Limit cycle analysis on a cubic Hamiltonian system with quintic perturbed terms

Abstract

This paper intends to explore bifurcation behavior of limit cycles for a cubic Hamiltonian system with quintic perturbed terms using both qualitative analysis and numerical exploration. To obtain the maximum number of limit cycles, a quintic perturbed function with the form of R(x, y, λ) = S(x, y, λ) = mx 2 + ny 2 + ky 4 − λ is added to a cubic Hamiltonian system, where m, n, k and λ are all variable. The investigation is based on detection functions which are particularly effective for the perturbed cubic Hamiltonian system. The study reveals that, for the Hamiltonian system [equation (1.5) in the introduction] with the perturbed terms mentioned above, there are 15 limit cycles if 15.1149 < λ < 15.1249; and 11 limit cycles if 15.1102 < λ < 15.1149. As numerical illustration, we numerically predict the detection curves and display graphically the distribution of limit cycles for the proposed perturbed Hamiltonian system.

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“…The study indicates that, for the system (1. The proof of this proposition can be found elsewhere [3,10]. For the sake of completeness, we briefly present the proof as below:…”

Bifurcation of Limit Cycles of a Perturbed Integrable Non-Hamiltonian System

“…The proof of this proposition can be found elsewhere [8,9]. For the sake of completeness, we briefly present the proof as below:…”