Ebook: Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension
- Genre: Physics // Mechanics: Nonlinear dynamics and chaos
- Tags: Probability Theory and Stochastic Processes, Dynamical Systems and Ergodic Theory, Applications of Mathematics
- Series: Probability and Its Applications
- Year: 1997
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
A hundred years ago it became known that deterministic systems can exhibit very complex behavior. By proving that ordinary differential equations can exhibit strange behavior, Poincare undermined the founda tions of Newtonian physics and opened a window to the modern theory of nonlinear dynamics and chaos. Although in the 1930s and 1940s strange behavior was observed in many physical systems, the notion that this phenomenon was inherent in deterministic systems was never suggested. Even with the powerful results of S. Smale in the 1960s, complicated be havior of deterministic systems remained no more than a mathematical curiosity. Not until the late 1970s, with the advent of fast and cheap comput ers, was it recognized that chaotic behavior was prevalent in almost all domains of science and technology. Smale horseshoes began appearing in many scientific fields. In 1971, the phrase 'strange attractor' was coined to describe complicated long-term behavior of deterministic systems, and the term quickly became a paradigm of nonlinear dynamics. The tools needed to study chaotic phenomena are entirely different from those used to study periodic or quasi-periodic systems; these tools are analytic and measure-theoretic rather than geometric. For example, in throwing a die, we can study the limiting behavior of the system by viewing the long-term behavior of individual orbits. This would reveal incomprehensibly complex behavior. Or we can shift our perspective: Instead of viewing the long-term outcomes themselves, we can view the probabilities of these outcomes. This is the measure-theoretic approach taken in this book.
Content:
Front Matter....Pages i-xv
Introduction....Pages 1-6
Preliminaries....Pages 7-28
Review of Ergodic Theory....Pages 29-73
The Frobenius-Perron Operator....Pages 74-95
Absolutely Continuous Invariant Measures....Pages 96-109
Other Existence Results....Pages 110-126
Spectral Decomposition of the Frobenius-Perron Operator....Pages 127-138
Properties of Absolutely Continuous Invariant Measures....Pages 139-173
Markov Transformations....Pages 174-208
Compactness Theorem and Approximation of Invariant Densities....Pages 209-225
Stability of Invariant Measures....Pages 226-251
The Inverse Problem for the Frobenius-Perron Equation....Pages 252-259
Applications....Pages 260-309
Back Matter....Pages 310-399
Content:
Front Matter....Pages i-xv
Introduction....Pages 1-6
Preliminaries....Pages 7-28
Review of Ergodic Theory....Pages 29-73
The Frobenius-Perron Operator....Pages 74-95
Absolutely Continuous Invariant Measures....Pages 96-109
Other Existence Results....Pages 110-126
Spectral Decomposition of the Frobenius-Perron Operator....Pages 127-138
Properties of Absolutely Continuous Invariant Measures....Pages 139-173
Markov Transformations....Pages 174-208
Compactness Theorem and Approximation of Invariant Densities....Pages 209-225
Stability of Invariant Measures....Pages 226-251
The Inverse Problem for the Frobenius-Perron Equation....Pages 252-259
Applications....Pages 260-309
Back Matter....Pages 310-399
....