How well can one resolve the state space of a chaotic map?

Symbolic dynamics;Noise;Chaos;Periodic orbits.

Lippolis, Domenico ; Physics

University:Georgia Institute of Technology

Department:Physics

关键词: Symbolic dynamics;    Noise;    Chaos;    Periodic orbits.;   

Others  :  https://smartech.gatech.edu/bitstream/1853/33841/1/lippolis_domenico_201005_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. My goal is to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additivewhite noise. That is achieved by computing the local eigenfunctions of the Fokker-Planck evolution operator in linearized neighborhoods of the periodic orbits of the corresponding deterministic system, and using overlaps oftheir widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. The method applies inprinciple to both continuous- and discrete-time dynamical systems. Numerical tests of such optimal partitions on unimodal maps support my hypothesis.