Analytical Estimation of the Maximal lyapunov Exponent in Oscillator Chains

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 $\epsilon$.