The subleading eigenvalues and associated eigenfunctions of the Perron-Frobenius operator for 2-dimensional area-preserving maps are numerically investigated. We closely examine validity so-called Ulam method, a numerical scheme believed to provide operator, both linear nonlinear on torus. For case, second-largest investigated by calculating Fokker-Planck with sufficiently small diffusivity. On...

For a piecewise linear intermittent map, the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the Frobenius-Perron operator P̂ are explicitly derived. The evolution of the averages are shown to be a superposition of the contributions from two simple eigenvalues 1 and λd ∈ (−1, 0), an...

A matrix is said to have the Perron-Frobenius property if its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are pro...

This article provides sufficient conditions for positive maps on the Schatten classes Jp; 1 p < 1 of bounded operators on a separable Hilbert space such that a corresponding Perron-Frobenius theorem holds. With applications in quantum information theory in mind sufficient conditions are given for a trace preserving, positive map on J1, the space of trace class operators, to have a unique, stric...

We provide a proof for a Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators based on the strong maximum principle. This proof is modeled after a variational proof of Perron’s theorem for matrices with positive entries that does not appeal to Perron-Frobenius theory.

We consider the collective dynamics in an ensemble of globally coupled chaotic maps. The transition to the coherent state with a macroscopic mean field is analyzed in the framework of the linear response theory. The linear response function for the chaotic system is obtained using the perturbation approach to the Frobenius-Perron operator. The transition point is defined from this function by v...

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...

Quantum and classical correlations are studied experimentally in model n-disk microwave billiards. The wave vector kappa autocorrelation C(kappa) of the quantum spectrum displays nonuniversal oscillations for large kappa, comparable to the universal random matrix theory behavior observed for small kappa. The nonuniversal behavior is shown to be completely determined by the classical Ruelle-Poll...

We consider the approximation of fluctuation induced almost invariant sets arising from stochastic dynamical systems. The dynamical evolution of densities is derived from the stochastic Frobenius-Perron operator. Given a stochastic kernel with a known distribution, approximate almost invariant sets are found by translating the problem into an eigenvalue problem derived from reversible Markov pr...

Definition 1. OG is the set of real symmetric V_V matrices with entries ai, j such that ai, j<0 if [i, j] # E and ai, j=0 if i{ j and [i, j] E. An operator A # OG has a non-degenerate first eigenvalue *1 (groundstate) if G is connected (Perron and Frobenius). The invariant +(G) is defined using multiplicities of the second eigenvalue *2 for some real symmetric matrix A # OG . Moreover, +(G) is ...