Journal of Statistical Physics

, Volume 50, Issue 5–6, pp 853–878 | Cite as

Liapunov spectra for infinite chains of nonlinear oscillators

  • Jean -Pierre Eckmann
  • C. Eugene Wayne


We argue that the spectrum of Liapunov exponents for long chains of nonlinear oscillators, at large energy per mode, may be well approximated by the Liapunov exponents of products of independent random matrices. If, in addition, statistical mechanics applies to the system, the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone. Under a certain isotropy hypothesis (which is not always satisfied), we argue that the Liapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix. This construction uses an integral equation first derived by Newman. We then derive and discuss a method to compute the spectrum of a typical random matrix. Putting the pieces together, we see that the Liapunov spectrum can be computed from the potential between the oscillators.

Key words

Liapunov exponents random matrices coupled oscillators 


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Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Jean -Pierre Eckmann
    • 1
  • C. Eugene Wayne
    • 2
  1. 1.Département de Physique ThéoriqueUniversité de GenèveGeneva 4Switzerland
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity Park

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