Bifurcations of small limit cycles in Liénard systems with cubic restoring terms

DOI10.1016/j.jde.2019.02.018zbMath1459.34098OpenAlexW2922438325MaRDI QIDQ2415284

Yun Tian, Mao'an Han, Fang Fang Xu

Publication date: 21 May 2019

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jde.2019.02.018

zbMATH Keywords

limit cycleHopf bifurcationsinvolutionspolynomial Liénard systems

Mathematics Subject Classification ID

Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Bifurcation theory for ordinary differential equations (34C23) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Discontinuous ordinary differential equations (34A36)

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Cites Work

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• Hopf bifurcation for two types of Liénard systems
• Classical Liénard equations of degree \(n\geqslant 6\) can have \([\frac{n-1}{2}+2\) limit cycles]
• Limit cycles of differential equations
• Small-amplitude limit cycles of some Liénard-type systems
• Averaging methods for finding periodic orbits via Brouwer degree.
• Mathematical problems for the next century
• Bifurcation theory for finitely smooth planar autonomous differential systems
• Hilbert's sixteenth problem for polynomial Liénard equations
• Averaging theory at any order for computing periodic orbits
• Averaging theory for discontinuous piecewise differential systems
• Limit cycles for \(m\)-piecewise discontinuous polynomial Liénard differential equations
• Center conditions in a switching Bautin system
• Limit cycles in discontinuous classical Liénard equations
• Cubic perturbations of elliptic Hamiltonian vector fields of degree three
• On the Number of Limit Cycles by Perturbing a Piecewise Smooth Liénard Model
• A Chebyshev criterion for Abelian integrals
• More limit cycles than expected in Liénard equations
• LIMIT CYCLES IN TWO TYPES OF SYMMETRIC LIÉNARD SYSTEMS
• HOPF BIFURCATION FOR NONSMOOTH LIÉNARD SYSTEMS
• Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces
• Centennial History of Hilbert's 16th Problem
• HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
• ON THE MAXIMUM NUMBER OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC EQUATIONS BY AVERAGE METHOD

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