The Fermi–Pasta–Ulam Problem and Its Underlying Integrable Dynamics: An Approach Through Lyapunov Exponents
Authors
Publication date
3 May 2018
Publisher
'Springer Science and Business Media LLC'
Doi
Abstract
open3noFPU models, in dimension one, are perturbations either of the linear model or of the Toda model; perturbations of the linear model include
the usual eta-model, perturbations of Toda include the usual alpha+eta model. In this paper we explore and compare two
families, or hierarchies, of FPU models, closer and closer to either the linear or the Toda model, by computing numerically, for each
model, the maximal Lyapunov exponent chi. More precisely, we consider statistically typical trajectories and study the asymptotics
of chi for large N (the number of particles) and small eps (the specific energy E/N), and find, for all models, asymptotic
power laws chisimeqCepsa, C and a depending on the model. The asymptotics turns out to be, in general, rather slow, and
producing accurate results requires a great computational effort. We also revisit and extend the analytic computation of chi introduced
by Casetti, Livi and Pettini, originally formulated for the eta-model. With great evidence the theory extends successfully to
all models of the linear hierarchy, but not to models close to Toda.openBenettin, G.*; Pasquali, S.; Ponno, A.Benettin, G.; Pasquali, S.; Ponno, A