Geometric Methods for Discrete Dynamical Systems

Erscheinungsjahr 1998
ISBN: 978-0-19-508545-7
Verlag: Oxford University Press

Buch, Englisch, Band 50, 176 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 429 g

Reihe: Oxford Engineering Science Series

This book is for those interested in dynamical systems. It assumes a solid undergraduate training in mathematics. Geometrical methods are developed to study the process of iteration, which involves taking the output of a function and feeding it back as input. Iteration processes are used to produce fractals and wavelets, and to numerically approximate solutions to ordinary and partical differential equations. Each iteration procedure generates a discrete dynamical system. These systems are at the heart of many numerical algorithms. Essentially all mathematical models of evolving physical systems can be viewed as discrete dynamical systems. This book attempts to present the fundamental ideas of discrete dynamical systems as clearly and geometrically as possible. Illustrative examples of dynamical systems are presented in the first chapter. The second chapter gives a review of the typology of metric spaces. The third presents basic results and establishes a philosophy of dynamics which is strongly influenced by the work of Charles Conley. The stable manifold and local structural stability theorems are presented in the fourth chapter. Invariant sets and isolating blocks are defined in the fifth. The sixth develops what is called the Conley Index in the context of discrete dynamics, and the final chpater covers measure-preserving and symplectic maps. The book would be suitable for use as a main text for a graduate course in dynamical systems, and as a reference for engineers and scientists.

This book is intended to be a useful reference for mathematicians, scientists, and engineers who are studying dynamical systems. Dynamics is viewed as an iteration process where the output of a function is fed back as input to determine the evolution in time of the initial state. Errors which arise from round-off in numerical simulations or from the inexactness of tthe mathematical models used to describe a process, or from the effects of external controls are incorporated into the theory. Conley's ideas about rough orbits and chain-recurrence play a central role. Generally the exposition is at the beginning graduate level and presupposes familiarity with basic courses in linear algebra and differential equations.

Autoren/Hrsg.

Easton, Robert W.

Fachgebiete

• Technische Wissenschaften Maschinenbau | Werkstoffkunde Technische Mechanik | Werkstoffkunde Statik, Dynamik, Kinetik, Kinematik
• Mathematik | Informatik Mathematik Geometrie Elementare Geometrie: Allgemeines
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Technische Wissenschaften Bauingenieurwesen Mathematische Methoden, Computeranwendungen (Bauingenieurwesen)
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie