Small limit cycles bifurcating in pendulum systems under trigonometric perturbations

2023/10/08 by Tian, Yun, Tingting Jing, Zhe Zhang +2
Mathematics · Medicine · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2310.05220

openalex publication_date 2023/10/08 · openalex created_date 2023/10/12 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the bifurcation of small-amplitude limit cycles near the origin in perturbed pendulum systems of the form x= y, y=-sin(x)+ε Q(x,y), where Q(x,y) is a smooth or piecewise smooth polynomial in the triple (sin(x),cos(x), y) with free coefficients. We obtain the sharp upper bound on the number of positive zeros of its associated first order Melnikov function near h=0 for Q(x,y) being smooth and piecewise smooth with the discontinuity at y=0, respectively.

Related

• On the number of limit cycles for perturbed pendulum equations
• Limit cycles by perturbing quadratic isochronous centers inside piecewise smooth polynomial differential systems
• Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop
• On the number of limit cycles for Bogdanov-Takens system under perturbations of piecewise smooth polynomials
• Bifurcation of ten small-amplitude limit cycles by perturbing a quadratic Hamiltonian system