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Bifurcation Analysis of the Nagumo–Sato Model and Its Coupled Systems

Department of Electronic Systems Engineering, The University of Shiga Prefecture, 2500 Hassaka-cho, Hikone-shi, Shiga 522-8533, Japan

E-mail Address: [email protected]

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Tetsushi Ueta

Center for Administration of Information Technology, Tokushima University, 2-1 Minami-Jousanjima, Tokushima-shi, Tokushima 770-8501, Japan

E-mail Address: [email protected]

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Takuji Kousaka

Department of Mechanical and Energy Systems Engineering, Oita University, 700 Dannoharu, Oita-shi, Oita 870-1192, Japan

E-mail Address: [email protected]

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Kazuyuki Aihara

Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba Meguro-ku, Tokyo 152-8505, Japan

E-mail Address: [email protected]

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https://doi.org/10.1142/S0218127416300068Cited by:3 (Source: Crossref)

Abstract

The Nagumo–Sato model is a simple mathematical expression of a single neuron, and it is categorized as a discrete-time hybrid dynamical system. To compute bifurcation sets in such a discrete-time hybrid dynamical system accurately, conditions for periodic solutions and bifurcations are formulated herewith as a boundary value problem, and Newton’s method is implemented to solve that problem. As the results of the analysis, the following properties are obtained: border-collision bifurcations play a dominant role in dynamical behavior of the model; chaotic regions are distinguished by tangent bifurcations; and multistable attractors are observed in its coupled system. We demonstrate several bifurcation diagrams and corresponding topological properties of periodic solutions.

Keywords:

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