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This book is devoted to the subject commonly called Chaotic Dynamics, namely the study of complicated behavior in time of maps and ?ows, called dynamical systems. The theory of chaotic dynamics has a deep impact on our understanding of - ture, and we sketch here our view on this question. The strength of this theory comes from its generality, in that it is not limited to a particular equation or scienti?c - main. It should be viewed as a conceptual framework with which one can capture properties of systems with complicated behavior. Obviously, such a general fra- work cannot describe a system down to its most intricate details, but it is a useful and important guideline on how a certain kind of complex systems may be understood and analyzed. The theory is based on a description of idealized systems, such as "hyperbolic" systems. The systems to which the theory applies should be similar to these idealized systems. They should correspond to a ?xed evolution equation, which, however, need to be neither modeled nor explicitly known in detail. Experimentally, this means that the conditions under which the experiment is performed should be as constant as possible. The same condition applies to analysis of data, which, say, come from the evolution of glaciations: One cannot apply "chaos theory" to systems under varying external conditions, but only to systems which have some self-generated chaos under ?xed external conditions.
List of contents
A Basic Problem.- Dynamical Systems.- Topological Properties.- Hyperbolicity.- Invariant Measures.- Entropy.- Statistics and Statistical Mechanics.- Other Probabilistic Results.- Experimental Aspects.
Summary
This book is devoted to the subject commonly called Chaotic Dynamics, namely the study of complicated behavior in time of maps and ?ows, called dynamical systems. The theory of chaotic dynamics has a deep impact on our understanding of - ture, and we sketch here our view on this question. The strength of this theory comes from its generality, in that it is not limited to a particular equation or scienti?c - main. It should be viewed as a conceptual framework with which one can capture properties of systems with complicated behavior. Obviously, such a general fra- work cannot describe a system down to its most intricate details, but it is a useful and important guideline on how a certain kind of complex systems may be understood and analyzed. The theory is based on a description of idealized systems, such as “hyperbolic” systems. The systems to which the theory applies should be similar to these idealized systems. They should correspond to a ?xed evolution equation, which, however, need to be neither modeled nor explicitly known in detail. Experimentally, this means that the conditions under which the experiment is performed should be as constant as possible. The same condition applies to analysis of data, which, say, come from the evolution of glaciations: One cannot apply “chaos theory” to systems under varying external conditions, but only to systems which have some self-generated chaos under ?xed external conditions.
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From the reviews:
"The book is a good introduction to the field of dynamical systems with a particular emphasis on statistical properties and applications. In particular, the relations both with real experiments with numerical simulations are discussed. … The book contains many figures that really help the understanding of the text. The book can be used as a text for an introductory course in dynamical systems (at the master’s or Ph.D. level). It is particularly suited for students with interests in applications (either physics, economy or biology)." (Carlangelo Liverani, Mathematical Reviews, Issue 2007 m)
"Two thoughts crossed my mind when I picked up this book. The first was: ‘what a physically attractive book.’ The second was: ‘what a short book to have on such a wide ranging topic.’ … It is a perfect size to carry in a knapsack, the print is clear and the layout of text, equations, and figures is marvelously done. … images are multi-colored stereo images, and allow the reader to ‘see’ a three dimensional effect that helps illustrate the phenomena." (David S. Mazel, MathDL, December, 2007)
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From the reviews:
"The book is a good introduction to the field of dynamical systems with a particular emphasis on statistical properties and applications. In particular, the relations both with real experiments with numerical simulations are discussed. ... The book contains many figures that really help the understanding of the text. The book can be used as a text for an introductory course in dynamical systems (at the master's or Ph.D. level). It is particularly suited for students with interests in applications (either physics, economy or biology)." (Carlangelo Liverani, Mathematical Reviews, Issue 2007 m)
"Two thoughts crossed my mind when I picked up this book. The first was: 'what a physically attractive book.' The second was: 'what a short book to have on such a wide ranging topic.' ... It is a perfect size to carry in a knapsack, the print is clear and the layout of text, equations, and figures is marvelously done. ... images are multi-colored stereo images, and allow the reader to 'see' a three dimensional effect that helps illustrate the phenomena." (David S. Mazel, MathDL, December, 2007)
