The Transition to Chaos

1. Front Matter

   Pages i-xvi

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2. Overview

   1. Overview

      • L. E. Reichl

      Pages 1-13

3. Classical Systems

   1. Fundamental Concepts

      • L. E. Reichl

      Pages 14-65

   2. Area Preserving Maps

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      Pages 66-155

   3. Global Properties

      • L. E. Reichl

      Pages 156-221

4. Quantum Systems

   1. Quantum Integrability

      • L. E. Reichl

      Pages 222-247

   2. Random Matrix Theory

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      Pages 248-286

   3. Observed Spectra

      • L. E. Reichl

      Pages 287-317

   4. Semi-Classical Theory — Path Integrals

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      Pages 318-381

   5. Driven Systems

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      Pages 382-444

5. Stochastic Systems

   1. Stochastic Systems

      • L. E. Reichl

      Pages 445-458

6. Back Matter

   Pages 459-551

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resonances. Nonlinear resonances cause divergences in conventional perturbation expansions. This occurs because nonlinear resonances cause a topological change locally in the structure of the phase space and simple perturbation theory is not adequate to deal with such topological changes. In Sect. (2.3), we introduce the concept of integrability. A sys­ tem is integrable if it has as many global constants of the motion as degrees of freedom. The connection between global symmetries and global constants of motion was first proven for dynamical systems by Noether [Noether 1918]. We will give a simple derivation of Noether's theorem in Sect. (2.3). As we shall see in more detail in Chapter 5, are whole classes of systems which are now known to be inte­ there grable due to methods developed for soliton physics. In Sect. (2.3), we illustrate these methods for the simple three-body Toda lattice. It is usually impossible to tell if a system is integrable or not just by looking at the equations of motion. The Poincare surface of section provides a very useful numerical tool for testing for integrability and will be used throughout the remainder of this book. We will illustrate the use of the Poincare surface of section for classic model of Henon and Heiles [Henon and Heiles 1964].

• Evolution
• chaos
• chemistry
• path integral
• physics

From the reviews of the second edition:

"This book is an expanded and updated version … from a previous edition and reviews results on the manifestation of chaos in classical and quantum mechanics. … A very wide range of topics is covered in the book, which thus can be used as preliminary reading for research areas … . The book can also be considered as a helpful companion both for mathematicians and for physicists. … Many technical details and background notions can be found in a rich complement of appendices." (Guido Gentile, Mathematical Reviews, Issue 2006 c)

• Center for Statistical Mechanics and Complex Systems, Department of Physics, University of Texas at Austin, Austin, USA

  L. E. Reichl

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