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The periodic solution bifurcated from homoclinic orbit for coupled ordinary differential equations
Author(s) -
Zhu Changrong,
Long Bin
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4200
Subject(s) - homoclinic orbit , mathematics , ordinary differential equation , homoclinic bifurcation , heteroclinic orbit , exponential dichotomy , mathematical analysis , lyapunov function , degenerate energy levels , linear subspace , differential equation , bifurcation , pure mathematics , physics , nonlinear system , quantum mechanics
We consider the problem of the periodic solutions bifurcated from a homoclinic orbit for a pair of coupled ordinary differential equations inR n × R m . Assume that the autonomous system has a degenerate homoclinic solution γ inR n × { 0 } . A functional analytic approach is used to consider the existence of periodic solution for the autonomous system with periodic perturbations. By exponential dichotomies and the method of Lyapunov–Schmidt, the bifurcation function defined between two finite dimensional subspaces is obtained, where the zeros correspond to the existence of periodic solutions for the coupled ordinary differential equations near { γ ( t ) | t ∈ R } × { 0 } . Copyright © 2016 John Wiley & Sons, Ltd.