Slow divergence integrals in classical Liénard equations near centers

DOI10.1007/s10884-014-9358-1zbMath1325.34040OpenAlexW2031392001MaRDI QIDQ2340290

Renato Huzak, Peter De Maesschalck

Publication date: 16 April 2015

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

Full work available at URL: https://doi.org/10.1007/s10884-014-9358-1

zbMATH Keywords

limit cyclesLienard equationgeometric singula perturbation theory

Mathematics Subject Classification ID

Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Singular perturbations for ordinary differential equations (34E15)

Related Items (14)

Fractal analysis of canard cycles with two breaking parameters and applications ⋮ On the limit cycles for a class of generalized Liénard differential systems ⋮ Limit cycles of the classical Liénard differential systems: a survey on the Lins Neto, de Melo and Pugh's conjecture ⋮ Normal forms of Liénard type for analytic unfoldings of nilpotent singularities ⋮ Cyclicity of periodic annulus and Hopf cyclicity in perturbing a hyper-elliptic Hamiltonian system with a degenerate heteroclinic loop ⋮ Limit cycles of polynomial Liénard systems via the averaging method ⋮ Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line ⋮ Fractal dimensions and two-dimensional slow-fast systems ⋮ Asymptotic lower bounds on Hilbert numbers using canard cycles ⋮ Phase portraits of the quadratic polynomial Liénard differential systems ⋮ Complete bifurcation diagram and global phase portraits of Liénard differential equations of degree four ⋮ The number of limit cycles for regularized piecewise polynomial systems is unbounded ⋮ Planar canards with transcritical intersections ⋮ Cyclicity of canard cycles with hyperbolic saddles located away from the critical curve

Cites Work

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• Classical Liénard equations of degree \(n\geqslant 6\) can have \([\frac{n-1}{2}+2\) limit cycles]
• Uniqueness of limit cycles for Liénard differential equations of degree four
• Polynomial Liénard equations near infinity
• Hilbert's sixteenth problem for polynomial Liénard equations
• Putting a boundary to the space of Liénard equations
• Slow divergence integral and balanced canard solutions
• Compactification and desingularization of spaces of polynomial Liénard equations
• Numerical Continuation Techniques for Planar Slow-Fast Systems
• More limit cycles than expected in Liénard equations

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