On Behavior of the Periodic Orbits of a Hamiltonian System of Bifurcation of Limit Cycles

Amor Menaceur,Mohamed Abdalla,Sahar Ahmed Idris,Ibrahim Mekawy,Sundarapandian Vaidyanathan

doi:10.1155/2021/1952706

Abstract

In light of the previous recent studies by Jaume Llibre et al. that dealt with the finite cycles of generalized differential Kukles polynomial systems using the first- and second-order mean theorem such as (Nonlinear Anal., 74, 1261–1271, 2011) and (J. Dyn. Control Syst., vol. 21, 189–192, 2015), in this work, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of Hamiltonian system using the averaging theory of first order.

Highlights

Control Syst., vol 21, 189–192, 2015), in this work, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of Hamiltonian system using the averaging theory of first order
Among the many interesting problems in the qualitative theory of planar polynomial differential systems is the study of their limit cycles
Y_ f(x, y), has a long history, where f(x, y) is a polynomial with real coefficients of degree n. Since it was first introduced in Kukles 1944, many researchers have concentrated on its maximum number of limit cycles and their location

Summary

Introduction

Among the many interesting problems in the qualitative theory of planar polynomial differential systems is the study of their limit cycles (see [1, 2]). Y_ f(x, y), has a long history, where f(x, y) is a polynomial with real coefficients of degree n Since it was first introduced in Kukles 1944, many researchers have concentrated on its maximum number of limit cycles and their location. In [6], Llibre and Mereu studied the maximum number of limit cycles using the averaging theory as follows:. Mathematical Problems in Engineering e number of limit cycles bifurcating from the center x_ − y2p− 1 and y_ x2q− 1, where p, q are positive integers, for the following two kinds of polynomial differential systems,. Consider system (5) with q lp, l is a positive integer, and |ε| sufficiently small; let H(ni, l) denote the maximum number of limit cycles of the polynomial differential system (5) bifurcating from the periodic orbits of the center x_ − y2p− 1 and y_ x2lp− 1 using the averaging theory of first order; ,. For more information about the averaging theory, see [10,11,12]

Proof of Theorem 1

Proof of Theorem 2

Conclusion