Nonlinearities in Action: Oscillations Chaos Order Fractals

ISBN-13: 9783642752940 / Angielski / Miękka / 2011 / 191 str.

Andrei V. Gaponov-Grekhov; Mikhail I. Rabinovich; E. F. Hefter

Opis

In the dynamics of mankind one can trace out a path of contemplation about the "world," leading from early speculations to today's natural sciences. The endeavour to understand how nature works has led to the construction, still in progress, of an abstract building of great com plexity. To the uninitiated it may look more like a scurrilous sculpture resting on many legs, among them such peculiar ones as probability, relativity, quantum mechanics .... At times problems with the stabil ity of the building or sculpture arise: known facts that won't fit and can no longer be ignored start to undermine the foundations. Then new footings are thought of, constructed and finally cast. In fact, the undermining and casting is often done in one step. This process has already been repeated many times and will un doubtedly repeat itself again and again. At present, one recognizable footing under construction goes by the name of "chaos theory." Physi cists seem to like the word chaos. When they came to recognize that the air is not just empty space but an obviously indescribable dance of myriads of molecules they called that "chaos." What else would fit? In the course of time the name was simplified to "gas." Thus the word chaos became free to serve for the next upsetting experience. That arose in the context of nonlinear dynamical systems, where peculiar motions were detected, ones seemingly beyond human comprehension."

Spis treści

1. Introduction.- 2. Nonlinear Oscillations and Waves. Classical Results.- 2.1 Oscillators.- 2.1.1 A Marble in the Chute.- 2.1.2 Spring Pendulum and Nonlinear Optics.- 2.1.3 Nonlinear Landau Damping and Amplification.- 2.2 Solitons.- 2.2.1 The Fermi-Pasta-Ulam Paradox.- 2.2.2 Solitons as Particles.- 2.2.3 Solitons and Shock Waves.- 2.3 Self-Excited Oscillations.- 2.3.1 Examples and Definitions.- 2.3.2 Competition and Synchronization.- 2.3.3 Self-Excited Oscillations in Chains and Continuous Systems.- 2.4 Bifurcations.- 2.4.1 Acquisition of a New Quality.- 2.4.2 Bifurcations of Equilibrium States.- 2.4.3 Bifurcations of Periodic Motion.- 2.4.4 Bifurcations — Changes of Stability in Periodic Motion.- 2.5 Modulation.- 2.5.1 The Role of Small Parameters.- 2.5.2 Running Mandelstam Lattices. Modulation of Waves by Waves.- 2.5.3 Generation of Modulation.- 2.5.4 Self-Modulation.- 2.5.5 Recurrence.- 2.5.6 Modulation Solitons.- 3. Chaos.- 3.1 Historical Remarks.- 3.2 Marble in an Oscillating Chute.- 3.3 Stochastic Self-Excited Oscillations.- 3.3.1 The Lorenz Attractor.- 3.3.2 Synchronization — Beats — Chaos.- 3.3.3 Autonomous Noise Generator.- 3.3.4 Scenarios for the Birth of Strange Attractors.- 3.4 Chaos and Noise.- 3.4.1 Dimension and Entropy.- 3.4.2 The Cantor Structure of a Strange Attractor.- 3.4.3 Dimension and Lyapunov Exponent.- 3.4.4 Deterministically Generated and Random Signals.- 4. Structures.- 4.1 Order and Disorder — Examples.- 4.2 Attractors and Spatial Patterns.- 4.2.1 Examples of Equations.- 4.2.2 Multistability. Defects.- 4.3 Self-Structures.- 4.3.1 Convective Self-Structures.- 4.3.2 Localization Mechanisms.- 4.3.3 Self-Structures in Three-Dimensional Media.- 4.3.4 Interaction of “Elementary Particles”.- 4.3.5 Birth and Interaction of Spiral Waves.- 4.4 Attractors — Memory — Learning.- 4.4.1 How to Remember.- 4.4.2 “Camera + TV + Feedback” Analogue.- 4.4.3 Critical Phenomena.- 4.4.4 Structures in Neuron-Like Media.- 5. Turbulence.- 5.1 Prehistory.- 5.2 Basic Models of Dynamic Theory.- 5.3 Turbulence and Structures in Two-Dimensional Fields.- 5.3.1 Experiments.- 5.3.2 Development of Turbulence and Multi-Dimensional Attractors.- 5.4 Spatial Evolution of Turbulence.- 5.4.1 Flow Dimension.- 5.4.2 Spatial Bifurcations.- 5.5 Discussion.- 6. Nonlinear Physics — Chaos and Order.- 6.1 The Where and the How.- 6.2 Randomness Born out of Nonrandomness.- 6.3 An Unstable Path and Steady Motion. Are They Incompatible?.- 6.4 Does Chance Rule the World?.- 6.5 What is the Character of Nature? Integer or Fractal?.- 6.6 Fractal Fingers.- 6.7 Self-Organizing Structures.- 6.8 Singles.- 6.9 The New Life of an Old Problem.- 6.10 Spatial Evolution of Disorder.- 6.11 What Does Your Camera See When It is Watching TV?.- 6.12 Multistability and Memory.- 6.13 Nonlinear Dynamics in Society.- Color Plates.- Literature.- Acknowledgements of the Figures.

O autorze

Order and disorder in nature, the birth of chaos in deterministic systems: This textbook provides a concise and comprehensive overview on nonlinear phenomena addressing all natural scientists. It also contains a beautifully illustrated color insert easily accessible to interested laypersons. Thus it is suitable for leisure reading and also for an introductory (under-)graduate course in nonlinear physics.