Two-parameter bifurcations in a discontinuous map with a variable gap

Bifurcation analysis is a useful tool for gaining a better understanding of nonlinear dynamics systems. In this paper, two-parameter bifurcation phenomenon in a piecewise linear map with a variable gap, which is both discontinuous and non-invertibility, is investigated. The results show that the variable gap can induce the rich dynamical behaviors including the various periods, coexistence of periods, coexistence of periods and chaotic band attractors. The bifurcation curves are obtained analytically by using the border collision bifurcations and stability analysis. The numerical results are in good agreement with the theoretical predictions. This analysis provides another opportunity to study the dynamics of networks with neurons possessing different types of multistability.