Upper Bounds for the Number of Limit Cycles for a Class of Polynomial Differential Systems

Abstract

One of the main open problems in the theory of ordinary differential equations is the

study of the existence and number of limit cycles, due to their fundamental role in under- standing the periodic behavior of dynamical systems. A limit cycle is an isolated periodic

orbit of the system and plays a central role in the qualitative analysis of differential equa- tions. This study falls within the framework of Hilbert’s 16th problem, specifically its

second part, which concerns the existence of a uniform upper bound on the number of limit cycles in polynomial differential systems of a given degree. In this thesis of master, we conducted a comprehensive review of the concept of limit cycles, focusing on their

identification within a specific class of polynomial differential systems arising from poly- nomial perturbations added to the linear center x ̇ = y, y ̇ = −x, These perturbations,

which involve small parameters ε, generate nonlinear dynamics and give rise to new limit cycles. We employed first- and second-order averaging theory to determine accurate upper

bounds on the number of limit cycles bifurcating from the periodic orbits of the unper- turbed system. This work is based on a detailed study of the scientific article authored

by Jaume Llibre and Clàudia Valls, entitled “On the number of limit cycles of a class of polynomial differential systems” [1], in which we reanalyzed their theoretical results and applied them to an original example. Our main contribution lies in providing an

original applied example that explicitly satisfies the conditions of the second-order av- eraging theory, illustrating the practical challenges involved in applying the theory and

complementing the theoretical results established in the referenced article.