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

TL;DR

This paper addresses the challenge of diagnosing chaotic dynamics in one-dimensional maps and in a neuron-inspired model by introducing two numerical tests: numerical transitivity and numerical LEO. The authors apply these tests to classical Lorenz-map families, including beta-transformations and expanding Lorenz maps, and to the Courbage–Nekorkin–Vdovin (CNV) neuron model in both its piecewise linear and nonlinear forms. They demonstrate that numerical transitivity provides a fast and reliable proxy for topological transitivity and that numerical LEO largely agrees with transitivity, yielding similar classifications with greater computational burden in most cases. By deriving invariant-interval conditions for CNV maps and mapping parameter regions where transitivity and LEO occur, the work connects dynamical-systems theory with computational neuroscience, offering practical diagnostics for chaotic neural-like dynamics and potentially informing spike-pattern analysis.

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.

Figures (14)

• Figure 1: The left panel presents an example of a Lorenz-like map, the middle panel an expanding Lorenz map, and the right panel a $\beta$-transformation.
• Figure 2: Results of the numerical transitivity test for classical $\beta$-transformations in the $\alpha$–$\beta$ parameter triangle $\mathcal{T}$. Horizontal red line: $\beta=\sqrt{2}$, left red line: $\alpha=1-1/\beta$, right red line: $\alpha=1+1/\beta-\beta$. Meshgrid = 800.
• Figure 3: Probability density functions and histograms for nontransitive (left panel) and transitive (right panel) $\beta$-transformations. Parameter values: $\alpha=0.4$, $\beta=1.2$ (left) and $\alpha=0.1$, $\beta=1.2$ (right).
• Figure 4: Comparison of results of numerical transitivity and numerical LEO tests for classical $\beta$-transformations in the $\alpha$–$\beta$ parameter plane. Meshgrid = 500.
• Figure 5: Differences between numerical transitivity and numerical LEO tests results for classical $\beta$-transformations in the $\alpha$-$\beta$ parameter plane. Blue points: numerical transitivity and not numerical LEO, Red points: numerical LEO and not numerical transitivity. Meshgrid = 600.

Theorems & Definitions (8)

• Proposition 4.1
• Theorem 4.1
• Conjecture 4.1
• Remark 4.2
• Proposition 4.2
• Theorem 6.1
• Conjecture 6.2
• Remark 6.3