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Lyapunov exponents from unstable periodic orbits

Roberto Franzosi*

Pietro Poggi†

Monica Cerruti-Sola‡

  • Dipartimento di Fisica Università di Pisa, and INFN, Sezione di Pisa, and INFM, Unità di Pisa, via Buonarroti 2, I-56127 Pisa, Italy

  • Dipartimento di Fisica, Universitá di Firenze, via Sansone 1, I-50019 Sesto Fiorentino, Italy

  • INAF–Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, and INFM, Unità di Firenze, Firenze, Italy

  • *Electronic address: Roberto.Franzosi@df.unipi.it
  • †Electronic address: pietro.poggi@unifi.it
  • ‡Electronic address: mcs@arcetri.astro.it
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Phys. Rev. E 71, 036218 – Published 21 March, 2005

DOI: https://doi.org/10.1103/PhysRevE.71.036218

Abstract

We propose a method that allows us to analytically compute the largest Lyapunov exponent of a Hamiltonian chaotic system from the knowledge of a few unstable periodic orbits (UPOs). In the framework of a recently developed theory for Hamiltonian chaos, by computing the time averages of the metric tensor curvature and of its fluctuations along analytically known UPOs, we have re-derived the analytic value of the largest Lyapunov exponent for the Fermi-Pasta-Ulam–β (FPU-β) model. The agreement between our results and the Lyapunov exponents obtained by means of standard numerical simulations confirms the point of view which attributes to UPOs the special role of efficient probes of general dynamical properties, among them chaotic instability.

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Phys. Rev. E 71, 036218– Published 21 March, 2005
  • Received 17 August 2004

© 2005 American Physical Society

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