Oseledets' splitting of standard-like maps

Chaos
M Sala, R Artuso

Abstract

For the class of differentiable maps of the plane and, in particular, for standard-like maps (McMillan form), a simple relation is shown between the directions of the local invariant manifolds of a generic point and its contribution to the finite-time Lyapunov exponents (FTLE) of the associated orbit. By computing also the point-wise curvature of the manifolds, we produce a comparative study between local Lyapunov exponent, manifold's curvature and splitting angle between stable/unstable manifolds. Interestingly, the analysis of the Chirikov-Taylor standard map suggests that the positive contributions to the FTLE average mostly come from points of the orbit where the structure of the manifolds is locally hyperbolic: where the manifolds are flat and transversal, the one-step exponent is predominantly positive and large; this behaviour is intended in a purely statistical sense, since it exhibits large deviations. Such phenomenon can be understood by analytic arguments which, as a by-product, also suggest an explicit way to point-wise approximate the splitting.

References

Dec 15, 1992·Physical Review. a· de Sousa Vieira MM A Lieberman
May 1, 1995·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·F Christiansen, A Politi
Oct 13, 2007·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Steven Tomsovic, Arul Lakshminarayan
Oct 13, 2007·Physical Review Letters·F GinelliA Politi
Nov 13, 2009·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Roberto Artuso, Cesar Manchein

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