Intrinsic properties of the chaotic Rulkov model

A comprehensive investigation concerning the intrinsic properties of the chaotic Rulkov neuron model is carried out by analyzing three distinct types of two-dimensional parameter-plane diagrams. The first type of parameter-plane diagram, established by qualitative analysis combined with the center manifold theorem and the normal form theory, systematically characterizes clearly the types of fixed point and possible bifurcation curves. The second diagram, generated numerically, visually delineates stability domains for periodic solutions, quasi-periodic solutions, and chaotic orbits. Notably, three types of classical routes to chaos, period-doubling bifurcation, intermittency, and quasi-periodicity are all identified in the chaotic Rulkov neuron model. The third diagram employs color-coded representations of the largest Lyapunov exponent values across parameter space, revealing a comb-shaped chaotic region interspersed with periodic windows of varying periods. Particularly significant is the observation of successive period-adding and period-doubling bifurcations within each periodic window, demonstrating complex dynamical transitions in the Rulkov neuron model.