Chaos Near Resonance

A unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, the book develops a general finite dimensional theory of homoclinic jumping, illustrating it with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context, incorporating previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds. The results are applied to a variety of different problems, which include applications from beam oscillations, surface wave dynamics, nonlinear optics, atmospheric science and fluid mechanics.

Contenu

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 silnikov 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 silnikov 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 silnikov 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 silnikov 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 silnikov 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 silnikov 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 silnikov-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 silnikov 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 silnikov Manifolds in the Full CNLS System.- 5.14 Bibliographical Notes.…