A Note on Small Amplitude Limit Cycles of Liénard Equations Theory

Abstract

In this paper, we demonstrate using a counterexample for a theorem of the small amplitude limit cycles in some Liénard systems and show that that there will be no solutions unless we add an extra condition. A new condition is derived for some specific Liénard systems where a violation of the small amplitude limit cycles theorem takes place.

1. Introduction

A lot of previous works consider studies on a limit cycles’ existence for Liénard systems [1–3]. It represents a very important class of nonlinear systems due to its appearance in some branches of science and engineering as well as in some ecological models, planar physical models, and even in some chemical models, where using a suitable transformation can change these systems into nonlinear Liénard systems. However, an extensive attention has been also devoted to the question of its uniqueness [4–6]; this uniqueness can be verified using different ways of methods based on Poincare–Bendixson theorem. In [4], Zhou . proposed a set of theorems for the limit cycles’ uniqueness for the Liénard systems; the proposed theorems represent a guarantee to complete the proof of some previous works’ propositions. In [7], Sabatini and Gabriele studied the uniqueness of limit cycles for a class of planar dynamical systems taking into account those which are equivalent to Liénard systems, and they have also proved a theorem for limit cycles of a class of plane differential systems. In the paper proposed by Li and Llibre [8], the authors proved that for any classical Liénard differential equation of degree four, there exists at most one hyperbolic limit cycle. In [9], a sufficient condition for the existence and the uniqueness of limit cycles for Liénard systems has been proposed for some applications.

In the theory of small amplitude limit cycles, Liénard systems have solutions, However, in this paper, we use a counterexample to demonstrate that the existence of solutions for some systems is not true unless we add an extra new condition.

We consider in our study the systems given by the following form: where and are polynomials of order of and . For several classes of such systems and in cases where the critical point is under perturbation of the coefficients in and , the maximum number of limit cycles that can bifurcate out can be formulated in terms of the degree of and [10–12].

2. Bendixon Criterion

We consider the following autonomous system:

Let be the vector field and .

Theorem 1. Let be a simply connected open subset of . If is of constant sign and not identically zero in , then the system defined by 2 has no periodic orbit lying entirely in the region .

Proof. If is a periodic orbit in , then on . Since the interior of is simply connected, we can apply Greenâs theorem to obtain the following:This is a contradiction since our hypothesis implies that the integral on the right cannot be zero.

Proof. If we suppose that the system given by 2 has a periodic solution of a period , then it has a closed orbit in . Let be the interior of , we can apply Greens theorem to obtain the following:Since is either or , then will not be zero; therefore, there are no periodic solutions.

3. A Note on Liénard Equations Theory

We consider the following system:where and are real coefficients.

Theorem 2 (see [1]). For the system of form (2), there are at most small-amplitudes limit cycles. If are so chosen thatthen there are exactly small-amplitudes limit cycles.

Proof. (counterexample).
We suppose the following system:whereBy putting , we obtainAsthenHowever,because

Theorem 3 (see [2]). We consider the following equation:

If the focus values given in equation (3) satisfy the following conditions:then the polynomial equation given by in equation (3) has positive real roots for .

Proof. (counterexample).
We consider the following equation:By putting , we obtainbecauseHowever,

4. Examples

In this section, by using the counterexample, we can demonstrate that Theorems 2 and 3 are not true. However, the previous theorems will be true if we add the following condition: , .

Example 1. We consider the following equation:We havebecauseHowever,where the system roots are given by

Example 2. Let us consider now the following system:with positive roots such asbecause

Example 3. We suppose the following system:By putting , we obtainand by applying the first-order averaging method [13, 14] on (14), we obtain implied , , , and , then there are exactly 4 small-amplitudes limit cycles ,
Note that for and for .

5. Conclusion

In this work, by using a counterexample for a theorem of the small amplitude limit cycles in some Liénard systems, we have shown that that there will be no solutions unless an extra condition is added. In addition, a new condition is derived for some specific Liénard systems where a violation of the small amplitude limit cycles theorem takes place. However, these theorems will be true if we add the following condition: , .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project under Grant no. (R.G.P-2/53/42).