The baker map is investigated by two different theories of irreversibility by Prigogine and his colleagues, namely, the Λ-transformation and complex spectral theories, and their structures are compared. In both theories, the evolution operatorU† of observables (the Koopman operator) is found to acquire dissipativity by restricting observables to an appropriate subspaceΦ of the Hilbert space L2 ...

This article is about the data-driven computation of optimal control for a class affine deterministic nonlinear systems. We assume that dynamical system model not available, and only information dynamics available in form time-series data. provide convex formulation problem (OCP) system. The relies on duality result system’s stability theory involving density function Perron–Frobenius operator....

We report the numerical observation of scarring, that is enhancement probability density around unstable periodic orbits a chaotic system, in eigenfunctions classical Perron-Frobenius operator noisy Anosov ("cat") maps, as well Bunimovich stadium. A parallel drawn between and quantum scars, based on unitarity or non-unitarity respective propagators. For uniformly hyperbolic systems such cat map...

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible...

Decomposition of the high dimensional conformational space of biomolecules into metastable subsets is used for data reduction of long molecular trajectories in order to facilitate chemical analysis and to improve convergence of simulations within these subsets. The metastability is identified by the Perron-Cluster Cluster Analysis of a Markov process that describes the thermodynamic distributio...

The paper presents the concept of a new type of algorithm for the numerical computation of what the authors call the essential dynamics of molecular systems. Mathematically speaking, such systems are described by Hamiltonian differential equations. In the bulk of applications, individual trajectories are of no specific interest. Rather, time averages of physical observables or relaxation times ...

We develop the connection between large deviation theory and more applied approaches to stochastic hybrid systems by highlighting a common underlying Hamiltonian structure. A stochastic hybrid system involves the coupling between a piecewise deterministic dynamical system in Rd and a time-homogeneous Markov chain on some discrete space Γ. We assume that the Markov chain on Γ is ergodic, and tha...

The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one cor...

This note presents a new proof of basic Perron{Frobenius theory of irreducible nonnegative matrices. ABSTRACT This note presents a new proof of basic Perron{Frobenius theory of irreducible nonnegative matrices.

We construct a two dimensional random sequence of low discrepancy, using the prime field F2 of characteristic 2, and the irreducible polynomial β +β+1 = 0 over F2. This method seems to be able to extend to higher dimensional cases. We have checked in computer experiments for several cases. But until now, we can not yet prove for dimensions greater than 2.