Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system

Abstract

The finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise. In the weak-noise approximation the neighborhoods of deterministic periodic orbits can be computed as distributions stationary under the action of a local Fokker-Planck operator and its adjoint. We derive explicit formulae for widths of these distributions in the case of chaotic dynamics, when the periodic orbits are hyperbolic. The resulting neighborhoods form a basis for functions on the attractor. The global stationary distribution, needed for calculation of long-time expectation values of observables, can be expressed in this basis.

6 pages, 3 figures

Cited by in corpus (4)

• Statistics of the stochastically-forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions
• Perturbation theory for the Fokker-Planck operator in chaos
• Computing Chaotic Time-Averages from Few Periodic or Non-Periodic Orbits
• Towards the resolution of a quantized chaotic phase space: The interplay of dynamics with noise