A Note on Small Amplitude Limit Cycles of Li´enard Equations Theory

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


Introduction
A lot of previous works consider studies on a limit cycles' existence for Liénard systems [1][2][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][5][6]; this uniqueness can be verified using different ways of methods based on Poincare-Bendixson theorem. In [4], Zhou et al. 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 n solutions, However, in this paper, we use a counterexample to demonstrate that the existence of n 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 F and g are polynomials of order n of x and y. For several classes of such systems and in cases where the critical point is under perturbation of the coefficients in F and g, the maximum number of limit cycles that can bifurcate out can be formulated in terms of the degree of F and g [10-12].

Bendixon Criterion
We consider the following autonomous system: Let X � (P, Q) be the vector field and div X � zP/zx + zQ/zy. Theorem 1. Let D be a simply connected open subset of R. If div X � zP/zx + zQ/zy is of constant sign and not identically zero in D, then the system defined by 2 has no periodic orbit lying entirely in the region D.

Proof.
If c is a periodic orbit in D, then P(x, y)dy − Q(x, y)dx � 0 on c. Since the interior U of c is simply connected, we can apply Greenâs theorem to obtain the following: is 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 T, then it has a closed orbit Γ in D. Let G be the interior of Γ, we can apply Greens theorem to obtain the following: Since divX is either >0 or <0, then J G divX dxdy will not be zero; therefore, there are no periodic solutions. □
Theorem 2 (see [1]). For the system of form (2), there are at most n small-amplitudes limit cycles. If a 1 , a 3 , . . . , a 2n+1 are so chosen that then there are exactly n small-amplitudes limit cycles.

Examples
In this section, by using the counterexample, we can demonstrate that eorems 2 and 3 are not true. However, the previous theorems will be true if we add the following condition: a 0 /a 2 ≪ a 2 /a 4 ≪ a 4 /a 6 ≪ a 6 /a 8 ≪ · · · ≪ a 2j−2 /a 2j , j � 1, . . . , n.

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 Mathematical Problems in Engineering 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: a 0 /a 2 ≪ a 2 /a 4 ≪ a 4 /a 6 ≪ a 6 /a 8 ≪ · · · ≪ a 2j−2 /a 2j , j � 1, . . . , n.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.