N-tree approximation for the largest Lyapunov exponent of a coupled-map lattice

TL;DR

This paper applies the N-tree approximation scheme to compute the maximum Lyapunov exponent in a coupled map lattice, revealing how phase transitions shift with tree depth and suggesting their possible disappearance.

Contribution

It introduces the N-tree approximation to coupled map lattices for Lyapunov exponent calculation, exploring effects of tree depth on phase transitions.

Findings

Phase transition shifts to larger coupling values with increased tree depth

Exact and numerical implementations for small and large tree depths

Conjecture that the phase transition may eventually disappear

Abstract

The N-tree approximation scheme, introduced in the context of random directed polymers, is here applied to the computation of the maximum Lyapunov exponent in a coupled map lattice. We discuss both an exact implementation for small tree-depth $n$ and a numerical implementation for larger $n$s. We find that the phase-transition predicted by the mean field approach shifts towards larger values of the coupling parameter when the depth $n$ is increased. We conjecture that the transition eventually disappears.