Probabilistic Properties of Deterministic Systems

This book shows how densities arise in simple deterministic systems. There has been explosive growth in interest in physical, biological and economic systems that can be profitably studied using densities. Due to the inaccessibility of the mathematical literature there has been little diffusion of the applicable mathematics into the study of these 'chaotic' systems. This book will help to bridge that gap. The authors give a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial differential equations. They have drawn examples from many scientific fields to illustrate the utility of the techniques presented. The book assumes a knowledge of advanced calculus and differential equations, but basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed.

1. Introduction; 2. The toolbox; 3. Markov and Frobenius–Perron operators; 4. Studying chaos with densities; 5. The asymptotic properties of densities; 6. The behaviour of transformations on intervals and manifolds; 7. Continuous time systems: an introduction; 8. Discrete time processes embedded in continuous time systems; 9. Entropy; 10. Stochastic perturbation of discrete time systems; 11. Stochastic perturbation of continuous time systems.