We present eecient techniques for the numerical approximation of complicated dynamical behavior. In particular, we develop numerical methods which allow to approximate SBR-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 diierent mathematical concepts are used: the ide...

We give a new proof of a result due to Ruelle about the existence and simplicity of a unique maximal eigenvalue for a Ruelle^Perron^Frobenius operator acting on some Ho« lder continuous function space. Mathematics Subject Classi¢cations (1991): Primary 58F23, Secondary 30C62. Key words: locally expanding, mixing, Ruelle^Perron^Frobenius operator, maximal eigenvalue. 1. Introduction Ruelle's The...

Some of the main results of the Perron-Frobenius theory of square nonnegative matrices over the reals are extended to matrices with elements in a real closed field. We use the results to prove the existence of a fractional power series expansion for the Perron-Frobenius eigenvalue and normalized eigenvector of real, square, nonnegative, irreducible matrices which are obtained by perturbing a (p...

We analyze the effects of stochastic perturbations in a physical example occurring as a higher-dimensional dynamical system. The physical model is that of a class- B laser, which is perturbed stochastically with finite noise. The effect of the noise perturbations on the dynamics is shown to change the qualitative nature of the dynamics experimentally from a stochastic periodic attractor to one ...

We prove comparison theorems for norms of iteration matrices in splittings of matrices in the setting of proper cones in a finite dimensional real space by considering cone linear absolute norms and cone max norms. Subject to mild additional hypotheses, we show that these comparison theorems can hold only for such norms within the class of cone absolute norms. Finally, in a Banach algebra setti...

This paper is concerned with properties of the algebraic degree of the Laplace–Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polyno...

The celebrated Perron–Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. H...

We define a (chaotic) deterministic variant of random multiplicative cascade models of turbulence. It preserves the hierarchical tree structure , thanks to the addition of infinitesimal noise. The zero-noise limit can be handled by Perron-Frobenius theory, just as the zero-diffusivity limit for the fast dynamo problem. Random multiplicative models do not possess Kolmogorov 1941 (K41) scaling be...

Basic properties in Perron-Frobenius theory are strict positivity, primitivity, and irreducibility. Whereas for nonnegative matrices, these properties are equivalent to elementary graph properties which can be checked in polynomial time, we show that for Kraus maps the noncommutative generalization of stochastic matrices checking strict positivity (whether the map sends the cone to its interior...

We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (In...