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A Dual Perturbation Series Analysis of Homoclinic Bifurcation for Autonomous Systems
Author(s) -
Smith P.,
Yorke J. M. E.
Publication year - 1992
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19920720707
Subject(s) - homoclinic orbit , homoclinic bifurcation , perturbation (astronomy) , duffing equation , mathematics , mathematical analysis , van der pol oscillator , nonlinear system , hamiltonian system , amplitude , bifurcation , computation , physics , quantum mechanics , algorithm
We consider the perturbation of a Hamiltonian second‐order differential equation which has at least one homoclinic path. The perturbations take the form of damping (either linear or nonlinear) and self‐excitation as can occur typically in the equation for the van der Pol‐Duffing oscillator. The perturbation is a two‐stage process which involves first finding the perturbed homoclinic path, and then using this as the first approximation for neighbouring paths. By a process of matching solutions, leading term approximations for the periods and amplitudes of limit‐cycles which bifurcate internally and externally from the homoclinic path can be found. The symbolic computation system REDUCE has been used to facilitate integration of terms in the series.