G. Haller

Chaos Near Resonance (eBook, PDF)

• Format: PDF

Produktbeschreibung

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of chaos near resonances is discussed in both the dissipative and the Hamiltonian context, incorporating previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds.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.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 430
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461215080
• Artikelnr.: 44058226

Inhaltsangabe

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.- 2.2 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 Silnikov 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 Silnikov 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.

Rezensionen

"An extensive bibliography and the many examples make this clearly-written book an excellent introduction to these techniques for identifying chaos in perturbations of systems with resonance." Applied Mechanics Reviews, Vol. 53/4, April 2000 "Haller makes a point of wanting to see dynamical systems theory fulfil "its long-standing promise to solve real-life problems". His book, through a wealth of detailed examples, delivers on this promise, and is certain ti become a standard text in this area. In particular, it is an excellent introduction to this research area, and contains a wealth of bibliographical and historical detail. Matthew Nicol, Bulletin of the LMS, No. 162, Vol. 33/3, May 2001

An extensive bibliography and the many examples make this clearly-written book an excellent introduction to these techniques for identifying chaos in perturbations of systems with resonance. Applied Mechanics Rezensions, Vol. 53/4, April 2000 "Haller makes a point of wanting to see dynamical systems theory fulfil "its long-standing promise to solve real-life problems". His book, through a wealth of detailed examples, delivers on this promise, and is certain ti become a standard text in this area. In particular, it is an excellent introduction to this research area, and contains a wealth of bibliographical and historical detail. Matthew Nicol, Bulletin of the LMS, No. 162, Vol. 33/3, May.