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Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model. (English) Zbl 1080.92505

Summary: Interior crises are understood as discontinuous changes of the size of a chaotic attractor that occurs when an unstable periodic orbit collides with the chaotic attractor. We present here numerical evidence and theoretical reasoning which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous. This occurs in the Hindmarsh-Rose model of a neuron, at the transition point between the bursting and spiking dynamics, which are two different dynamic behaviors that this system is able to present. Moreover, besides the change in attractor size, other significant properties of the system undergoing the transitions do change in a relevant qualitative way. The mechanism for such transition is understood in terms of a simple one-dimensional map whose dynamics undergoes a crossover between two different universal behaviors.

MSC:

92C20 Neural biology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N25 Dynamical systems in biology
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