Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows

Abstract

4.3.3. A system of coordinates on Λ * ε 4.3.4. The Hamiltonian flow in Λ * ε 4.3.5. Averaging theory 4.3.6. K.A.M. theory 4.3.7. Some approximate expressions for the KAM tori 4.4. The perturbed scattering map 4.4.1. Introduction and overview Date: July 14 2004. AMADEU DELSHAMS, RAFAEL DE LA LLAVE, AND TERE M. SEARA 4.4.2. Calculation of the scattering map 4.4.3. Intersection of KAM tori with the image of KAM tori under the scattering map 42 4.4.4. The Melnikov potential 4.5. Existence of transition chains and orbits of unbounded energy 4.5.1. Whiskered tori 4.5.2. Transition paths 4.5.3. Existence of transition paths for systems as in Theorem 3.4 4.5.4. Existence of orbits shadowing transition paths 4.5.5. Proof of Theorem 4.27 Acknowledgements Appendix A. Proof of Theorem 4.8 A.1. Diophantine properties A.2. Spaces of functions A.3. Some canonical transformations A.3.1. Transformations close to identity A.4. Proof of an analytic version of the KAM theorem A.4.1. Estimates for a step of the iterative procedure A.4.2. Repeating the iterative step and convergence A.5. Proof of a differentiable version of the KAM theorem A.5.1. Proof of Theorem A.26 Appendix B. Proof of Theorem 4.1 B.1. System of coordinates B.2. Extensions of the equations B.3. The Poincaré map B.4. Equations for the invariant manifolds B.5. Solution of the fixed point equations B.5.1. The operator T is well defined B.5.2. The range of the operator T B.5.3. Contraction properties of T B.5.4. Sharp regularity B.6. The stable and unstable manifolds ofΛ B.6.1. The stable and unstable bundles ofΛ B.6.2. The stable/unstable manifolds for a point B.7. Uniqueness of the manifoldΛ. Invariance.

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…The criteria for the existence of trajectories of the energy that grows up to infinity are known for sufficiently large initial energies [10,24,29,30,44,45,80,81]. The results of the present paper can be used to establish the generic existence of orbits of unbounded energy for all possible values of initial energy.…”

“…Arnold's paper inspired a large number of studies in the longtime stability of actions, the problem which is known as "Arnold diffusion". It has been attracting significant attention recently and we refer the reader to papers [11,31,33,[40][41][42][43]54,57,[60][61][62][68][69][70]76,82,84] for a more detailed discussion.…”

“…A system of this type is called a priori unstable. The drift of orbits along the cylinder has been actively studied in the last decade [3][4][5][7][8][9]12,[18][19][20]22,27,28,[31][32][33][34]70,87,88], including the problem of genericity of this phenomenon and instability times. It should be noted that the Arnold diffusion can be much faster in this case.…”

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

“…This excludes possibly interesting applications. Settings where a noncompact, general geometric version of normal hyperbolicity may be useful include chemical reaction dynamics [Uze+02] and problems in classical and celestial mechanics [DLS06].…”

“…It seems to be a well-known belief by many experts that the theory of normal hyperbolicity can be extended to a general noncompact setting [DLS06,p. 165].…”