Knowing when to stop: How noise frees us from determinism

Abstract

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...

Perturbation theory for the Fokker–Planck operator in chaos

“…Time is referenced by n. i and j range from 1 to d. Here we shall consider the evolution of densities of trajectories, according to the Fokker-Planck picture [21]. In discrete time, a distribution moves one time step according to the operator [22,23]…”

“…the relative insensitivity to fluctuations demonstrated constitutes evidence that a chaotic system acquires a finite resolution when noise is introduced, and it can be modelled via a transfer operator of finite degrees of freedom. This problem is discussed extensively in [22,23,31]; 2. the analysis presented sets bounds for the robustness of a model, and it is straightforward enough to be implemented in any algorithm that creates a template out of a time series. It is noted that low-dimensional, noisy discrete-time mappings are still widely used as models in several fields of science and engineering [32,33,34].…”

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

“…This problem was solved in refs. [8,9] for repelling periodic orbits with no contracting directions, by balancing the stochastic diffusion against the contraction by the adjoint Fokker-Planck operator. The resulting covariance matrix defines the stochastic neighborhood for a repelling orbit, while the Ornstein-Uhlenbeck covariance defines it for a stable orbit.…”

“…All of these approaches (see ref. [9] for a review) are based on global averages, and assume that granularity is uniform across the state space. In contrast, the main, computationally precise lesson of our work is that even when the external noise is white, additive, and globally homogenous, the interplay of noise and nonlinear dynamics always results in a local stochastic neighborhood, whose covariance depends on both the past and the future noise integrated and non-linearly convolved with deterministic evolution along the trajectory.…”

“…In other words, the two covariance matrices, (i) the deterministically transported Q a → M a Q a M a , and (ii) the noise diffusion tensor ∆ a , add together in the usual manner, as squares of errors. Similarly, density evolution for dynamics with strictly expanding Jacobian matrices M a can be described by the action of the adjoint Fokker-Planck operator [8,9], with kernel…”