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\(1:3\) resonance and chaos in a discrete Hindmarsh-Rose model. (English) Zbl 1442.37101

Summary: \(1:3\) resonance of a two-dimensional discrete Hindmarsh-Rose model is discussed by normal form method and bifurcation theory. Numerical simulations are presented to illustrate the theoretical analysis, which predict the occurrence of a closed invariant circle, period-three saddle cycle, and homoclinic structure. Furthermore, it also displays the complex dynamical behaviors, especially the transitions between three main dynamical behaviors, namely, quiescence, spiking, and bursting.

MSC:

37N25 Dynamical systems in biology
92C20 Neural biology
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