Chaos Near Resonance

Resonances are ubiquitous in dynamical systems with many degrees of freedom. They have the basic effect of introducing slow-fast behavior in an evolutionary system which, coupled with instabilities, can result in highly irregular behavior. This book gives a unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, a general finite dimensional theory of homoclinic jumping is developed and illustrated with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context. Previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds are described. 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. The theory is further used to study resonances in Hamiltonian systems with applications to molecular dynamics and rigid body motion. The final chapter contains an infinite dimensional extension of the finite dimensional theory, with application to the perturbed nonlinear Schrödinger equation and coupled NLS equations.

Résumé
A unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping.

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 PoincaréCartan 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 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 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 ofNonlinear 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 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 Stokes's Theorem.- B Some Facts From Analysis.- B.1 Fourier Series.- B.2 Gronwall Inequality.- B.3 Banach and Hilbert Spaces.- B.4Differentiation 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.