 4), and q of Eq. (20), the corresponding coefficientsa1, b1 = 1, α1 = 1, β1 = 1, a2, b2, c2 = –q, α2, β2 = –1 of system (16) exist if and only if

Here,

F xε ε,( ) 11ε o ε( )+

1 xε+ 3--------------------------.=

x xy13--- ε–⎝ ⎠

⎛ ⎞ x y,+ +=

y xy– y2 13---x–

13---y–+=

x F x( ) x G x( )+ + 0,=

F x( ) Ax B+( )x x 1+ q 2– ,=

G x( ) C1x3 C2x2 C3x C4+ + +( )xx 1+ 2q

x 1+( )3-------------------,=

xβ1

b1-----x, y

α1

b1-----y, t

tα1-----.= = =

xα1β1a1

b1-----------------x2 xy x y.+ + +=

B A–( )2q 1–( )2

---------------------- 1 q–( )B 3q 2–( )A+( ) 2C2 3C1– C3,–=

B A–( )2q 1–( )2

---------------------- B 2 q 1–( )A+( ) C2 2C1– C4.–=

180


LEONOV

These relations follow immediately from equalities (17) and (18).

5. DESCRIPTION OF THE GENERAL BIFURCATION OF SIMULTANEOUSLY ARISING CYCLES IN A NEIGHBORHOOD OF TWO EQUILIBRIUM STATES

Consider system (4) with functions (17) and (18). In the general case, we can assume that c1 = 0, b1 = 1,α1 = 1, and β1 = 1.

For the existence of a cycle birth bifurcation in a neighborhood of x = y = 0, it is necessary that the unper-turbed system (for which the control scalar parameter is ε = 0) satisfy the condition α1 = –β2, i.e., β2 = –1.

It is also necessary that, in the other equilibrium state (x = x2, y = 0),

(21)

Here,

Let us introduce the quantities

In the general case, we can assume that L1 ≠ 0 and L2 ≠ 0.A necessary condition for the bifurcation under consideration is also the fulfillment of the inequalities

g'(0) > 0 and g'(x2) > 0.Now, let us change the parameters α1, β2, and α2 as

Here, the pj are some numbers satisfying the relations

(22)

(23)

(24)

System (23), (24) is solvable with respect to p1 and p2 if c2 ≠ –1/2.If the pj satisfy relation (22), then, for the equilibrium state of the perturbed system, we have

This relation, (23), and (24) imply that, in the case under consideration, the conditions of eitherTheorem 1 or Corollary 1 are satisfied.

Thus, the bifurcation of simultaneously arising cycles in a neighborhood of two equilibrium states occursunder a variation of one parameter ε.

In numerically searching for cycles of quadratic systems, it seems to be reasonable to fix the parametersof system (16) so that α1 = –β2 and (21) holds. In this case, the existence of two small cycles can be achievedby a small perturbation. Such small perturbations do not affect the existence of large limit cycles, and there-fore, if a numerical search for large limit cycles yields a number K of cycles, then it follows at once that thesystem has K + 2 cycles.

a1 1B A–2q 1–---------------,+=

a2 q 1+( )a12– Aa1– C1,–=

b2 A– a1 2q 1+( ),–=

α2 a12 2a1– A a1 1–( ) 2C1 C2–( ).+ +=

f x2( ) F x2 0,( ) 0, g x2( ) G x2 0,( ) 0.= == =

x2

b2 1– 2c2– 2a1+( )b2 2a1c2– a1+( )

-----------------------------------------------.–=

L1 f '' 0( )g ' 0( ) f ' 0( )g '' 0( ),–=

L2 f '' x2( )g ' x2( ) f ' x2( )g ' x2( ).–=

α1 1 p1ε, β2+ 1– p2ε, α2 = –1 p3ε.+ += =

p3

p1 b2x2 β2+( ) x2 1+( ) 2c2x2 a1x2 α1+( )–[ ] p2 a1x2 α1+( ) x2 1+( )+

x2 1+( )2-------------------------------------------------------------------------------------------------------------------------------------------------------------------,=

p1 p2+ L1,sgn=

p2 2 p1c2–( )x2 p1 p2+( )+ L2.sgn=

xε x2 O ε2( ).+=



ACKNOWLEDGMENTSThis work was supported by the Russian Foundation for Basic Research (project no. 07-01-00151),

Dutch Russian research cooperation NWO, and the program “Leading Scientific Schools” (projectno. 4609.2006.1).

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existence criteria for cycles in two-dimensional quadratic systems

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