Melnikov functions in quadratic perturbations of generalized Lotka-Volterra systems

DOI10.1007/s10883-015-9282-7zbMath1333.34051OpenAlexW1118857587MaRDI QIDQ745888

Henryk Żołądek

Publication date: 15 October 2015

Published in: Journal of Dynamical and Control Systems (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10883-015-9282-7

zbMATH Keywords

limit cycleMelnikov functionLotka-Volterra systemquadratic plane vector field

Mathematics Subject Classification ID

Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Explicit solutions, first integrals of ordinary differential equations (34A05) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) (34C08)

Related Items (10)

New criteria for the monotonicity of the ratio of two abelian integrals ⋮ The maximal number of limit cycles bifurcating from a Hamiltonian triangle in quadratic systems ⋮ The smallest upper bound on the number of zeros of abelian integrals ⋮ Limit cycle bifurcations in perturbations of a reversible quadratic system with a non-rational first integral ⋮ Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center ⋮ Irreducibility of the Picard-Fuchs equation related to the Lotka-Volterra polynomial \(x^2y^2(1-x-y)\) ⋮ Hilbert’s 16th problem on a period annulus and Nash space of arcs ⋮ The period function of quadratic generalized Lotka-Volterra systems without complex invariant lines ⋮ Hopf cyclicity of a family of generic reversible quadratic systems with one center ⋮ Perturbation theory of the quadratic Lotka–Volterra double center

Cites Work

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• Unfolding of the Hamiltonian triangle vector field
• Quadratic systems with center and their perturbations
• On cyclicity of triangles and segments in quadratic systems
• The cyclicity of the period annulus of the quadratic Hamiltonian triangle
• Pseudo-Abelian integrals along Darboux cycles
• The infinitesimal 16th Hilbert problem in the quadratic case

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