Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system

DOI10.1016/j.jde.2008.06.039zbMath1168.34029OpenAlexW2083318543MaRDI QIDQ2378186

Hong Zang, Junmin Yang, Mao'an Han

Publication date: 7 January 2009

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

Full work available at URL: https://doi.org/10.1016/j.jde.2008.06.039

zbMATH Keywords

bifurcationlimit cyclecuspidal loopMelnikov functionHamiltonian system

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) Homoclinic and heteroclinic solutions to ordinary differential equations (34C37) Hyperbolic singular points with homoclinic trajectories in dynamical systems (37G20)

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Cites Work

• Unnamed Item
• Unnamed Item
• Limit cycles near homoclinic and heteroclinic loops
• Melnikov functions and perturbation of a planar Hamiltonian system
• Bifurcations of invariant tori and subharmonic solutions for periodic perturbed systems
• Cyclicity of planar homoclinic loops and quadratic integrable systems
• Abelian integrals for cubic vector fields
• Cyclicity of homoclinic loops and degenerate cubic Hamiltonians
• On the number of limit cycles in double homoclinic bifurcations
• Melnikov functions for period annulus, nondegenerate centers, heteroclinic and homoclinic cycles.
• Some bifurcation methods of finding limit cycles
• On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields
• HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
• Perturbations from an elliptic Hamiltonian of degree four. II: Cuspidal loop