| Description |
1 online resource (xxiii, 255 pages) |
| Series |
Springer series in synergetics, 0172-7389 ; 10 |
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Springer complexity |
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Springer series in synergetics ; 10.
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Springer complexity.
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| Contents |
880-01 Introduction -- Hamiltonian Systems of Few Degrees of Freedom -- Local and Global Stability of Motion -- Normal Modes, Symmetries and Stability -- Efficient Indicators of Ordered and Chaotic Motion -- FPU Recurrences and the Transition from Weak to Strong Chaos -- Localization and Diffusion in Nonlinear One-Dimensional Lattices -- The Statistical Mechanics of Quasi-stationary States -- Conclusions, Open Problems and Future Outlook |
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880-01/(S Machine generated contents note: 1. Introduction -- 1.1. Preamble -- 1.2. Lyapunov Stability of Dynamical Systems -- 1.3. Hamiltonian Dynamical Systems -- 1.4. Complex Hamiltonian Dynamics -- 2. Hamiltonian Systems of Few Degrees of Freedom -- 2.1. Case of N = 1 Degree of Freedom -- 2.2. Case of N = 2 Degrees of Freedom -- 2.2.1. Coordinate Transformations and Solution by Quadratures -- 2.2.2. Integrability and Solvability of the Equations of Motion -- 2.3. Non-autonomous One Degree of Freedom Hamiltonian Systems -- 2.3.1. Duffing Oscillator with Quadratic Nonlinearity -- 2.3.2. Duffing Oscillator with Cubic Nonlinearity -- Exercises -- Problems -- 3. Local and Global Stability of Motion -- 3.1. Equilibrium Points, Periodic Orbits and Local Stability -- 3.1.1. Equilibrium Points -- 3.1.2. Periodic Orbits -- 3.2. Linear Stability Analysis -- 3.2.1. Analytical Criterion for "Weak" Chaos -- 3.3. Lyapunov Characteristic Exponents and "Strong" Chaos -- 3.3.1. Lyapunov Spectra and Their Convergence -- 3.3.1.1. Lyapunov Spectra and the Thermodynamic Limit -- 3.4. Distinguishing Order from Chaos -- 3.4.1. SALI Method -- 3.4.2. GALI Method -- Exercises -- Problems -- 4. Normal Modes, Symmetries and Stability -- 4.1. Normal Modes of Linear One-Dimensional Hamiltonian Lattices -- 4.2. Nonlinear Normal Modes (NNMs) and the Problem of Continuation -- 4.3. Periodic Boundary Conditions and Discrete Symmetries -- 4.3.1. NNMs as One-Dimensional Bushes -- 4.3.2. Higher-Dimensional Bushes and Quasiperiodic Orbits -- 4.4. Group Theoretical Study of Bushes -- 4.4.1. Subgroups of the Parent Group and Bushes of NNMs -- 4.4.2. Bushes in Modal Space and Stability Analysis -- 4.5. Applications to Solid State Physics -- 4.5.1. Bushes of NNMs for a Square Molecule -- 4.5.2. Bushes of NNMs for a Simple Octahedral Molecule -- Exercises -- Problems -- 5. Efficient Indicators of Ordered and Chaotic Motion -- 5.1. Variational Equations and Tangent Map -- 5.2. SALI Method -- 5.3. GALI Method -- 5.3.1. Theoretical Results for the Time Evolution of GALI -- 5.3.1.1. Exponential Decay of GALI for Chaotic Orbits -- 5.3.1.2. Evaluation of GALI for Ordered Orbits -- 5.3.2. Numerical Verification and Applications -- 5.3.2.1. Low-Dimensional Hamiltonian Systems -- 5.3.2.2. High-Dimensional Hamiltonian Systems -- 5.3.2.3. Symplectic Maps -- 5.3.2.4. Motion on Low-Dimensional Tori -- Appendix A Wedge product -- Appendix B Example algorithms for the computation of the SALI and GALI chaos indicators -- Exercises -- Problems -- 6. FPU Recurrences and the Transition from Weak to Strong Chaos -- 6.1. Fermi Pasta Ulam Problem -- 6.1.1. Historical Remarks -- 6.1.2. Concept of q-breathers -- 6.1.3. Concept of q-tori -- 6.2. Existence and Stability of q-tori -- 6.2.1. Construction of q-tori by Poincare-Linstedt Series -- 6.2.2. Profile of the Energy Localization -- 6.3. Numerical Study of FPU Trajectories -- 6.3.1. Long Time Stability near q-tori -- 6.4. Diffusion and the Breakdown of FPU Recurrences -- 6.4.1. Two Stages of the Diffusion Process -- 6.4.2. Rate of Diffusion of Energy to the Tail Modes -- 6.4.3. Time Interval for Energy Equipartition -- Exercises -- Problems -- 7. Localization and Diffusion in Nonlinear One-Dimensional Lattices -- 7.1. Introduction and Historical Remarks -- 7.1.1. Localization in Configuration Space -- 7.2. Discrete Breathers and Homoclinic Dynamics -- 7.2.1. How to Construct Homoclinic Orbits -- 7.3. Method for Constructing Discrete Breathers -- 7.3.1. Stabilizing Discrete Breathers by a Control Method -- 7.4. Disordered Lattices -- 7.4.1. Anderson Localization in Disordered Linear Media -- 7.4.2. Diffusion in Disordered Nonlinear Chains -- 7.4.2.1. Two Basic Models -- 7.4.2.2. Regimes of Wave Packet Spreading -- 7.4.2.3. Numerical Results -- Exercises -- Problems -- 8. Statistical Mechanics of Quasi-stationary States -- 8.1. From Deterministic Dynamics to Statistical Mechanics -- 8.1.1. Nonextensive Statistical Mechanics and q-Gaussian pdfs -- 8.2. Statistical Distributions of Chaotic QSS and Their Computation -- 8.3. FPU π-Mode Under Periodic Boundary Conditions -- 8.3.1. Chaotic Breathers and the FPU π-Mode -- 8.4. FPU SPO1 and SPO2 Modes Under Fixed Boundary Conditions -- 8.5. q-Gaussian Distributions for a Small Microplasma System -- 8.6. Chaotic Quasi-stationary States in Area-Preserving Maps -- 8.6.1. Time-Evolving Statistics of pdfs in Area-Preserving Maps -- 8.6.2. Perturbed MacMillan Map -- 8.6.2.1. ε = 0.9, μ = 1.6 Class of Examples -- 8.6.2.2. ε = 1.2, μ = 1.6 Class of Examples -- Exercises -- Problems -- 9. Conclusions, Open Problems and Future Outlook -- 9.1. Conclusions -- 9.2. Open Problems -- 9.2.1. Singularity Analysis: Where Mathematics Meets Physics -- 9.2.2. Nonlinear Normal Modes, Quasiperiodicity and Localization -- 9.2.3. Diffusion, Quasi-stationary States and Complex Statistics -- 9.3. Future Outlook -- 9.3.1. Anomalous Heat Conduction and Control of Heat Flow -- 9.3.2. Complex Soliton Dynamics in Nonlinear Photonic Structures -- 9.3.3. Kinetic Theory of Hamiltonian Systems and Applications to Plasma Physics |
| Summary |
This book introduces and explores modern developments in the well established field of Hamiltonian dynamical systems. It focuses on high degree-of-freedom systems and the transitional regimes between regular and chaotic motion. The role of nonlinear normal modes is highlighted and the importance of low-dimensional tori in the resolution of the famous FPU paradox is emphasized. Novel powerful numerical methods are used to study localization phenomena and distinguish order from strongly and weakly chaotic regimes. The emerging hierarchy of complex structures in such regimes gives rise to particularly long-lived patterns and phenomena called quasi-stationary states, which are explored in particular in the concrete setting of one-dimensional Hamiltonian lattices and physical applications in condensed matter systems.¡ The self-contained and pedagogical approach is blended with a unique balance between mathematical rigor, physics insights and concrete applications. End of chapter exercises and (more demanding) research oriented¡ problems provide many opportunities to deepen the reader's insights into specific aspects of the subject matter. ¡ Addressing a broad audience of graduate students, theoretical physicists and applied mathematicians, this text combines the benefits of a reference work with those of a self-study guide for newcomers to the field |
| Analysis |
Mathematics |
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Mathematical physics |
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Mechanics |
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Engineering |
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Statistical Physics, Dynamical Systems and Complexity |
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Complexity |
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Information and Communication, Circuits |
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Mathematical Methods in Physics |
| Bibliography |
Includes bibliographical references and index |
| Subject |
Hamiltonian systems.
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MATHEMATICS -- Differential Equations -- General.
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Physique.
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Astronomie.
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Hamiltonian systems
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| Form |
Electronic book
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| Author |
Skokos, Haris
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| ISBN |
9783642273056 |
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364227305X |
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3642273041 |
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9783642273049 |
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