Analytical Estimation of the Maximal lyapunov Exponent in Oscillator Chains

TL;DR

This paper derives an analytical formula for the maximal Lyapunov exponent in oscillator chains, validating it with simulations and exploring its scaling behavior at high energy densities.

Contribution

It provides a new analytical expression for the maximal Lyapunov exponent in Fermi-Pasta-Ulam oscillator chains based on modulational instability analysis.

Findings

Analytical expression matches numerical results across various energy densities.

At high energy densities, the Lyapunov exponent scales as the square root of energy density.

For certain potentials, the Lyapunov exponent exhibits a specific power law scaling.

Abstract

An analytical expression for the maximal Lyapunov exponent $\lambda_1$ in generalized Fermi-Pasta-Ulam oscillator chains is obtained. The derivation is based on the calculation of modulational instability growth rates for some unstable periodic orbits. The result is compared with numerical simulations and the agreement is good over a wide range of energy densities $\epsilon$. At very high energy density the power law scaling of $\lambda_1$ with $\epsilon$ can be also obtained by simple dimensional arguments, assuming that the system is ruled by a single time scale. Finally, we argue that for repulsive and hard core potentials in one dimension $\lambda_1 \sim \sqrt{\epsilon}$ at large $\epsilon$.