Numerical Transitivity and Numerical LEO for Lorenz Maps with Applications to Courbage–Nekorkin–Vdovin Neuron Model
TL;DR
This research investigates the dynamic behavior of one-dimensional systems using two computational criteria: the numerical transitivity and the numerical Locally Eventually Onto (LEO) tests, and highlights the practical potential of these numerical approaches as diagnostic tools for studying complex dynamical systems arising in nonlinear science and mathematical neuroscience.
Abstract
This research investigates the dynamic behavior of one-dimensional systems using two computational criteria: 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 [Formula: see text]-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’s 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.