Daisuke Ito, Tetsushi Ueta, Takuji Kousaka, Kazuyuki Aihara, Bifurcation Analysis of the Nagumo-Sato Model and its Coupled Systems, World Scientific, International Journal of Bifurcation and Chaos, Vol.26, No.3, pp.1630006-1-1630006-11, Mar. 2016.
Abstract: The Nagumo-Sato model is a simple mathematical expression of a single neuron, and it is catego- rized 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 multi-stable attractors are observed in its coupled system. We demonstrate several bifurcation diagrams and corresponding topological properties of periodic solutions.