Author | Haller, G. author |
---|---|
Title | Chaos Near Resonance [electronic resource] / by G. Haller |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1999 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1508-0 |
Descript | XVI, 430 p. online resource |
1 Concepts From Dynamical Systems -- 1.1 Flows, Maps, and Dynamical Systems -- 1.2 Ordinary Differential Equations as Dynamical Systems -- 1.3 Liouvilleโs Theorem -- 1.4 Structural Stability and Bifurcation -- 1.5 Hamiltonian Systems -- 1.6 PoincarรฉโCartan Integral Invariant -- 1.7 Generating Functions -- 1.8 Infinite-Dimensional Hamiltonian Systems -- 1.9 Symplectic Reduction -- 1.10 Integrable Systems -- 1.11 KAM Theory and Whiskered Tori -- 1.12 Invariant Manifolds -- 1.13 Stable and Unstable Manifolds -- 1.14 Stable and Unstable Foliations -- 1.15 Strong Stable and Unstable Manifolds -- 1.16 Weak Hyperbolicity -- 1.17 Homoclinic Orbits and Homoclinic Manifolds -- 1.18 Singular Perturbations and Slow Manifolds -- 1.19 Exchange Lemma -- 1.20 Exchange Lemma and Observability -- 1.21 Normal Forms -- 1.22 Averaging Methods -- 1.23 Lambda Lemma and the Homoclinic Tangle -- 1.24 Smale Horseshoes and Symbolic Dynamics -- 1.25 Chaos -- 1.26 Hyperbolic Sets, Transient Chaos, and Strange Attractors -- 1.27 Melnikov Methods -- 1.28 ล ilnikov Orbits -- 2 Chaotic Jumping Near Resonances: Finite-Dimensional Systems -- 2.1 Resonances and Slow Manifolds -- 2.2 Assumptions and Definitions -- 2.3 Passage Lemmas -- 2.4 Tracking Lemmas -- 2.5 Energy Lemmas -- 2.6 Existence of Multipulse Orbits -- 2.7 Disintegration of Invariant Manifolds Through Jumping -- 2.8 Dissipative Chaos: Generalized ล ilnikov Orbits -- 2.9 Hamiltonian Chaos: Homoclinic Tangles -- 2.10 Universal Homoclinic Bifurcations in Hamiltonian Applications -- 2.11 Heteroclinic Jumping Between Slow Manifolds -- 2.12 Partially Slow Manifolds of Higher Codimension -- 2.13 Bibliographical Notes -- 3 Chaos Due to Resonances in Physical Systems -- 3.1 Oscillations of a Parametrically Forced Beam -- 3.2 Resonant Surface-Wave Interactions -- 3.3 Chaotic Pitching of Nonlinear Vibration Absorbers -- 3.4 Mechanical Systems With Widely Spaced Frequencies -- 3.5 Irregular Particle Motion in the Atmosphere -- 3.6 Subharmonic Generation in an Optical Cavity -- 3.7 Intermittent Bursting in Turbulent Boundary Layers -- 3.8 Further Problems -- 4 Resonances in Hamiltonian Systems -- 4.1 Resonant Equilibria -- 4.2 The Classical Water Molecule -- 4.3 Dynamics Near Intersecting Resonances -- 4.4 An Example From Rigid Body Dynamics -- 4.5 Resonances in A Priori Unstable Systems -- 5 Chaotic Jumping Near Resonances: Infinite-Dimensional Systems -- 5.1 The Main Examples -- 5.2 Assumptions and Definitions -- 5.3 Invariant Manifolds and Foliations -- 5.4 Passage Lemmas -- 5.5 Tracking Lemmas -- 5.6 Energy Lemmas -- 5.7 Multipulse Homoclinic Orbits in Sobolev Spaces -- 5.8 Disintegration of Invariant Manifolds Through Jumping -- 5.9 Generalized ล ilnikov Orbits -- 5.10 The Purely Hamiltonian Case -- 5.11 Homoclinic Jumping in the Perturbed NLS Equation -- 5.12 Partially Slow Manifolds of Higher Codimension -- 5.13 Homoclinic Jumping in the CNLS System -- 5.14 Bibliographical Notes -- A Elements of Differential Geometry -- A.1 Manifolds -- A.2 Tangent, Cotangent, and Normal Bundles -- A.3 Transversality -- A.4 Maps on Manifolds -- A.5 Regular and Critical Points -- A.6 Lie Derivative -- A.7 Lie Algebras, Lie Groups, and Their Actions -- A.8 Orbit Spaces -- A.9 Infinite-Dimensional Manifolds -- A.10 Differential Forms -- A.11 Maps and Differential Forms -- A.12 Exterior Derivative -- A.13 Closed and Exact Forms -- A.14 Lie Derivative of Forms -- A.15 Volume Forms and Orientation -- A.16 Symplectic Forms -- A.17 Poisson Brackets -- A.18 Integration on Manifolds and Stokesโs Theorem -- B Some Facts From Analysis -- B.1 Fourier Series -- B.2 Gronwall Inequality -- B.3 Banach and Hilbert Spaces -- B.4 Differentiation and the Mean Value Theorem -- B.5 Distributions and Generalized Derivatives -- B.6 Sobolev Spaces -- B.8 Factorization of Functions With a Zero -- References -- Symbol Index