Schomerus, H. and Titov, M. (2002) Statistics of finite-time Lyapunov exponents in a random time-dependent potential. Physical Review E, 66. 066207.
The sensitivity of trajectories over finite-time intervals t to perturbations of the initial conditions can be associated with a finite-time Lyapunov exponent Î», obtained from the elements Mij of the stability matrix M. For globally chaotic dynamics, Î» tends to a unique value (the usual Lyapunov exponent Î»â��) for almost all trajectories as t is sent to infinity, but for finite t it depends on the initial conditions of the trajectory and can be considered as a statistical quantity. We compute for a particle moving in a randomly time-dependent, one-dimensional potential how the distribution function P(Î»;t) approaches the limiting distribution P(Î»;â��)=Î´(Î»-Î»â��). Our method also applies to the tail of the distribution, which determines the growth rates of moments of Mij. The results are also applicable to the problem of wave-function localization in a disordered one-dimensional potential.
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