Fluctuations of separation of trajectories in chaos and correlation dimension

Itzhak Fouxon, Siim Ainsaar, Jaan Kalda

Published 2019-04-29Version 1

We study the generalized Lyapunov exponent that gives the logarithmic growth exponents of the moments of the distance between two infinitesimally close trajectories of a chaotic system. The Legendre transform of the exponent is a large deviations function. That gives the probability of rare fluctuations where the trajectories separate much faster or much slower than prescribed by the first Lyapunov exponent. The function has a non-trivial zero at minus the correlation dimension of the attractor which for incompressible flows equals the space dimension. We describe properties of the generalized exponent and the Gallavotti-Cohen type relations that hold when there is symmetry under time-reversal. This demands studying joint growth rates of infinitesimal distances and volumes. We describe efficient schemes for approximating the generalized exponent and the correlation dimension. The derived relations allow to fix quartic polynomial approximations that improve the quadratic Grassberger-Procaccia estimates by removing effective time-reversibility and short correlation time. Violation of time-reversibility for the Lagrangian trajectories of the incompressible Navier-Stokes turbulence below the viscous scale is considered. We describe a different approximation scheme for finding the correlation dimension from expansion in the flow compressibility.