State Space Partitions of Stochastic Chaotic Maps
Heninger, Jeffrey M.
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The finest resolution that can be achieved in any real chaotic system is limited by the presence of noise. This noise can be used to define neighborhoods of the deterministic periodic orbits using the local eigenfunctions of the Fokker-Planck operator and its adjoint. We extend the work of D. Lippolis to include hyperbolic periodic orbits. The dynamics along the stable and unstable directions are separated. The neighborhoods on the stable and unstable manifolds can be defined in the same way as the neighborhoods for entirely stable or entirely unstable orbits. The neighborhoods are then returned to the original coordinates. The Fokker-Planck evolution can be described as a finite Markov transition graph between these neighborhoods. Its spectral determinant is used to calculate the time averages of observables. We apply this technique to calculate long time observables of the Lozi map.