ISSN 1028-3358, Doklady Physics, 2009, Vol. 54, No. 5, pp. 238–241. © Pleiades Publishing, Ltd., 2009.Original Russian Text © G.A. Leonov, 2009, published in Doklady Akademii Nauk, 2009, Vol. 426, No. 1, pp. 47–50.

238

Many mechanical, electromechanical, and electronsystems are described by the Lienard differential equa-tion [1, 2]

(1)

The classical criteria of existence and uniqueness ofthe limit cycle of Eq. (1) [2] with smooth nonlinearcoefficients

f

and

g

are well known.

In this work, we formulate the criteria of existencefor four limit cycles of Eq. (1), when

(2)

where

ϕ

(

x

)

and

ψ

(

x

)

are the smooth functions on(

−∞

,

+

∞

),

ψ

(0)

= 0,

and

q

∈

(–1, 1). In particular, anarbitrary quadratic two-dimensional systems [3–5] istransformed to Eq. (1) with functions (2).

Equation (1) is equivalent to the system

(3)

First, we obtain the estimates for solutions of system(3) for large

|

x

|

. We assume here that

(4)

Under the assumption that

(5)

x f x( ) x g x( )+ + 0.=

f x( ) ϕ x( ) x 1+ q 2– , g x( ) ψ x( ) x 1+ 2q

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

x y,=

y f x( )y– g x( ).–=

f x( ) A O1x

-----⎝ ⎠⎛ ⎞+⎝ ⎠

⎛ ⎞ x q,=

g x( ) C O1x

-----⎝ ⎠⎛ ⎞+⎝ ⎠

⎛ ⎞ x x 2q.=

C 0, 4C q 1+( ) A2>>

we introduce the designation

We fix certain numbers

a

> –1 and

ε

> 0 and introduceinto consideration a reasonably large number

R

.

Lemma 1.

For solution of system (3) with the initialdata x

(0) =

a

,

y

(0) =

R, there exists the number T > 0such that

We present the scheme of the proof of Lemma 1.System (3) is equivalent to the equation

which can be presented in the form

For large

x

, the solution

F

(

x

)

of this equation at thefinite fixed ranges of variation of

x

is close to a certainsolution

G

(

x

)

of the equation

Making the replacement

z

=

x

q

+ 1

, we pass to the equa-tion

λ A–2 q 1+( )--------------------, ω 4 q 1+( )C A2–

2 q 1+( )-----------------------------------------.= =

x T( ) a, y T( )= 0, x t( ) a, t∀ 0 T,( ),∈><

Rλπω------ ε–⎝ ⎠

⎛ ⎞exp y T( ) Rλπω------ ε+⎝ ⎠

⎛ ⎞ .exp< <

FdFdx------- f x( )F g x( )+ + 0,=

FdF

A O1x

-----⎝ ⎠⎛ ⎞+

q 1+--------------------------Fd xq 1+( )+

+

C O1x

-----⎝ ⎠⎛ ⎞+

q 1+--------------------------xq 1+ d xq 1+( ) 0.=

GdGA

q 1+------------Gd xq 1+( ) C

q 1+------------xq 1+ d xq 1+( )+ + 0.=

GdGdz------- A

q 1+------------G

Cq 1+------------z+ + 0,=

Limit Cycles of the Lienard Equation with Discontinuous Coefficients

Corresponding Member of the Russian Academy of Sciences G. A. LeonovReceived December 10, 2008

PACS numbers: 02.30.Hg, 45.50.Dd

DOI: 10.1134/S102833580905005X

St. Petersburg State University, St. Petersburg, 199164Russiae-mail: leonov@math.spbu.ru

MECHANICS

DOKLADY PHYSICS Vol. 54 No. 5 2009

LIMIT CYCLES OF THE LIENARD EQUATION WITH DISCONTINUOUS COEFFICIENTS 239

which is equivalent to the linear equation

For solution of this equation with the initial dataz(0) = 0, (0) = R, under assumptions (5), the followingrelations are fulfilled:

From here, the statement of Lemma 1 follows at reason-ably large R.

We now introduce a certain number c < –1.Similarly to the proof of Lemma 1, the following

statement can be obtained.Lemma 2. For solution of system (3) with the initial

data x(0) = c, y(0) = –R, there exists the number T > 0such that

Now the behavior of the solutions of system (3) inthe vicinity of the discontinuity x = –1, y ∈ R1 is con-sidered.

In this case, we single out the “principal parts” ofthe functions f and g:

(6)

Let us first consider the case x > –1. We assume herethat

(7)

We consider the linear equation

(8)

The following the simple statement takes place.Lemma 3. Let (7) be fulfilled. For solution of

Eq. (8) with the initial data z(0) = 0, (0) = –R, there isno number t < 0 such that z(t) = 0, and

Lemma 4. Let conditions (7) be fulfilled. For solu-tion of system (3) with the initial data x(0) = a, y(0) =–R, there exists the number T > 0 such that

We present the scheme of the proof of Lemma 4.

zA

q 1+------------ z

Cq 1+------------z+ + 0.=

z

zπω----⎝ ⎠

⎛ ⎞ 0, zπω----⎝ ⎠

⎛ ⎞ Rλπω------.exp–= =

x T( ) c, y T( ) 0, x t( ) c, t∀ 0 T,( ),∈<>=

Rλπω------ ε–⎝ ⎠

⎛ ⎞ y T( ) Rλπω------ ε+⎝ ⎠

⎛ ⎞ .exp< <exp

f x( ) P O x 1+( )+( ) x 1+ q 2– ,=

g x( ) Q O x 1+( )+( ) x 1+ 2q

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

P 0, P2 4Q q 1–( ) 0.> > >

zP

q 1–----------- z

Qq 1–-----------z+ + 0.=

z

z t( )t ∞–→lim 0, z t( )

t ∞–→lim 0.= =

x T( ) a, 0 y T( ) εR,< <=

x t( ) 1– a,( ), t∀ 0 T,( ).∈ ∈

System (3) is equivalent to the equation

which can be presented here as

Making the replacement z = (x + 1)q – 1, we rewrite thisequation in the following form:

For large z, the solution F(z) of this equation at the finitefixed intervals of variation of z is close to a certain solu-tion G(z) of the equation

which is, in turn, equivalent to Eq. (8). From here, thestatement of Lemma 4 follows from Lemma 3.

The case of x < –1 is considered similarly.The following statement follows from Lemmas 1–4.Theorem 1. Let conditions (4)–(7) be fulfilled. Then

the behavior shown in Fig. 1 corresponds to the trajec-tories of system (3) with reasonably large initial data|x(0)| + |y(0)| 1, x(0) ≠ –1.

Consequence. Let conditions (4)–(7) be fulfilled, thefunction g(x) have unique zeros x = 0 and x = x1 corre-spondingly at (–1, +∞) and (–∞, –1), and the equilibriumstates x = y = 0, x = x1, and y = 0 be the locally unstableequilibrium states. Then system (3) has two limit cycles.One of them is located in the region x < –1, y ∈ R1, andthe second is in the region x > –1, y ∈ R1.

We apply the consequence formulated here to thecase when

FdFdx------- f x( )F g x( )+ + 0,=

FdFP O x 1+( )+

q 1–----------------------------------Fd x 1+( )q 1–+

+Q O x 1+( )+

q 1–----------------------------------- x 1+( )q 1– d x 1+( )q 1– 0.=

FdFP

q 1–----------- O z

1q 1–-----------

⎝ ⎠⎛ ⎞+⎝ ⎠

⎛ ⎞ Fdz+

+ Qq 1–----------- O z

1q 1–-----------

⎝ ⎠⎛ ⎞+⎝ ⎠

⎛ ⎞ zdz 0, z a 1+( )q 1– .≥=

GdGdz------- P

q 1–-----------G

Qq 1–-----------z+ + 0,=

y

–1 x

Fig. 1. Qualitative behavior of trajectories for large |x(0)| +|y(0)|.

240

DOKLADY PHYSICS Vol. 54 No. 5 2009

LEONOV

(9)

In this case the following equalities are fulfilled:

which are the necessary and sufficient condition for theexistence of the inverse transformation from Lienardsystem (3), (9), to the quadratic system [6, 7]:

In this case,

Under the assumptions made, the first Lyapunovvalue of the point x = y = 0

is zero and the following relation is fulfilled for the sec-ond Lyapunov value:

Because here

condition (7) is fulfilled for all considered parameters A,b, and q.

Condition (5) with C = C1 can be written in the fol-lowing form:

(10)

Thus, if condition (10) is fulfilled, according to The-orem 1, the trajectories are arranged as shown in Fig. 1.

We assume that g(x) has only one zero x1 at (–∞, –1).Then the inequality g(–b) > 0 is the condition thatf (x1) < 0. Here

Because it is necessary that b(q + 1) < 1 in (10), it iseasy to understand that it suffices that (10) and the ine-qualities

(11)

be fulfilled for the positivity of g(–b) and L2(0).Thus, from conditions (10) and (11), it follows that

the equilibrium states x = y = 0 and x = x1, y = 0, arelocally unstable. From here and from the consequence,the following result follows.

Theorem 2. Let function g(x) have a unique zero atthe intervals (–∞, –1) and (−1, +∞). Let A < 0, b > 1,q ∈ (–1, 0), and (10), (11), also be fulfilled. Then sys-tem (3) has two limit cycles.

It is well-known [8–11] that it is possible to singleout the classes of systems having two small limit cyclesin the vicinity of the point x = y = 0 by small parameterperturbations in the quadratic systems (and, conse-quently, also systems (3)). For such small perturba-tions, the “large” limit cycles established in Theorem 2are retained.

Thus, the conditions formulated in Theorem 2 plusthe above special perturbations of parameters single outclasses of systems (1) and (3) with four limit cycles.

ϕ x( ) = A x2 bx+( ), ψ x( ) = C1x4 C2x3 C3x2 x, A 0, b 1, q 1 0,–( ),∈><+ + +

C12q 1–( )2 b 1 3q– 4q3+( )– A2b b2 1 q+( ) 3b– 2 q–+( )–

b 2q 1–( )2--------------------------------------------------------------------------------------------------------------------------------------------,=

C22 b– 2bq–( ) 2q 1–( )2 A2b b 1–( )+( )

b 2q 1–( )2--------------------------------------------------------------------------------------------, C3

1 b – bq+b

-------------------------.= =

A2 b 1–( )2q 1–( )2

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

A2 b 1–( )2q 1–( )2

----------------------- b 2 q 1–( )+( ) C2 2C1– 1,–=

x a1x2 b1xy c1y2 α1x β1y,+ + + +=

y a2x2 b2xy c2y2 α2x β2y.+ + + +=

b1 β1 α1 1, β2 1, c1– 0,= = = = =

c2 q, a1– 1 A b 1–( )2q 1–

---------------------+ ,= =

a2 q 1+( )a12– Aa1– C1,–=

b2 A– a1 2q 1+( ), α2– 2.–= =

L1 0( ) πA4

------- bC3 bq b– 1–+( )=

L2 0( ) πA b 1–( ) 4b 2bq 5–+( ) A2b b q 1+( ) 2–( ) 1 3q– 4q3+ +( )24b 2q 1–( )2

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

P A 1 b–( ), QA2q b 1–( )2

2q 1–( )2----------------------------,= =

A2 q 1+( ) 2q 1–( )2 1 b q 1+( )–( )

b 1 q+( )b32---–⎝ ⎠

⎛ ⎞ 2--------------------------------------------------------------------------.<

g b–( )

= b2 b 1–( )2q 1– A2b b q 1+( ) 2–( ) 1 3q– 4q3+ +( )

2q 1–( )2---------------------------------------------------------------------------------------------------------------------.

52--- b– b q 1+( ) 1< <

DOKLADY PHYSICS Vol. 54 No. 5 2009

LIMIT CYCLES OF THE LIENARD EQUATION WITH DISCONTINUOUS COEFFICIENTS 241

The limiting case, when 2b(q + 2) = 5, has beenstudied intensively in many works. One of the firstworks in which four limit cycles were found in this casewas [12]. Here the inequality b(q + 1) < 1 is fulfilled forq ∈ (–1, –1/3), and inequality (10) takes the form

(12)

where B = bA = A5/2(q + 2). In Fig. 2, the area Ω, wherecondition (12) is fulfilled, is singled out on the plane ofparameters B, q. The shaded area representing the sys-tem of parameters (B, q), where there are four cycles(after small perturbations of parameters), was obtainedin [12]. It should be noted that this area is wholly con-tained in Ω.

In [7] we carried out computer experiments in whichthe limit cycles for the parameters from the area Ω werecalculated.

B2 5 q 1+( ) 1 3q+( ),–<

Conjecture. Apparently, there are three “large” limitcycles if the conditions of (10) are fulfilled and b – 1 aresmall in the quadratic system. In the same time it is pos-sible to achieve the existence of five limit cycles aftersmall perturbations of parameters.

REFERENCES

1. A. A. Andronov, A. A. Witt, and S. É. Khaikin, Theory ofOscillators (Fizmatgiz, Moscow, 1959; Pergamon Press,Oxford, 1966).

2. S. Lefshets, Differential Equations: Geometric Theory(Inostrannaya Literatura, Moscow, 1961; Interscience,New York, 1957).

3. G. A. Leonov, Different. Equats. and Dyn. Syst. 5 (3/4),289 (1997).

4. G. A. Leonov, Ukr. Mat. Zhurn. 50 (1), 48 (1998).5. G. A. Leonov, Vestn. S.-Peterburg. Gos. Univ.: Ser. 1,

Matemat., Mekhan., Astron., No. 4, 48 (2006).6. G. A. Leonov, Intern. J. Bifurcation and Chaos 18 (3),

877 (2008).7. G. A. Leonov, N. V. Kuznetsov, and E. V. Kudryashova,

Vestn. S.-Peterburg. Gos. Univ.: Ser. 1, Matemat.,Mekhan., Astron., No. 3 (2008).

8. J. Chavarriga and M. Gran, Sci. Ser. A 9, 1 (2003).9. J. Li, Intern. J. Bifurcation and Chaos 13 (1) 47 (2003).

10. S. Lynch, in Differential Equations with Symbolic Com-putations, Ser. Trend in Mathematics, London, p. 1(2005).

11. P. Yu and G. Chen, Nonlinear Dynamics, No. 3, 409(2008).

12. S. L. Shi, Sci. Sin. 23, 153 (1980).

Translated by V. Bukhanov

–0.2

–0.4

–0.6

–0.8

–1.0

–1.2

–1.4–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 B

q

Ω

Fig. 2. Region of existence of two “large” limit cycles.