Bifurcation structure of interval maps with orbits homoclinic to a saddle-focus

Carter Hinsley; James Scully; Andrey L. Shilnikov

doi:10.3842/umzh.v75i12.7706

Carter Hinsley Department of Mathematics and Statistics, Georgia State University, Atlanta, USA
James Scully Neuroscience Institute, Georgia State University, Atlanta, USA
Andrey L. Shilnikov Neuroscience Institute and Department of Mathematics and Statistics, Georgia State University, Atlanta, USA

DOI: https://doi.org/10.3842/umzh.v75i12.7706

Keywords: Interval map, Shilnikov homoclinic bifurcation, Belyakov bifurcation, Dynamical systems, Chaos, Ordinary Differential Equations

Abstract

UDC 517.9

We study homoclinic bifurcations in an interval map associated with a saddle-focus of (2, 1)-type in $Z_2$-symmetric systems. Our study of this map reveals a homoclinic structure of the saddle-focus, with bifurcation unfolding guided by the codimension-two Belyakov bifurcation. We consider three parameters of the map corresponding to the saddle quantity, splitting parameter, and the focal frequency of the smooth saddle-focus in a neighborhood of homoclinic bifurcations. We symbolically encode the dynamics of the map in order to find stability windows and locate homoclinic bifurcation sets in a computationally efficient manner. The organization and possible shapes of homoclinic bifurcation curves in the parameter space are examined, taking into account the symmetry and discontinuity of the map. Sufficient conditions for stability and local symbolic constancy of the map are presented. This study provides insights into the structure of homoclinic bifurcations of the saddle-focus map, furthering comprehension of low-dimensional chaotic systems.

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