Dynamical Analysis of a Soliton Neuron Model: Bifurcations, Quasi-Periodic Behaviour, Chaotic Patterns, and Wave Solutions

Adel Elmandouh ()
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Adel Elmandouh: Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 12, 1-26

Abstract: This research explores the dynamic characteristics of the soliton neuron model, a mathematical approach used to describe various complicated processes in neuroscience, including the unclear mechanisms of numerous anesthetics. An appropriate wave transformation converts the neuron model into a two-dimensional dynamical system, which takes the form of a conservative Hamiltonian system with a single degree of freedom. This study utilizes qualitative methods from planar integrable systems theory to analyze and interpret phase portraits. The conditions under which periodic, super-periodic, and solitary wave solutions exist are clearly defined and organized into theorems. These solutions are obtained analytically, with several examples depicted through 2D- and 3D-dimensional graphical illustrations. The research also examines how key physical parameters, such as frequency and sound velocity, affect the nature of these solutions, specifically on the width and the amplitude of those solutions. In addition, by inserting a generalized periodic external force, the model exhibits quasi-periodic and chaotic dynamics. These complicated dynamics are visualized using 2D and 3D phase portraits and time series plots. To further assess chaotic behavior, Lyapunov exponents are calculated. Numerical results indicate that the system’s overall behavior is strongly impacted by changes in the external force’s frequency and amplitude.

Keywords: bifurcation theory; wave solutions; soliton neuron model; quasi-periodic; chaotic pattern (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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