New insights into chaos in Rulkov map

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Dynamical Systems Theory

Background:

• The Rulkov map is a key model for simulating biological neuron bursting and spiking.
• Understanding the complex dynamics of neuron models is crucial for neuroscience.
• Existing models may lack sufficient mathematical tractability for rigorous analysis.

Purpose of the Study:

• To introduce a novel, simplified Rulkov fast subsystem for analyzing complex neuronal behaviors.
• To rigorously analyze the stability, bifurcations, and chaotic dynamics of this new subsystem.
• To analytically confirm the existence of chaos and infinite periodic orbits within the modified Rulkov map.

Main Methods:

• Formulation of a new one-dimensional Rulkov fast subsystem.
• Analysis of local stability and bifurcation properties using control parameters.
• Application of inverse mapping techniques and Marotto's theorem to prove chaos.
• Rigorous mathematical proof of the existence of periodic points for all integer periods.

Main Results:

• The new subsystem provides a transparent framework for analyzing dynamics near the trivial fixed point.
• Explicit parameter conditions for snap-back repellers and chaos (via Marotto's theorem) were identified.
• It was rigorously proven that the subsystem possesses periodic points of every positive integer period.
• The coexistence of infinitely many periodic orbits with all possible periods was established.

Conclusions:

• The new Rulkov fast subsystem offers new analytical insights into neuron models.
• The findings provide analytical evidence for the rich structure underlying chaotic dynamics in neuronal simulations.
• This work complements numerical studies and deepens the mathematical understanding of chaos in neuroscience.