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Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 209--222 | DOI:10.5890/DNC.2016.09.002

Tanushree Roy$^{1}$, Roy Choudhury$^{1}$, and Ugur Tanriver$^{2}$

$^{1}$ Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364 USA

$^{2}$ Department of Mathematics, Texas A&M University, Texarkana, TX 75505

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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 periodic 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.

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