Lyapunov exponent in the Vicsek model

TL;DR

This paper investigates the chaotic behavior of the Vicsek model of self-propelled particles by computing the largest Lyapunov exponent, revealing a chaotic regime and its relation to phase transitions.

Contribution

It introduces a smooth version of the Vicsek model to enable direct computation of Lyapunov exponents and analyzes the chaotic dynamics near phase transition points.

Findings

Chaotic regime identified in the Vicsek model via Lyapunov exponent.

Lyapunov exponent varies significantly near phase transition points.

Proposed smooth model facilitates Lyapunov analysis of SPP systems.

Abstract

The well-known Vicsek model describes the dynamics of a flock of self-propelled particles (SPPs). Surprisingly, there is no direct measure of the chaotic behavior of such systems. Here, we discuss the dynamical phase transition present in Vicsek systems in light of the largest Lyapunov exponent (LLE), which is numerically computed by following the dynamical evolution in tangent space for up to two million SPPs. As discontinuities in the neighbor weighting factor hinder the computations, we propose a smooth form of the Vicsek model. We find a chaotic regime for the collective behavior of the SPPs based on the LLE. The dependence of LLE with the applied noise, used as a control parameter, changes sensibly in the vicinity of the well-known transition points of the Vicsek model.