Self-organization in Trees and Motifs of Two-Dimensional Chaotic Maps with Time Delay

TL;DR

This paper investigates how two-dimensional chaotic maps coupled with time delay on scale-free trees exhibit diverse dynamical behaviors, including localization, periodicity, and strange attractors, with implications for understanding collective dynamics.

Contribution

It introduces a study of chaotic map networks on scale-free trees with time delay, revealing new dynamical phenomena and the role of motifs in collective behavior.

Findings

Identification of dynamical localization and emergent periodicity.

Observation of strange non-chaotic attractors and long-range correlations.

Quantitative link between small-scale motif dynamics and global network behavior.

Abstract

We study two-dimensional chaotic standard maps coupled along the edges of scale-free trees and tree-like subgraph (4-star) with a non-symplectic coupling and time delay between the nodes. Apart from the chaotic and regular 2-periodic motion, the coupled map system exhibits variety of dynamical effects in a wide range of coupling strengths. This includes dynamical localization, emergent periodicity, and appearance of strange non-chaotic attractors. Near the strange attractors we find long-range correlations in the intervals of return-times to specified parts of the phase space. We substantiate the analysis with the finite-time Lyapunov stability. We also give some quantitative evidence of how the small-scale dynamics at 4-star motifs participates in the genesis of the collective behavior at the whole network.