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Chaos : from simple models to complex systems /
Chaos : from simple models to complex systems aims to guide science and engineering students through chaos and nonlinear dynamics from classical examples to the most recent fields of research. The first part, intended for undergraduate and graduate students, is a gentle and self-contained introducti...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Hackensack, NJ :
World Scientific,
©2010.
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| Colección: | Series on advances in statistical mechanics ;
v. 17. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Introduction to dynamical systems and chaos. 1. First encounter with chaos. 1.1. Prologue. 1.2. The nonlinear pendulum. 1.3. The damped nonlinear pendulum. 1.4. The vertically driven and damped nonlinear pendulum. 1.5. What about the predictability of pendulum evolution? 1. 6. Epilogue. 2. The language of dynamical systems. 2.1. Ordinary Differential Equations (ODE). 2.2. Discrete time dynamical systems : maps. 2.3. The role of dimension. 2.4. Stability theory. 2.5. Exercises. 3. Examples of chaotic behaviors. 3.1. The logistic map. 3.2. The Lorenz model. 3.3. The Hénon-Heiles system. 3.4. What did we learn and what will we learn? 3.5. Closing remark. 3.6. Exercises. 4. Probabilistic approach to chaos. 4.1. An informal probabilistic approach. 4.2. Time evolution of the probability density. 4.3. Ergodicity. 4.4. Mixing. 4.5. Markov chains and chaotic maps. 4.6. Natural measure. 4.7. Exercises. 5. Characterization of chaotic dynamical systems. 5.1. Strange attractors. 5.2. Fractals and multifractals. 5.3. Characteristic Lyapunov exponents. 5.4. Exercises. 6. From order to chaos in dissipative systems. 6.1. The scenarios for the transition to turbulence. 6.2. The period doubling transition. 6.3. Transition to chaos through intermittency : Pomeau-Manneville scenario. 6.4. A mathematical remark. 6.5. Transition to turbulence in real systems. 6.6. Exercises. 7. Chaos in Hamiltonian systems. 7.1. The integrability problem. 7.2. Kolmogorov-Arnold-Moser theorem and the survival of tori. 7.3. Poincaré-Birkhoff theorem and the fate of resonant tori. 7.4. Chaos around separatrices. 7.5. Melnikov's theory. 7.6. Exercises
- Advanced topics and applications : from information theory to turbulence. 8. Chaos and information theory. 8.1. Chaos, randomness and information. 8.2. Information theory, coding and compression. 8.3. Algorithmic complexity. 8.4. Entropy and complexity in chaotic systems. 8.5. Concluding remarks. 8.6. Exercises. 9. Coarse-grained information and large scale predictability. 9.1. Finite-resolution versus infinite-resolution descriptions. 9.2. [symbol]-entropy in information theory : lossless versus lossy coding. 9.3. [symbol]-entropy in dynamical systems and stochastic processes. 9.4. The finite size lyapunov exponent (FSLE). 9.5. Exercises. 10. Chaos in numerical and laboratory experiments. 10.1. Chaos in silico. 10.2. Chaos detection in experiments. 10.3. Can chaos be distinguished from noise? 10.4. Prediction and modeling from data. 11. Chaos in low dimensional systems. 11.1. Celestial mechanics. 11.2. Chaos and transport phenomena in fluids. 11.3. Chaos in population biology and chemistry. 11.4. Synchronization of chaotic systems. 12. Spatiotemporal chaos. 12.1. Systems and models for spatiotemporal chaos. 12.2. The thermodynamic limit. 12.3. Growth and propagation of space-time perturbations. 12.4. Non-equilibrium phenomena and spatiotemporal chaos. 12.5. Coarse-grained description of high dimensional chaos. 13. Turbulence as a dynamical system problem. 13.1. Fluids as dynamical systems. 13.2. Statistical mechanics of ideal fluids and turbulence phenomenology. 13.3. From partial differential equations to ordinary differential equations. 13.4. Predictability in turbulent systems. 14. Chaos and statistical mechanics : Fermi-Pasta-Ulam a case study. 14.1. An influential unpublished paper. 14.2. A random walk on the role of ergodicity and chaos for equilibrium statistical mechanics. 14.3. Final remarks.