Numerical Transitivity and Numerical Leo Properties for Lorenz Maps with Applications to Courbage-Nekorkin-Vdovin Neuron Model

TL;DR

This paper introduces and compares two numerical algorithms, transitivity and LEO tests, for analyzing chaotic behavior in one-dimensional maps, with applications to a neuron model exhibiting complex dynamics.

Contribution

It presents a systematic application and comparison of the numerical transitivity and LEO algorithms to Lorenz maps and a neuron model, demonstrating their effectiveness in detecting chaos.

Findings

Both algorithms accurately identify transitions to chaos.

The methods are effective on both linear and nonlinear neuron models.

Numerical tests reliably detect qualitative changes in system dynamics.

Abstract

This research investigates the dynamic behavior of one dimensional discrete systems using two computational algorithms, the numerical transitivity and the numerical locally eventually onto (LEO) tests. Both algorithms are systematically applied to a variety of interval maps, including classical examples such as beta transformations and expanding Lorenz maps, in order to assess and characterize their chaotic dynamics. We perform a detailed comparison of the two methods in terms of accuracy, computational efficiency, and their sensitivity in detecting transitions between regular and chaotic regimes. Particular emphasis is placed on the Courbage Nekorkin Vdovin (CNV) model of a single neuron, known for its rich, spiking like dynamics and its mathematical reducibility to Lorenz type maps. By analyzing both the piecewise linear and nonlinear versions of the CNV model, we illustrate how the proposed numerical tests reliably capture qualitative changes in the system dynamics, focusing on the onset of chaos and chaotic regimes. The results highlight the practical potential of these numerical approaches as diagnostic tools for studying complex dynamical systems arising in nonlinear science and mathematical neuroscience.