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December  2009, 2(4): 829-850. doi: 10.3934/dcdss.2009.2.829

Homoclinic clusters and chaos associated with a folded node in a stellate cell model

Martin Wechselberger 1, and  Warren Weckesser 1,

1. 

Department of Mathematics and Statistics, University of Sydney, Sydney, Australia

Received  September 2008 Revised  May 2009 Published  September 2009

Acker et al (J. Comp. Neurosci., 15, pp.71-90, 2003) developed a model of stellate cells which reproduces qualitative oscillatory patterns known as mixed mode oscillations observed in experiments. This model includes different time scales and can therefore be viewed as a singularly perturbed system of differential equations. The bifurcation structure of this model is very rich, and includes a novel class of homoclinic bifurcation points. The key to the bifurcation analysis is a folded node singularity that allows trajectories known as canards to cross from a stable slow manifold to an unstable slow manifold as well as a node equilibrium of the slow flow on the unstable slow manifold. In this work we focus on the novel homoclinic orbits within the bifurcation diagram and show that the return of canards from the unstable slow manifold to the funnel of the folded node on the stable slow manifold results in a horseshoe map, and therefore gives rise to chaotic invariant sets. We also use a one-dimensional map to explain why many homoclinic orbits occur in "clusters'' at exponentially close parameter values.
Keywords: homoclinic orbit, canard, folded node, stellate cell., mixed-mode oscillations, chaos.
Mathematics Subject Classification: Primary: 34D15, 34C26, 34C37; Secondary: 92C2.
Citation: Martin Wechselberger, Warren Weckesser. Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 829-850. doi: 10.3934/dcdss.2009.2.829
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