G. Haller: Chaos Near Resonance, Kartoniert / Broschiert
Chaos Near Resonance
(soweit verfügbar beim Lieferanten)
- Verlag:
- Springer US, 10/2012
- Einband:
- Kartoniert / Broschiert, Paperback
- Sprache:
- Englisch
- ISBN-13:
- 9781461271727
- Artikelnummer:
- 3490556
- Umfang:
- 452 Seiten
- Sonstiges:
- XVI, 430 p.
- Ausgabe:
- Softcover reprint of the original 1st edition 1999
- Copyright-Jahr:
- 2012
- Gewicht:
- 678 g
- Maße:
- 235 x 155 mm
- Stärke:
- 24 mm
- Erscheinungstermin:
- 17.10.2012
- Hinweis
-
Achtung: Artikel ist nicht in deutscher Sprache!
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Beschreibung
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of chaos near resonances is discussed in both the dissipative and the Hamiltonian context, incorporating previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds.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.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 ilnikov 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 ilnikov 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 ilnikov 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 ilnikov 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 ilnikov 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
Klappentext
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.
