A heteroclinic point and basin boundaries in a piecewise linear chaotic neuron model

It has been shown that a high-dimensional discrete-time chaotic neuron model whose output is described by the logistic function exhibits bursting phenomena. When the bursting occurs, the model has heteroclinic loops connecting between fixed points and two-periodic points. In this paper, we consider a two-dimensional discrete-time chaotic neuron model with a piecewise linear output function. We derive occurrence conditions for the bursting, and show that, when the bursting occurs, there do not exist heteroclinic loops but rather heteroclinic points due to transversal intersections of unstable and stable manifolds of fixed and two-periodic points, which is a unique feature of piecewise linear model. Moreover, we show that boundaries of a set of initial conditions for the bursting are caused by piecewise linearity.