New insights into chaos in Rulkov map

The Rulkov map has been widely employed to mimic the bursting and spiking behaviors of biological neurons, offering both biological relevance and mathematical tractability. In this paper, a new formulation of the Rulkov fast subsystem is introduced by incorporating the fixed-point information (x∗,y∗)=(σ,σ-α1+σ2) of the original map. This auxiliary one-dimensional system, while not capturing the full two-dimensional dynamics, provides a mathematically transparent framework for rigorously analyzing the mechanisms underlying complex behaviors near the trivial fixed point. The local stability and bifurcation properties of this new fast subsystem are analyzed, revealing how the number and stability of fixed points evolve as the control parameters α and σ vary, and these findings are corroborated by numerical simulations. Using inverse mapping techniques, explicit parameter conditions are identified under which the fixed point x∗=σ serves as a snap-back repeller, thereby confirming the existence of chaos via Marotto's theorem. More importantly, going beyond the existence of chaos, we rigorously prove that the new fast subsystem contains periodic points of every positive integer period. This result not only aligns with the periodic windows observed in numerical bifurcation diagrams but also establishes a fundamental property of the system's dynamical complexity: the coexistence of infinitely many periodic orbits with all possible periods provides analytical evidence of the rich structure underlying the chaotic dynamics. These findings offer new analytical insights into the Rulkov map, complementing existing numerical studies and deepening the mathematical understanding of chaos in neuron models.