Call number:
AWI A17-92-0413
;
PIK M 102-01-0315
In:
Texts in applied mathematics
Description / Table of Contents:
Contents: Series Preface. - Preface. - 0 Introduction. - 1 The Geometrical Point of View of Dynamical Systems: Background Material, Poincare Maps, and Examples. - 1.1 Background Material from Dynamical Systems Theory. - 1.1A Equilibrium solutions: Linearized Stability. - 1.1B Liapunov Functions. - 1.1c Invariant Manifolds: Linear and Nonlinear Systems. - 1.1D Periodic Solutions. - 1.1E Integrable Vector Fields on Two-Manifolds. - 1.1F Index Theory. - 1.1G Some General Properties of Vector Fields: Existence, Uniqueness, Differentiability, and Flows. - 1.1H Asymptotic Behavior. - 1.1I The Poincare-Bendixson Theorem. - Exercises. - 1.2 Poincare Maps: Theory, Construction, and Examples. - 1.2A Poincare Maps: Examples. - 1.2B Varying the Cross-Section: Conjugacies of Maps. - 1.2c Structural Stability, Genericity, and Transversality. - 1.2D Construction of the Poincare Map. - 1.2E Application to the Dynamics of the Damped, Forced Duffing Oscillator. - Exercises. - 2 Methods for Simplifying Dynamical Systems. - 2.1 Center Manifolds. - 2.1A Center Manifolds for Vector Fields. - 2.1B Center Manifolds Depending on Parameters. - 2.1c The Inclusion of Linearly Unstable Directions. - 2.1D Center Manifolds for Maps. - 2.iE Properties of Center Manifolds. - 2.2 Normal Forms. - 2.2A Normal Forms for Vector Fields. - 2.2B Normal Forms for Vector Fields with Parameters. - 2.2c Normal Forms for Maps. - 2.2D Conjugacies and Equivalences of Vector Fields. - 2.3 Final Remarks. - Exercises. - 3 Local Bifurcations. - 3.1 Bifurcation of Fixed Points of Vector Fields. - 3.1A A Zero Eigenvalue. - 3.1B A Pure Imaginary Pair of Eigenvalues: The Poincare-Andronov-Hopf Bifurcation. - 3.1c Stability of Bifurcations Under Perturbations. - 3.1D The Idea of the Codimension of a Bifurcation Appendix 1: Versal Deformations of Families of Matrices. - 3.1E The Double-Zero Eigenvalue. - 3.1F A Zero and a Pure Imaginary Pair of Eigenvalues. - 3.2 Bifurcations of Fixed Points of Maps. - 3.2A An Eigenvalue of 1. - 3.2B An Eigenvalue of -1. - 3.2c A Pair of Eigenvalues of Modulus 1: The Naimark-Sacker Bifurcation. - 3.2D The Codimension of Local Bifurcations of Maps. - 3.3 On the Interpretation and Application of Bifurcation Diagrams: A Word of Caution. - Exercises. - 4 Some Aspects of Global Bifurcation and Chaos. - 4.1 The Smale Horseshoe. - 4.1A Definition of the Smale Horseshoe Map. - 4.1B Construction of the Invariant Set. - 4.1c Symbolic Dynamics. - 4.1D The Dynamics on the invariant set. - 4.1E Chaos. - 4.2 Symbolic Dynamics. - 4.2A The Structure of the Space of Symbol Sequences. - 4.2B The Shift Map. - 4.3 The Conley-Moser Conditions, or "How to Prove That a Dynamical System is Chaotic". - 4.3A The main theorem. - 4.3B Sector bundles. - 4.3C Hyperbolic invariant sets. - 4.4 Dynamics near homoclinic points of two-dimensional maps. - 4.5 Melnikov's method for homoclinic orbits in two-dimensional, Time-Periodic Vector Fields. - 4.5A The General Theory. - 4.5B Poincare Maps and the Geometry of the Melnikov Function. - 4.5c Some Properties of the Melnikov Function. - 4.5D Relationship with the Subharmonic Melnikov Function. - 4.5E Homoclinic and Subharmonic Bifurcations. - 4.5F Application to the Damped, Forced Duffing Oscillator. - 4.6 Geometry and Dynamics in the Tangle. - 4.6A Pips and Lobes. - 4.6B Transport in Phase Space. - 4.6c Technical Details. - 4.6D Application to the Melnikov Theory to Transport. - 4.7 Homoclinic Bifurcations: Cascades of Period-Doubling and Saddle-Node Bifurcations. - 4.8 Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields. - 4.8A Orbits Homoclinic to a saddle-point with purely real eigenvalues. - 4.8B Orbits homoclinic to a saddle-focus. - 4.9 Global bifurcations arising from local codimension-two bifurcations. - 4.9A The double-zero eigenvalue. - 4.9B A zero and a pure imaginary pair of eigenvalues. - 4.10 Liapunov exponents. - 4.11 Chaos and strange attractors exercises. - Bibliography. - Index.
Type of Medium:
Monograph available for loan
Pages:
XIV, 672 S.: Ill.
Edition:
2. corr. print.
ISBN:
0387970037
Series Statement:
Texts in applied mathematics 2
Location:
A 18 - must be ordered
Branch Library:
AWI Library
Branch Library:
PIK Library
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