M Combescure 2005 J. Phys. A: Math. Gen. 38 5380 doi:10.1088/0305-4470/38/23/B02
Hamiltonian Chaos and Fractional Dynamics
FREE ARTICLEM Combescure
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This book provides an introduction and discussion of the main issues in the current understanding of classical Hamiltonian chaos, and of its fractional space-time structure. It also develops the most complex and open problems in this context, and provides a set of possible applications of these notions to some fundamental questions of dynamics: complexity and entropy of systems, foundation of classical statistical physics on the basis of chaos theory, and so on.
Starting with an introduction of the basic principles of the Hamiltonian theory of chaos, the book covers many topics that can be found elsewhere in the literature, but which are collected here for the readers' convenience. In the last three parts, the author develops topics which are not typically included in the standard textbooks; among them are:
- the failure of the traditional description of chaotic dynamics in terms of diffusion equations;
- he fractional kinematics, its foundation and renormalization group analysis;
- `pseudo-chaos', i.e. kinetics of systems with weak mixing and zero Lyapunov exponents;
- directional complexity and entropy.
The purpose of this book is to provide reearchers and students in physics, mathematics and engeenering with an overview of many aspects of chaos and fractality in Hamiltonian dynamical systems. In my opinion it achieves this aim, at least provided researchers and students (mainly those involved in mathematical physics) can complement this reading with comprehensive material from more specialized sources which are provided as references and `further reading'.
Each section contains introductory pedagogical material, often illustrated by figures coming from several numerical simulations which give the feeling of what's going on, and thus is very useful to the reader who is not very familiar with the topics presented. Some problems are included at the end of most sections to help the reader to go deeper into the subject.
My one regret is that the book does not mention the famous `Shadowing Lemma' of Anosov and Bowen for hyperbolic systems.
- PACS
- MSC
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81S40 Path integrals (See also 58D30)
81T13 Yang-Mills and other gauge theories (See also 53C07, 58E15)
- Subjects
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Statistical physics and nonlinear systems
- Dates
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Issue 23 ( 10 June 2005)
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Hamiltonian Chaos and Fractional Dynamics
M Combescure 2005 J. Phys. A: Math. Gen. 38 5380
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