Stationary Density Computation of the Frobenius-Perron Operators Based on the Dirac Delta Function

The statistical study of chaotic dynamical systems has received a great deal of attention in the past several decades. As a branch of applied mathematics, its application has been found in various fields in science and engineering, while the theory and methods for the existence and computation of absolutely invariant measures have played an important role in this field. In this study, we focus on the computation of a nontrivial fixed point of Frobenius-Perron operators (F-P operators). Let S: [0,1] → [0,1] be a piecewise monotonic mapping, and let PS : [0,1] → [0,1] be the Frobenius-Perron operators associated with S, which is defined by PSf(x) = d/dx ∫S-1([0,x]) fdm, x ∈ [0,1] a.e., where m is the Lebesgue measure of [0,1]. Suppose that PS: [0,1] → [0,1] has a stationary density f*. By using Ulam\u27s method, which he proposed based on a probability argument, approximating the fixed density function f* can be constructed by piecewise constant functions with respect to a partition of [0,1]. From another argument, we propose a different form for the definition of the Frobenius-Penon operator by combining the properties of the Dirac delta function. We can prove that the two definitions for Frobenius-Perron operators are equivalent. Then, we find that by approximating the Dirac delta function, we can exactly obtain the famous Ulam\u27s method again. For the computation of fixed density functions we use the quasi- Monte Carlo method. We partition [0,1] into n subintervals, and for each subinterval we take N equal distance test points. Numerical results are given for several one dimensional test mappings

A boundary integral formalism for stochastic ray tracing in billiards

Determining the flow of rays or non-interacting particles driven by a force or velocity field is fundamental to modelling many physical processes. These include particle flows arising in fluid mechanics and ray flows arising in the geometrical optics limit of linear wave equations. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain

Finite Element Maximum Entropy Method for Approximating Absolutely Continuous Invariant Measures

In a chaotic dynamical system, the eventual behavior of iterates of initial points of a map is unpredictable even though the map is deterministic. A system which is chaotic in a deterministic point of view may be regular in a statistical viewpoint. The statistical viewpoint requires the study of absolutely continuous invariant measure (ACIM) of a map with respect to the Lebesgue measure. An invariant density of the Frobenius-Perron (F-P) operator associated with a nonsingular map is employed to evaluate an ACIM of the map. The ACIM is a key factor for studying the eventual behavior of iterates of almost all initial points of the map. It is difficult to obtain an invariant density of the F-P operator in an exact mathematical form except for some simple maps. Different numerical schemes have been developed to approximate such densities. The maximum entropy principle gives a criterion to select a least-biased density among all densities satisfying a system of moment equations. In this principle, a least-biased density maximizes the Boltzmann entropy. In this dissertation, piecewise quadratic functions and quadratic splines are used in the maximum entropy method to calculate the L1 errors between the exact and the approximate invariant densities of the F-P operator associated with nonsingular maps defined from [0;1] to itself. The numerical results are supported by rigorous mathematical proofs. The L1 errors between the exact and approximate invariant densities of the Markov operator associated with Markov type position dependent random maps, defined from [0;1] to itself, are calculated by using the piecewise linear polynomials maximum entropy method

Identifying almost invariant sets in stochastic dynamical systems

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 processes. Analytic and computational examples of the methods are used to illustrate the technique, and are shown to reveal the probability transport between almost invariant sets in nonlinear stochastic systems. Both small and large noise cases are considered. © 2008 American Institute of Physics

A boundary integral method for modelling vibroacoustic energy distributions in uncertain built up structures

A phase-space boundary integral method is developed for modelling stochastic high-frequency acoustic and vibrational energy transport in both single and multi-domain problems. The numerical implementation is carried out using the collocation method in both the position and momentum phase-space variables. One of the major developments of this work is the systematic convergence study, which demonstrates that the proposed numerical schemes exhibit convergence rates that could be expected from theoretical estimates under the right conditions. For the discretisation with respect to the momentum variable, we employ spectrally convergent basis approximations using both Legendre polynomials and Gaussian radial basis functions. The former have the advantage of being simpler to apply in general without the need for preconditioning techniques. The Gaussian basis is introduced with the aim of achieving more efficient computations in the weak noise case with near-deterministic dynamics. Numerical results for a series of coupled domain problems are presented, and demonstrate the potential for future applications to larger scale problems from industry

Solutions of the Inverse Frobenius-Perron Problem

The Frobenius-Perron operator describes the evolution of density functions in a dynamical system. Finding the �fixed points of this operator is referred to as the Frobenius-Perron problem. This thesis discusses the inverse Frobenius-Perron problem (IFPP), which seeks the transformation that generates a prescribed invariant probability density. In particular, we present in detail five different ways of solving the IFPP, including approaches using conjugation and differential equation, and two matrix solutions. We also generalize Pingel's method to the case of two-pieces maps

Improved approximation of phase-space densities on triangulated domains using Discrete Flow Mapping with p-refinement

We consider the approximation of the phase-space flow of a dynamical system on a triangulated surface using an approach known as Discrete Flow Mapping. Such flows are of interest throughout statistical mechanics, but the focus here is on flows arising from ray tracing approximations of linear wave equations. An orthogonal polynomial basis approximation of the phase-space density is applied in both the position and direction coordinates, in contrast with previous studies where piecewise constant functions have typically been applied for the spatial approximation. In order to improve the tractability of an orthogonal polynomial approximation in both phase-space coordinates, we propose a careful strategy for computing the propagation operator. For the favourable case of a Legendre polynomial basis we show that the integrals in the definition of the propagation operator may be evaluated analytically with respect to position and via a spectrally convergent quadrature rule for the direction coordinate. A generally applicable spectral quadrature scheme for integration with respect to both coordinates is also detailed for completeness. Finally, we provide numerical results that motivate the use of p-refinement in the orthogonal polynomial basis

Uncertainty quantification for phase-space boundary integral models of ray propagation

Vibrational and acoustic energy distributions of wave fields in the high-frequency regime are often modeled using flow transport equations. This study concerns the case when the flow of rays or non-interacting particles is driven by an uncertain force or velocity field and the dynamics are determined only up to a degree of uncertainty. A boundary integral equation description of wave energy flow along uncertain trajectories in finite two-dimensional domains is presented, which is based on the truncated normal distribution, and interpolates between a deterministic and a completely random description of the trajectory propagation. The properties of the Gaussian probability density function appearing in the model are applied to derive expressions for the variance of a propagated initial Gaussian density in the weak noise case. Numerical experiments are performed to illustrate these findings and to study the properties of the stationary density, which is obtained in the limit of infinitely many reflections at the boundary

INVARIANT MEASURES OF STOCHASTIC PERTURBATIONS OF DYNAMICAL SYSTEMS USING FOURIER APPROXIMATIONS

We consider dynamical system τ : [0, 1] → [0, 1] and its stochastic perturbations , N ≥ 1. Using Fourier approximation, we construct a finite dimensional approximation PN to a perturbed Perron–Frobenius operator. Let be an invariant density of τ and be a fixed point of PN. We show that converges in L1 to