Fluctuations of the mean field of a globally coupled dynamical systems are discussed. The origin of hidden coherence is related with the instability of the fixed point solution of the self-consistent Perron-Frobenius equation. Collective dynamics in globally coupled tent maps are re-examined, both with the help of direct simulation and the Perron-Frobenius equation. Collective chaos in a single...

The spectral properties of the Frobenius-Perron operator of one-dimensional maps are studied when approaching a weakly intermittent situation. Numerical investigation of a particular family of maps shows that the spectrum becomes extremely dense and the eigenfunctions become concentrated in the vicinity of the intermittent fixed point. Analytical considerations generalize the results to a broad...

Conditions are investigated which guarantee exactness for measurable maps on measure spaces. The main application is to certain piecewise continuous maps T on [0,1] for which 7"(0) > 1. We assume [0,1] can be broken into intervals on which T is continuous and convex and at the left end of these intervals T = 0 and dt/dx > 0. Such maps have an invariant absolutely continuous density which is exact.

A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius prope...

A basic systems question concerns the concept of closure, meaning autonomomy (closed) in the sense of describing the (sub)system as fully consistent within itself. Alternatively, the system may be nonautonomous (open) meaning it receives influence from an outside coupling subsystem. Information flow, and related causation inference, are tenant on this simple concept. We take the perspective of ...

A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius prope...

Recently, Storm [10] defined the Ihara-Selberg zeta function of a hypergraph, and gave two determinant expressions of it by the Perron-Frobenius operator of a digraph and a deformation of the usual Laplacian of a graph. We present a new determinant expression for the Ihara-Selberg zeta function of a hypergraph, and give a linear algebraic proof of Storm’s Theorem. Furthermore, we generalize the...

We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the Perron-Frobenius operator of an expanding map T of [0, 1] with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of f ◦ T , when f belongs to a large class of unbounded functions from [0, 1] to R. We also prove other limit theorems and momen...

This paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius-Perron operator for such maps. The method is applied to es...

For a transformation F on a measure space (X,μ), we show that the Perron-Frobenius operator of F can be written by a representation (L2(X,μ), π) of the Cuntz-Krieger algebra OA associated with F when F satisfies some assumption. Especially, when OA is the Cuntz algebra ON and (L2(X,μ), π) in the above is some irreducible representation of ON , then there is an F -invariant measure on X which is...