Analytical Prediction Of Homoclinic Bifurcations Following A Supercritical Hopf Bifurcation

Keywords

Collision with neighboring saddle-point; Homoclinic orbit formation; Post-Hopf regimes; Supercritical Hopf bifurcations

Abstract

An analytical approach to homoclinic bifurcations at a saddle fixed point is developed in this paper based on high-order, high-accuracy approximations of the stable periodic orbit created at a supercritical Hopf bifurcation of a neighboring fixed point. This orbit then expands as the Hopf bifurcation parameter(s) is(are) varied beyond the bifurcation value, with the analytical criterion proposed for homoclinic bifurcation being the merging of the pe riodic orbit with the neighboring saddle. Thus, our approach is applicable in any situation where the homoclinic bifurcation at any saddle fixed point of a dynamical system is associated with the birth or death of a periodic orbit. We apply our criterion to two systems here. Using approximations of the stable, post-Hopf periodic orbits to first, second, and third orders in a multiple-scales perturbation expansion, we find that, for both systems, our proposed analytical criterion indeed reproduces the numerically-obtained parameter values at the onset of homoclinic bifurcation very closely.

Publication Date

1-1-2016

Publication Title

Discontinuity, Nonlinearity, and Complexity

Volume

5

Issue

3

Number of Pages

209-222

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.5890/DNC.2016.09.002

Socpus ID

85020306558 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85020306558

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