On the equivalence of the Melnikov functions method and the averaging method

DOI10.1007/s12346-016-0216-xzbMath1392.34052OpenAlexW2543855378MaRDI QIDQ1641972

Adriana Buică

Publication date: 20 June 2018

Published in: Qualitative Theory of Dynamical Systems (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s12346-016-0216-x

zbMATH Keywords

averaging methodMelnikov functionsperturbation of a period annuluspolar-like coordinates

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) Averaging method for ordinary differential equations (34C29) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Perturbations, asymptotics of solutions to ordinary differential equations (34E10)

Related Items (12)

Bifurcation of limit cycles for a perturbed piecewise quadratic differential systems ⋮ Analysis of degenerate fold-Hopf bifurcation in a three-dimensional differential system ⋮ Limit cycles via higher order perturbations for some piecewise differential systems ⋮ The local cyclicity problem: Melnikov method using Lyapunov constants ⋮ Zero-Hopf periodic orbit of a quadratic system of differential equations obtained from a third-order differential equation ⋮ An algorithmic approach to small limit cycles of nonlinear differential systems: the averaging method revisited ⋮ Limit cycles of polynomial Liénard systems via the averaging method ⋮ On the critical points of the flight return time function of perturbed closed orbits ⋮ Perturbed normalizers and Melnikov functions ⋮ On the Number of Limit Cycles in Generalized Abel Equations ⋮ Melnikov functions of arbitrary order for piecewise smooth differential systems in \(\mathbb{R}^n\) and applications ⋮ Limit cycles in piecewise polynomial systems allowing a non-regular switching boundary

Cites Work

• Limit cycles of differential equations
• Averaging methods for finding periodic orbits via Brouwer degree.
• Cyclicity of a family of vector fields
• Averaging theory at any order for computing periodic orbits
• Equivalence of the Melnikov function method and the averaging method
• Averaging methods of arbitrary order, periodic solutions and integrability
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