Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom (ams-208).

The first complete proof of Arnold diffusion--one of the most important problems in dynamical systems and mathematical physicsArnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical phy...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Kaloshin, Vadim
Otros Autores: Zhang, Ke
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Princeton : Princeton University Press, 2020.
Colección:Annals of mathematics studies.
Temas:
Hamiltonian systems.
Diffusion > Mathematical models.
Systèmes hamiltoniens.
Diffusion (Physique) > Modèles mathématiques.
SCIENCE / Mechanics / Dynamics.
Diffusion > Mathematical models
Hamiltonian systems
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Title
  • Copyright
  • Dedication
  • Contents
  • Preface
  • Acknowledgments
  • I Introduction and the general scheme
  • 1 Introduction
  • 1.1 Statement of the result
  • 1.2 Scheme of diffusion
  • 1.3 Three regimes of diffusion
  • 1.4 The outline of the proof
  • 1.5 Discussion
  • 2 Forcing relation
  • 2.1 Sufficient condition for Arnold diffusion
  • 2.2 Diffusion mechanisms via forcing equivalence
  • 2.3 Invariance under the symplectic coordinate changes
  • 2.4 Normal hyperbolicity and Aubry-Mather type
  • 3 Normal forms and cohomology classes at single resonances
  • 5.4 Jump from non-simple homology to simple homology
  • 5.5 Forcing equivalence at the double resonance
  • II Forcing relation and Aubry-Mather type
  • 6 Weak KAM theory and forcing equivalence
  • 6.1 Periodic Tonelli Hamiltonians
  • 6.2 Weak KAM solution
  • 6.3 Pseudographs, Aubry, Mañé, and Mather sets
  • 6.4 The dual setting, forward solutions
  • 6.5 Peierls barrier, static classes, elementary solutions
  • 6.6 The forcing relation
  • 6.7 The Green bundles
  • 7 Perturbative weak KAM theory
  • 7.1 Semi-continuity
  • 7.2 Continuity of the barrier function
  • 7.3 Lipschitz estimates for nearly integrable systems
  • 7.4 Estimates for nearly autonomous systems
  • 8 Cohomology of Aubry-Mather type
  • 8.1 Aubry-Mather type and diffusion mechanisms
  • 8.2 Weak KAM solutions are unstable manifolds
  • 8.3 Regularity of the barrier functions
  • 8.4 Bifurcation type
  • III Proving forcing equivalence
  • 9 Aubry-Mather type at the single resonance
  • 9.1 The single maximum case
  • 9.2 Aubry-Mather type at single resonance
  • 9.3 Bifurcations in the double maxima case
  • 9.4 Hyperbolic coordinates
  • 9.5 Normally hyperbolic invariant cylinder
  • 9.6 Localization of the Aubry and Mañé sets
  • 9.7 Genericity of the single-resonance conditions
  • 10 Normally hyperbolic cylinders at double resonance
  • 10.1 Normal form near the hyperbolic fixed point
  • 10.2 Shil'nikov's boundary value problem
  • 10.3 Properties of the local maps
  • 10.4 Periodic orbits for the local and global maps
  • 10.5 Normally hyperbolic invariant manifolds
  • 10.6 Cyclic concatenations of simple geodesics
  • 11 Aubry-Mather type at the double resonance
  • 11.1 High-energy case
  • 11.2 Simple non-critical case
  • 11.3 Simple critical case

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