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. We propose to determine the 'finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise, by computing the local eigenfunctions of the adjoint Fokker-Planck operator along each periodic point, 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. Numerical tests of such 'optimal partition' of a onedimensional repeller support our hypothesis.

This corrects the article DOI: 10.1103/PhysRevE.93.040201.

Deterministic chaotic dynamics presumes that the state space can be partitioned arbitrarily finely. In a physical system, the inevitable presence of some noise sets a finite limit to the finest possible resolution that can be attained. Much previous research deals with what this attainable resolution might be, all of it based on a global averages over stochastic flow. We show how to compute the locally optimal partition, for a given dynamical system and given noise, in terms of local eigenfunctions of the Fokker-Planck operator and its adjoint. We first analyze the interplay of the deterministic dynamics with the noise in the neighborhood of a periodic orbit of a map, by using a discretized version of Fokker-Planck formalism. Then we propose a method to determine the 'optimal resolution' of the state space, based on solving Fokker-Planck's equation locally, on sets of unstable periodic orbits of the deterministic system. We test our hypothesis on unimodal maps. Source: ChaosBook.orgFIGURE 2. (a) A deterministic partition of state space M . (b) Noise blurs the partition boundaries. At some level of deterministic partitioning some of the boundaries start to overlap, preventing further local refinement of adjecent neighborhoods. (c) The fixed point 1 = {x 1 } and the two-cycle 01 = {x 01 , x 10 } are examples of the shortest period periodic points within the partition regions of (a). The optimal partition hypothesis: (d) Noise blurs periodic points into cigar-shaped trajectory-centered densities explored by the Langevin noise. The optimal partition hypothesized in this paper consists of the maximal set of resolvable periodic point neighborhoods.of periodic orbits increase, the diffusion always wins, and successive refinements of a deterministic partition of the state space stop at the finest attainable partition, beyond which the diffusive smearing exceeds the size of any deterministic subpartition.There is a considerable literature (reviewed here in remark 5.1) on interplay of noise and chaotic deterministic dynamics, and the closely related problem of limits on the validity of the semi-classical periodic orbit quantization. All of this literature implicitly assumes uniform hyperbolicity and seeks to define a single, globally averaged, diffusion induced average resolution (Heisenberg time, in the context of semi-classical quantization). However, the local diffusion rate differs from a trajectory to a trajectory, as different neighborhoods merge at different times, so there is no one single time beyond which noise takes over. Nonlinear dynamics interacts with noise in a nonlinear way, and methods for implementing the optimal partition for a given noise still need to be developed. This paper is an attempt in this direction. Here we follow and expand upon the Fokker-Planck approach to the 'optimal partition hypothesis' introduced in ref. [7].What is novel here is that we show how to compute the locally optimal partition, for a given dynamical system and given noise, in terms of local eigenfunctions of the forward-back...

A quantized chaotic map exhibiting localization of wave function intensity is opened. We investigate how such patterns as scars in the Husimi distributions are influenced by the losses through a number of numerical experiments. We find the scars to relocate on the stable or unstable manifolds depending on the position of the opening, and provide a classical argument to explain the observations. For asymmetrically introduced openings mode interaction contributes to determine the localization patterns. We finally show examples of similar localization in a simulated dielectric microcavity.

Coupled map lattices have been widely used as models in several fields of physics, such as chaotic strings, turbulence, and phase transitions, as well as in other disciplines, such as biology (ecology, evolution) and information processing. This paper investigates properties of periodic orbits in two coupled Tchebyscheff maps. Then zeta function cycle expansions are used to compute dynamical averages appearing in Beck's theory of chaotic strings. The results show close agreement with direct simulation for most values of the coupling parameter, and yield information about the system complementary to that of direct simulation.