Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 Introduction
- Chapter 2 The toolbox
- Chapter 3 Markov and Frobenius–Perron operators
- Chapter 4 Studying chaos with densities
- Chapter 5 The asymptotic properties of densities
- Chapter 6 The behavior of transformations on intervals and manifolds
- Chapter 7 Continuous time systems: an introduction
- Chapter 8 Discrete time processes embedded in continuous time systems
- Chapter 9 Entropy
- Chapter 10 Stochastic perturbation of discrete time systems
- Chapter 11 Stochastic perturbation of continuous time systems
- References
- Notation and symbols
- Index
Chapter 7 - Continuous time systems: an introduction
Published online by Cambridge University Press: 05 March 2012
- Frontmatter
- Contents
- Preface
- Chapter 1 Introduction
- Chapter 2 The toolbox
- Chapter 3 Markov and Frobenius–Perron operators
- Chapter 4 Studying chaos with densities
- Chapter 5 The asymptotic properties of densities
- Chapter 6 The behavior of transformations on intervals and manifolds
- Chapter 7 Continuous time systems: an introduction
- Chapter 8 Discrete time processes embedded in continuous time systems
- Chapter 9 Entropy
- Chapter 10 Stochastic perturbation of discrete time systems
- Chapter 11 Stochastic perturbation of continuous time systems
- References
- Notation and symbols
- Index
Summary
In previous chapters we concentrated on discrete time systems because they offer a convenient way of introducing many concepts and techniques of importance in the study of irregular behaviors in model systems. Now we turn to a study of continuous time systems.
Continuous and discrete systems differ in several important and interesting ways, which we will touch on throughout the remainder of this book. For example, in a continuous time system, complicated irregular behaviors are possible only if the dimension of the phase space of the system is three or greater. As we have seen, this is in sharp contrast to discrete time processes that can have extremely complicated dynamics in only one dimension. Further, continuous time processes in a finite dimensional phase space are in general invertible, which immediately implies that exactness is a property that will not occur for these systems (recall that noninvertibility is a necessary condition for exactness). However, systems in an infinite dimensional phase space, namely, time delay equations and some partial differential equations, are generally not invertible and, thus, may display exactness.
This chapter is devoted to an introduction of the concept of continuous time systems, an extension of many properties developed previously for discrete time systems, and the development of tools and techniques specifically designed for studying continuous time systems.
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- Chapter
- Information
- Probabilistic Properties of Deterministic Systems , pp. 163 - 218Publisher: Cambridge University PressPrint publication year: 1985