Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2412.12134

Chaotic Dynamics and Fractal Geometry in Ring Lattice Systems of Nonchaotic Rulkov Neurons

Brandon B. Le
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper investigates the complex dynamics and fractal attractors that arise in a 60-dimensional ring lattice system of electrically coupled nonchaotic Rulkov neurons. While networks of chaotic Rulkov neurons have been widely studied, systems of nonchaotic Rulkov neurons have not been extensively explored due to the piecewise complexity of the nonchaotic Rulkov map. Here, we find that rich dynamics emerge from the electrical coupling of regular-spiking Rulkov neurons, including chaotic spiking, synchronized chaotic bursting, and synchronized hyperchaos. By systematically varying the electrical coupling strength between neurons, we also uncover general trends in the maximal Lyapunov exponent across the system's dynamical regimes. By means of the Kaplan-Yorke conjecture, we examine the fractal geometry of the ring system's high-dimensional chaotic attractors and find that these attractors can occupy as many as 45 of the 60 dimensions of state space. We further explore how variations in chaotic behavior - quantified by the full Lyapunov spectra - correspond to changes in the attractors' fractal dimensions. This analysis advances our understanding of how complex collective behavior can emerge from the interaction of multiple simple neuron models and highlights the deep interplay between dynamics and geometry in high-dimensional systems.

Explore related subjects

nlin.CDmath.DSq-bio.NC

Keep this discovery

BibTeXRIS

Brandon B. Le. 2024-12-06. Chaotic Dynamics and Fractal Geometry in Ring Lattice Systems of Nonchaotic Rulkov Neurons. https://doi.org/10.3390/fractalfract9090584

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Linear Response Predicts Cusp-Pair Births in Networks with a Localized Cubic

Linear response is cheap to measure; the bistability boundaries it organizes are not. For a passive network with one localized cubic, the driving-point receptance $G$ fixes the period-one cusp set at fundamental-harmonic order: cusps lie on a fixed phase contour of $G$, a tangency of that contour under parameter variation creates a pair, and its curvature separates a gap opening from an isolated loop. For a two-mode absorber the linear prediction locates a benchmark birth coupling to $0.3\%$, and to $0.03\%$ once a third-harmonic correction of scale $|G(3\Omega)/G(\Omega)|$ is included.

nlin.CD↗
arXiv2026

Dynamics Creation through Neural Dynamical Transfer Learning

Data-driven machine learning has established a robust foundation for reconstructing nonlinear dynamical systems from observations, primarily for the purposes of forecasting and control. However, most existing efforts focus on recovering specific observed dynamics rather than the generative synthesis of new ones. Inspired by image fusion and style transfer, we introduce a neural network framework termed Neural Dynamical Transfer Learning (NDTL) to create new systems with prescribed dynamics from pairs of parent nonlinear dynamical systems. By computing fundamental dynamical signatures, including the intrinsic dimension, the Kaplan-Yorke dimension, the invariant measure statistics, and the Lyapunov spectrum, we demonstrate that NDTL preserves key features inherited from the parent models while simultaneously generating novel dynamics. Beyond these validation examples, NDTL induces a criterion for dynamics classification, creates stable oscillatory coexistence in the Hastings-Powell food chain model, produces interpretable epidemiological models, and provides a chaotic source for image encryption.

nlin.CD↗
arXiv2026

The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincar\'e Sections

A Poincar\'e section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the R\"ossler system, a 32-mode Kuramoto--Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$--$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$--$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.

nlin.CD↗