非線性振動(dòng)，動(dòng)力學(xué)系統(tǒng)和矢量場(chǎng)的分叉

內(nèi)容概要

Problems in dynamics have fascinated physical scientists (and mankind in general) for thousands of years. Notable among such problems are those of celestial mechanics, especially the study of the motions of the bodies in the solar system. Newton's attempts to understand and model their observed motions incorporated Kepler's laws and led to his development of the calculus. With this the study of models of dynamical problems as differential equations began.

書(shū)籍目錄

CHAPTER 1  Introduction: Differential Equations and Dynamical Systems  1.0. Existence and Uniqueness ofSolutions  1.1. The Linear System x = Ax  1.2. Flows and Invariant Subspaces  1.3. The Nonlinear System x = f(x)  1.4. Linear and Nonlinear Maps  1.5. Closed Orbits. Poincare Maps and Forced Oscillations  1.6. Asymptotic Behavior  1.7. Equivalence Relations and Structural Stability  1.8. Two-Dimensional Flows  1.9. Peixoto's Theorem for Two-Dimensional FlowsCHAPTER 2  An Introduction to Chaos: Four Examples  2.1. Van der Pol's Equation  2.2. Duffing's Equation  2.3. The Lorenz Equations  2.4. The Dynamics ofa Bouncing Ball  2.5. Conclusions: The Moral ofthe TalesCHAPTER 3  Local Bifurcations  3.1. Bifurcation Problems  3.2. Center Manifolds  3.3. Normal Forms  3.4. Codimension One Bifurcations of Equilibria  3.5. Codimension One Bifurcations ofMaps and Periodic OrbitsCHAPTER 4  Averaging and Perturbation from a Geometric Viewpoint  4.1. Averaging and Poincare Maps  4.2. Examples of Averaging  4.3. Averaging and Local Bifureations  4.4. Averaging, Hamiltonian Systems, and Global Behavior: Cautionary Notes  4.5. Melnikov's Method: Perturbations ofPlanar Homoclinic Orbits  4.6. Melnikov's Method: Perturbations of Hamiltonian Systems and Subharmonic Orbits  4.7. Stability of Subharmonic Orbits  4.8. Two Degree of Freedom Hamiltonians and Area Preserving Maps of the PlaneCHAPTER 5  Hyperbolic Sets, Symbolic Dynamics, and Strange Attractors  5.0. Introduction  5.1. The Smale Horseshoe: An Example ofa Hyperbolic Limit Set  5.2. Invariant Sets and Hyperbolicity  5.3. Markov Partitions and Symbolic Dynamics  5.4. Strange Attractors and the Stability Dogma  5.5. Structurally Stable Attractors  5.6. One-Dimensional Evidence for Strange Attractors  5.7. The Geometric Lorenz Attractor  5.8. Statistical Propenies: Dimension. Entropy and Liapunov ExponentsCHAPTER 6  Global Bifurcations  6.1. Saddle Connections  6.2. Rotation Numbers  6.3. Bifurcations of One-Dimensional Maps  6.4. The Lorenz Bifurcations  6.5. Homoclinic Orbits in Three-Dimensional Flows: Silnikov's Example  6.6. Homoclinic Bifurcations of Periodic Orbits  6.7. Wild Hyperbolic Sets  6.8. Renormalization and UniversalityCHAPTER7  Local Codimension Two Bifurcations of Flows  7.1. Degeneracy in Higher-Order Terms  7.2. A Note on k-Sels and Determinacy  7.3. The Double Zero Eigenvalue  7.4. A Pure Imaginary Pair and a Simple Zero Eigenvalue  7.5. Two Pure Imaginary Pairs of Eigenvalues without Resonance  7.6. Applications to Large SystemsAPPENDIXSuggestions for Further ReadingPostscript Added at Second PrintingGlossaryReferencesIndex

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