On the Approximation of Complicated Dynamical Behavior

Abstract

We present efficient techniques for the numerical approximation of complicated dynamical behavior. In particular, we develop numerical methods which allow us to approximate Sinai-Ruelle-Bowen (SRB)-measures as well as (almost) cyclic behavior of a dynamical system. The methods are based on an appropriate discretization of the Frobenius-Perron operator, and two essentially different mathematical concepts are used: our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of SRB-measures by invariant measures of stochastically perturbed systems. The efficiency of the methods is illustrated by several numerical examples.Key words. computation of invariant measures, approximation of the Frobenius-Perron operator, computation of SRB-measures, almost invariant set, cyclic behavior that is, the dynamics modulo complex (unpredictable) behavior which is due to the presence of chaos.From a practical point of view the most important invariant measures are the socalled SRB-measures. The reason is that for these measures the spatial and temporal averages of observables are identical for a set of initial conditions which has positive Lebesgue measure. The introduction of the underlying concept goes back to Y. Sinai (see [27]), and the existence of these measures has been shown for Axiom A systems by D. Ruelle and R. Bowen [26,3]. In this article we suggest a numerical method for the approximation of SRB-measures, and in this context their stochastic stability is particularly important: first we use this fact as an analytical tool in our main convergence result in section 4, and second it is of practical importance if we view the numerical approximation as a small random perturbation. Indeed, stochastic stability of SRB-measures is guaranteed for Axiom A systems [17,18].More precisely there are two essential mathematical ingredients which allow us to develop a numerical method for the approximation of SRB-measures. We use a result of Yu. Kifer on the convergence of invariant measures in stochastic perturbations of the underlying dynamical system to the SRB-measure (see [17]) and combine this with results on the convergence of eigenspaces of discretized compact operators (see, e.g., [25]). The same technique is used for the approximation of the subsets in state space which are (almost) cyclically permuted by the dynamical process. With respect to the approximation of SRB-measures a similar result has previously been obtained by F. Hunt (see [15]). However, our methods are quite different from the ones used in that work. In particular, the results stated here cover the important situation where the random perturbations have a probability distribution with local support. In fact, this is the relevant case having in mind that the round off error in the numerical approximation can be interpreted as such a local perturbation. Another approach for the computation of SRB-measures-avoiding the approximation of the Frobeniu...

A scalable approach to the computation of invariant measures for high-dimensional Markovian systems

“…Lorenz-63 is a deterministic dynamical system often used as a benchmark 2,3,23,24 . We use the standard setting of Lorenz model parameters ( σ  = 10, ρ  = 28, β  = 8/3) and generate the observational data with a time step τ  = 10 −3 and an overall simulation time S  = 10 3 by means of the adaptive Runge-Kutta-method.…”

Topological Chaos and Periodic Braiding of Almost-Cyclic Sets

“…We compute the AIS, and hence the ACS, using the discretized Perron-Frobenius transfer operator, P t; f , which we approximate using a multidimensional Ulam method [16,26]. From P t; f we form a reversible matrix R t; f and determine its eigenspectrum [27].…”

“…Almost-cyclic sets [16] are closely related to almost-invariant sets (AIS) [17], which define macroscopic structures preserved by the dynamics. This generalization is an important step in making the analysis of topological chaos using the TNCT applicable to a wider range of problems, including more complex fluid systems [18] and other dynamical systems that can be represented as surface homeomorphisms [2,4].…”

Gaussian Markov transition models of molecular kinetics

“…13,7 Roughly speaking, the discretization in the MSM method can be improved in two ways. The first way is to increase the number of bins by using, e.g., subdivision 14,15 and cell-mapping 16,17 techniques, so that the state space can be finely discretized with small truncation error. But this way suffers from the "curse of dimensionality" when applied to macromolecules.…”