In this chapter we introduce the concept of Frobenius-Perron operators, that is used to study the stochastic aspects of chaotic dynamical systems. We study basic properties and some existence results on fixed densities of this operator. Ulam's method as well as the direct iteration method are also introduced for the computation of the fixed density which gives an absolutely continuous invariant measure underlying the statistical property of the dynamics. The theory and methods are applied to investigate chaotic behavior of electrical circuits.

11.1 Introduction

Studying statistical properties of chaotic deterministic dynamical systems leads to the concept of Frobenius-Perron operators, which has been investigated from both theoretical and numerical viewpoints in the past several decades ^[1], ^[2]. In this chapter we introduce the concept of Frobenius-Perron operators and study their properties. For the purpose of applying this class of operators to many physical science fields, such as chaotic electrical circuits which will be investigated in detail later, we will describe the direct iteration method and the famous Ulam's piecewise constant approximations method for the computation of the fixed density of the Frobenius-Perron operator.

The study of absolutely continuous invariant measures has its origin in the examination of the Boltzmann ergodic hypothesis in statistical physics, which is related to the investigation of measure-preserving...