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

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 additive white 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 of their 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 in principle to both continuous- and discrete-time dynamical systems. Numerical tests of such optimal partitions on unimodal maps support my hypothesis.

Department dc:contributor.department

Physics

Grantor dc:publisher

Georgia Institute of Technology

Year dc:date.issued

2010

Author dc:creator

• Lippolis, Domenico

Advisor dc:contributor.advisor

• Cvitanović, Predrag

Committee members dc:contributor.committeemember

• Bellissard, Jean
• Weiss, Howard
• Wiesenfeld, Kurt
• Zhou, Hao Min

• Symbolic dynamics
• Noise
• Chaos
• Periodic orbits.

Handle dc:identifier.uri

http://hdl.handle.net/1853/33841

OAI identifier oai:identifier

oai:repository.gatech.edu:1853/33841