Bifurcation and Chaos presents a collection of especially written articles describing the theory and application of nonlinear dynamics to a wide variety of problems encountered in physics and engineering. Each chapter is self-contained and includes an elementary introduction, an exposition of the present state of the art, and details of recent theoretical, computational and experimental results. Included among the practical systems analysed are: hysteretic circuits, Josephson circuits, magnetic systems, railway dynamics, rotor dynamics and nonlinear dynamics of speech. This book contains important information and ideas for all mathematicians, physicists and engineers whose work in R&D or academia involves the practical consequence of chaotic dynamics.

Klappentext

A collection of especially written articles describing the theory and application of nonlinear dynamics to a wide variety of problems encountered in physics and engineering. Each chapter is self-contained and includes an elementary introduction, an exposition of the state of the art, as well as details of recent theoretical, computational and experimental results. Included among the practical systems analysed are: hysteretic circuits, Josephson circuits, magnetic systems, railway dynamics, rotor dynamics and nonlinear dynamics of speech. This book provides important information and ideas for all mathematicians, physicists and engineers whose work in R & D or academia involves the practical consequences of chaotic dynamics.

Inhalt

Quantum Chaos and Ergodic Theory.- 1. Introduction.- 2. Definition of Quantum Chaos.- 3. The Time Scales of Quantum Dynamics.- 4. The Quantum Steady State.- 5. Concluding Remarks.- References.- On the Complete Characterization of Chaotic Attractors.- 1. Introduction.- 2. Scaling Behavior.- 2.1 Scale Invariance.- 2.2 Non-unified Approach.- 3. Unified Approach.- 3.1 The Generalized Entropy Function.- 3.2 Hyperbolic Models with Complete Grammars.- 4. Extensions.- 4.1 The Need for Extensions.- 4.2 Convergence Properties.- 4.3 Nonhyperbolicity and Phase-Transitions.- 5 Conclusions.- References.- New Numerical Methods for High Dimensional Hopf Bifurcation Problems.- 1. Introduction.- 2. Static Bifurcation and Pseudo-Arclength Method.- 3. The Numerical Methods for Hopf Bifurcation.- 4. Examples.- References.- Catastrophe Theory and the Vibro-Impact Dynamics of Autonomous Oscillators.- 1. Introduction.- 2. Generalities on Vibro-Impact Dynamics.- 3. The Geometry of Singularity Subspaces.- 4. Continuity of the Poincaré Map of the S/U Oscillator.- References.- Codimension Two Bifurcation and Its Computational Algorithm.- 1. Introduction.- 2. Bifurcations of Fixed Point.- 2.1 The Poincaré Map and Property of Fixed Points.- 2.2 Codimension One Bifurcations.- 2.3 Codimension Two Bifurcations.- 3. Computational Algorithms.- 3.1 Derivatives of the Poincaré Map.- 3.2 Numerical Method of Analysis.- 4. Numerical Examples.- 4.1 Circuit Model for Chemical Oscillation at a Water-Oil Interface.- 4.2 Coupled Oscillator with a Sinusoidal Current Source.- 5. Concluding Remarks.- References.- Chaos and Its Associated Oscillations in Josephson Circuits.- 1. Introduction.- 2. Model of Josephson Junction.- 3. Chaos in a Forced Oscillation Circuit.- 4. Autonomous Josephson Circuit.- 4.1 Introduction.- 4.2 Results of Calculation.- 5. Distributed Parameter Circuit.- 6. Conclusion.- References.- Chaos in Systems with Magnetic Force.- 1. Introduction.- 2. System of Two Conducting Wires.- 2.1 Formulation of Dynamical Equations.- 2.2 Analytical Procedure.- 2.3 Numerical Simulation of Chaos.- 3. Multi-Equilibrium Magnetoelastic Systems.- 3.1 Theoretical Models.- 3.2 Numerical Simulation.- 3.3 Experiment.- 4. Magnetic Levitation Systems.- 4.1 Formulation of Dynamic Equations.- 4.2 Linearization in Terms of Manifolds.- 4.3 Numerical Simulation.- 4.4 Conclusion.- References.- Bifurcation and Chaos in the Helmholtz-Duffing Oscillator.- 1. Mechanical System and Mathematical Model.- 2. Behaviour Chart and Characterization of Chaotic Response.- 3. Prediction of Local Bifurcations of Regular Solutions.- 4. Geometrical Description of System Response Using Attractor-Basin Portraits and Invariant Manifolds.- 5. Conclusions.- References.- Bifurcations and Chaotic Motions in Resonantly Excited Structures.- 1. Introduction.- 2. Nonlinear Structural Members.- 2.1 Strings.- 2.2 Beams.- 2.3 Cylindrical Shells and Rings.- 2.4 Plates.- 3. Resonant Motions of Rectangular Plates with Internal and External Resonances.- 3.1 Equations of Motion.- 3.2 Averaged Equations.- 3.3 Steady-State Constant Solutions.- 3.4 Stability Analysis of Constant Solutions.- 3.5 Periodic and Chaotic Solutions of Averaged Equations.- 4. Summary and Conclusions.- References.- Non-Linear Behavior of a Rectangular Plate Exposed to Airflow.- 1. Introduction.- 2. Mathematical Model.- 3. Threshold Determination of Periodic Oscillations.- 4. Dynamics Past the Hopf Bifurcation Point.- 5. Summary and Concluding Remarks.- References.