Bifurcation and Chaos

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.