Chaotic fluctuations in graphs with amplification

TL;DR

This paper models chaotic diffusion with amplification on graphs, exploring power-law tails in invariant measures and their relation to Lyapunov exponents, with implications for chaotic systems like lasing networks.

Contribution

It introduces a mathematical framework for understanding power-law distributions and intermittency in chaotic systems with amplification and escape dynamics.

Findings

Power-law tails occur when $L(q)=r$, linking Lyapunov exponents to escape rates.

Stationary power-law distributions are observed in open chaotic maps.

The model provides insights into Lévý statistics in active chaotic cavities.

Abstract

We consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of the interval. We investigate the possibility of having power-law tails in the invariant measure by approximate solution of the Perron-Frobenius equation and discuss the connection with the generalized Lyapunov exponents $L(q)$. We then consider the case of open maps where trajectories escape and demonstrate that stationary power-law distributions occur when $L(q)=r$, with $r$ being the escape rate. The proposed system is a toy model for coupled active chaotic cavities or lasing networks and allows to elucidate in a simple mathematical framework the conditions for observing L\'evy statistical regimes and chaotic intermittency in such systems.