New phenomena in Rulkov map based on Poincaré cross section

Abstract

Different from analyzing the time-series directly, the Poincaré cross-section method is employed in the dynamical study of the Rulkov mapping model, more importance is that some new phenomena of the model are reported in this paper. First, it is proved that x = −1 is the Poincaré cross-section for all non-converging-to-fixed-point trajectories based on fast-slow dynamics. Then, some new dynamics are presented by means of the Poincaré cross-section method. In detail, a " lightning-shaped" periodic divider of the original model is introduced, which is caused by the different selection of the initial value of slow variable. The Period 3 on the Poincaré cross-section is proposed near the chaos region, which may be not reported in other literatures according to our understanding. In some parameter region, it will cause the pseudo chaos phenomenon, here the trajectories spend a lot time converging to a periodic solution. Besides, in the view of mean-field of action potential, the sensitiveness with small parameter's perturbations is analyzed, which the period changes frequently in the bursts of spikes region, and it has robustness in the normal spikes region.