Chaos Near Resonance

1 Concepts From Dynamical Systems.- 1.1 Flows, Maps, and Dynamical Systems.- 1.2 Ordinary Differential Equations as Dynamical Systems.- 1.3 Liouville’s Theorem.- 1.4 Structural Stability and Bifurcation.- 1.5 Hamiltonian Systems.- 1.6 Poincaré—Cartan 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 Šilnikov Orbits.- 2 Chaotic Jumping Near Resonances: Finite-Dimensional Systems.- 2.1 Resonances and Slow Manifolds.- 2.1.1 The Main Examples.- 2.2 Assumptions and Definitions.- 2.2.1 An Important Class of ODEs.- 2.2.2 N-Chains of Homoclinic Orbits.- 2.2.3 Partially Slow Manifolds.- 2.2.4 N-Pulse Homoclinic Orbits.- 2.3 Passage Lemmas.- 2.3.1 Fenichel Normal Form.- 2.3.2 Entry Conditions and Passage Time.- 2.3.3 Local Estimates.- 2.4 Tracking Lemmas.- 2.4.1 The Local Map.- 2.4.2 The Global Map.- 2.4.3 A Note on the Purely Hamiltonian Case.- 2.5 Energy Lemmas.- 2.5.1 Energy as a Coordinate.- 2.5.2 Energy of Entry Points.- 2.5.3 Improved Local Estimates.- 2.5.4 Energy of Projected Entry Points.- 2.6 Existence of Multipulse Orbits.- 2.6.1 Main Ideas.- 2.6.2 Existence Theorem.- 2.6.3 Remarks on Applications of the Main Theorem.- 2.6.4 The Most Frequent Case: Chain-Independent Energy Functions.- 2.6.5 Formulation With Other Invariants.- 2.7 Disintegration of Invariant Manifolds Through Jumping.- 2.8 Dissipative Chaos: Generalized Šilnikov Orbits.- 2.9 Hamiltonian Chaos: Homoclinic Tangles.- 2.9.1 Orbits Homoclinic to Invariant Spheres.- 2.9.2 The Case of n = 0: Orbits Heteroclinic to Slow m-Tori.- 2.9.3 The Case of n = 0, m = 1: Orbits Homoclinic to Slow Periodic Solutions.- 2.9.4 Resonant Energy Functions.- 2.9.5 Phase Shifts of Opposite Sign.- 2.10 Universal Homoclinic Bifurcations in Hamiltonian Applications.- 2.11 Heteroclinic Jumping Between Slow Manifolds.- 2.11.1 Partially Broken Heteroclinic Structures.- 2.11.2 Cat’s Eyes Heteroclinic Structures.- 2.12 Partially Slow Manifolds of Higher Codimension.- 2.12.1 Setup.- 2.12.2 Passage Lemmas.- 2.12.3 Tracking Lemmas.- 2.12.4 Energy Lemmas.- 2.12.5 Existence Theorem for Multipulse Orbits.- 2.12.6 Multipulse Šilnikov Manifolds.- 2.13 Bibliographical Notes.- 3 Chaos Due to Resonances in Physical Systems.- 3.1 Oscillations of a Parametrically Forced Beam.- 3.1.1 The Mechanical Model.- 3.1.2 The Modal Approximation.- 3.1.3 The Integrable Limit.- 3.1.4 Homoclinic Bifurcations in the Purely Forced Modal Equations.- 3.1.5 Structurally Stable Heteroclinic Connections for the Forced—Damped Beam.- 3.1.6 Chaos: Generalized Šilnikov Orbits and Cycles for the Forced—Damped Beam.- 3.1.7 Numerical Study.- 3.2 Resonant Surface-Wave Interactions.- 3.2.1 Derivation of the Amplitude Equations.- 3.2.2 The ? = 0 Limit.- 3.2.3 Chaotic Dynamics for ? > 0: Generalized Šilnikov Cycles.- 3.2.4 Passage to the Limit $$
\in = \sqrt \mu $$.- 3.2.5 The Inclusion of the $$
\mathcal{O}\left( {{\mu ^v}} \right)$$ Time-Dependent Terms.- 3.2.6 Comparison With the Simonelli—Gollub Experiment.- 3.3 Chaotic Pitching of Nonlinear Vibration Absorbers.- 3.3.1 The Mechanical Model.- 3.3.2 A More General Class of Problems.- 3.4 Mechanical Systems With Widely Spaced Frequencies.- 3.4.1 A Two-Mode Model.- 3.4.2 The Geometry of Energy Transfer.- 3.4.3 An Example.- 3.5 Irregular Particle Motion in the Atmosphere.- 3.5.1 The Model.- 3.5.2 Phase Space Geometry and Its Physical Meaning.- 3.6 Subharmonic Generation in an Optical Cavity.- 3.6.1 A Two-Mode Model.- 3.6.2 The Ideal Cavity (? = 0).- 3.6.3 Chaotic Dynamics for ? > 0.- 3.7 Intermittent Bursting in Turbulent Boundary Layers.- 3.7.1 Modal Equations With Weak O(2) ? D4 Symmetry Breaking.- 3.7.2 The Slow Manifold.- 3.7.3 Fast Heteroclinic Cycles.- 3.8 Further Problems.- 4 Resonances in Hamiltonian Systems.- 4.1 Resonant Equilibria.- 4.1.1 Birkhoff Normal Form.- 4.1.2 A Class of 1 : 2 : k Resonances.- 4.1.3 Geometry of the Normal Form.- 4.1.4 Homoclinic Orbits in the Two-Degree-of-Freedom Subsystem.- 4.1.5 Homoclinic Jumping in the Normal Form.- 4.1.6 Homoclinic Jumping and Chaos in the Full Problem.- 4.2 The Classical Water Molecule.- 4.2.1 The Normal Form.- 4.2.2 Homoclinic Chaos and Energy Transfer.- 4.3 Dynamics Near Intersecting Resonances.- 4.3.1 Arnold Diffusion in Near-Integrable Systems.- 4.3.2 Cross-Resonance Diffusion.- 4.3.3 Normal Form for Weak—Strong Double Resonances.- 4.3.4 The Pendulum-Type Hamiltonian.- 4.3.5 Dynamics in the Full Normal Form.- 4.4 An Example From Rigid Body Dynamics.- 4.5 Resonances in A Priori Unstable Systems.- 4.5.1 A Physical Example.- 4.5.2 Whiskered Tori.- 4.5.3 Resonances on Invariant Manifolds.- 4.5.4 Cross-Resonance Diffusion, Homoclinic Bifurcations, and Horseshoes.- 5 Chaotic Jumping Near Resonances: Infinite-Dimensional Systems.- 5.1 The Main Examples.- 5.2 Assumptions and Definitions.- 5.2.1 The Phase Space and the Evolution Equation.- 5.2.2 Regularity and Geometric Assumptions.- 5.2.3 N-Chains of Homoclinic Orbits.- 5.3 Invariant Manifolds and Foliations.- 5.3.1 Partially Slow Manifold.- 5.3.2 Preliminary Normal Form.- 5.3.3 Smooth Foliations for Wlocs(M?,k) and Wlocu(M?,k).- 5.3.4 N-Pulse Homoclinic Orbits.- 5.4 Passage Lemmas.- 5.4.1 Fenichel Normal Form.- 5.4.2 Entry Conditions and Passage Time.- 5.4.3 Local Estimates.- 5.5 Tracking Lemmas.- 5.5.1 The Local Map.- 5.5.2 The Global Map.- 5.6 Energy Lemmas.- 5.6.1 Energy as a Coordinate.- 5.6.2 Energy of Entry Points.- 5.6.3 Improved Local Estimates.- 5.6.4 Energy of Projected Entry Points.- 5.7 Multipulse Homoclinic Orbits in Sobolev Spaces.- 5.7.1 Definitions and Notation.- 5.7.2 Existence Theorem.- 5.7.3 Remarks on Applications of the Main Theorem.- 5.7.4 Chain-Independent Energy Functions.- 5.7.5 Formulation With Other Invariants.- 5.8 Disintegration of Invariant Manifolds Through Jumping.- 5.9 Generalized Šilnikov Orbits.- 5.10 The Purely Hamiltonian Case.- 5.10.1 Universal Homoclinic Bifurcations.- 5.11 Homoclinic Jumping in the Perturbed NLS Equation.- 5.11.1 Homoclinic Tree in the Forced NLS Equation $$
\hat D \equiv 0$$.- 5.11.2 N -Pulse Orbits in the Damped—Forced NLS Equation.- 5.11.3 Šilnikov-Type Orbits in the Damped—Forced NLS Equation.- 5.12 Partially Slow Manifolds of Higher Codimension.- 5.12.1 Setup.- 5.12.2 Existence Theorem for Multipulse Orbits.- 5.12.3 Multipulse Šilnikov Manifolds.- 5.13 Homoclinic Jumping in the CNLS System.- 5.13.1 Homoclinic Jumping in the Forced CNLS Equations (?k = ?k = 0).- 5.13.2 N-Pulse Jumping Orbits in the Damped—Forced CNLS System ?k.- 5.13.3 N-Pulse Šilnikov Manifolds in the Full 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 Stokes’s 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.