Arnold diffusion for smooth systems of two and a half degrees of freedom

Arnold diffusion for smooth systems of two and a half degrees of freedom /

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Bibliographic Details
Main Authors: Kaloshin, Vadim Yu., 1975- (Author), Zhang, Ke, (Mathematician) (Author)
Corporate Author: JSTOR (Organization)
Format: Electronic eBook
Language:English
Published: Princeton, New Jersey : Princeton University Press 2020.
Series:Annals of mathematics studies ; no. 208.
Subjects:
Hamiltonian systems.
Diffusion > Mathematical models.
Online Access:Connect to this title online (unlimited simultaneous users allowed)
Table of Contents:
  • Machine generated contents note: 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. 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
  • 3.1. Resonant component and non-degeneracy conditions
  • 3.2. Normal form
  • 3.3. resonant component
  • 4. Double resonance: geometric description
  • 4.1. slow system
  • 4.2. Non-degeneracy conditions for the slow system
  • 4.3. Normally hyperbolic cylinders
  • 4.4. Local maps and global maps
  • 5. Double resonance: forcing equivalence
  • 5.1. Choice of cohomologies for the slow system
  • 5.2. Aubry-Mather type at a double resonance
  • 5.3. Connecting to Γk1, K2 and Γk1SR
  • 5.3.1. Connecting to the double resonance point
  • 5.3.2. Connecting single and double resonance
  • 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, Mane, and Mather sets
  • 6.4. dual setting, forward solutions
  • 6.5. Peierls barrier, static classes, elementary solutions
  • 6.6. forcing relation
  • 6.7. 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. 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 Mane 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 geodesies
  • 11. Aubry-Mather type at the double resonance
  • 11.1. High-energy case
  • 11.2. Simple non-critical case
  • 11.3. Simple critical case
  • 11.3.1. Proof of Aubry-Mather type using local coordinates
  • 11.3.2. Construction of the local coordinates
  • 12. Forcing equivalence between kissing cylinders
  • 12.1. Variational problem for the slow mechanical system
  • 12.2. Variational problem for original coordinates
  • 12.3. Scaling limit of the barrier function
  • 12.4. jump mechanism
  • IV. Supplementary topics
  • 13. Generic properties of mechanical systems on the two-torus
  • 13.1. Generic properties of periodic orbits
  • 13.2. Generic properties of minimal orbits
  • 13.3. Non-degeneracy at high-energy
  • 13.4. Unique hyperbolic minimizer at very high energy
  • 13.5. Generic properties at the critical energy
  • 14. Derivation of the slow mechanical system
  • 14.1. Normal forms near maximal resonances
  • 14.2. AfRne coordinate change, rescaling, and energy reduction
  • 14.3. Variational properties of the coordinate changes
  • 15. Variational aspects of the slow mechanical system
  • 15.1. Relation between the minimal geodesies and the Aubry sets
  • 15.2. Characterization of the channel and the Aubry sets
  • 15.3. width of the channel
  • 15.4. case E = 0.

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