Limit Cycles of a Class of Polynomial Differential Systems Bifurcating from the Periodic Orbits of a Linear Center

Amor Menaceur,Salah Boulaaras,Salem Alkhalaf,Shilpi Jain

doi:10.3390/sym12081346

Abstract

In this paper, we study the number of limit cycles of a new class of polynomial differential systems, which is an extended work of two families of differential systems in systems considered earlier. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a center using the averaging theory of first and second order.

Highlights

One of the more difficult problems in the qualitative theory of polynomial differential equations in the plane R2 is the study of their limit cycles
Much has been written on Kolmogorov systems, Liénard systems and Kukles systems, that is, systems of the form ẋ = −y, ẏ = x + λy + g( x, y), (1)
Christopher and Lloyd [5] presented some systems that yield at most five limit cycles bifurcating from the origin

Summary

Introduction

One of the more difficult problems in the qualitative theory of polynomial differential equations in the plane R2 is the study of their limit cycles. By using the averaging theory, we shall study in this work the maximum number of limit cycles which can bifurcate from the periodic orbits of a linear center perturbed inside the following differential systems ẋ = −y + ∑ εl f nl l ( x ). Suppose that ml = 2k l or ml = 2k l − 1 and k l ≥ 2 for |ε| sufficiently small the maximum number of limit cycles of the polynomial differential systems (4) bifurcating from the periodic orbits of the linear centre ẋ = y, ẏ = − x, using averaging theory nh i o (a) of first order is λ1 = max n12−1 , k1 − 1 limit cycles (b) of second order is λ2.

The Averaging Theory of First and Second Order

Proof of Theorem 2