 
Summary: Lyapunov exponents in disordered chaotic systems: Avoided crossing and level statistics
Volker Ahlers, Ru¨diger Zillmer, and Arkady Pikovsky
Department of Physics, University of Potsdam, Postfach 601553, D14415 Potsdam, Germany
Received 8 May 2000; revised manuscript received 24 October 2000; published 26 February 2001
The behavior of the Lyapunov exponents LEs of a disordered system consisting of mutually coupled
chaotic maps with different parameters is studied. The LEs are demonstrated to exhibit avoided crossing and
level repulsion, qualitatively similar to the behavior of energy levels in quantum chaos. Recent results for the
coupling dependence of the LEs of two coupled chaotic systems are used to explain the phenomenon and to
derive an approximate expression for the distribution functions of LE spacings. The depletion of the level
spacing distribution is shown to be exponentially strong at small values. The results are interpreted in terms of
the random matrix theory.
DOI: 10.1103/PhysRevE.63.036213 PACS number s : 05.45.Jn, 02.10.Ud
The Lyapunov exponent LE , which measures the insta
bility of dynamical trajectories, is a standard tool in the stud
ies of chaotic systems 1 . The positiveness of the maximal
LE serves as the criterion for chaos; the inverse LE is a
characteristic time of mixing and of correlation decay. For an
Ndimensional chaotic system, N LEs corresponding to dif
ferent directions in the phase space can be defined.
There are different methods to calculate LEs numerically
