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Diffusing orbits along chains of cylinders

  • *Corresponding author: Marian Gidea

    *Corresponding author: Marian Gidea 

The first author is supported by NSF grants DMS-0635607, DMS-1515851 and DMS-1814543, and by the Alfred P. Sloan Foundation grant G-2016-7320

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  • We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on $ {{\mathbb A}}^3 $. Our approach can also be applied to more general Hamiltonians that are not necessarily convex.

    The main geometric objects in our framework are $ 3 $–dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections.

    Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the tilt map and the homoclinic maps associated to its essential invariant circles.

    We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.

    Mathematics Subject Classification: Primary: 37J40, 70H08; Secondary: 37E40.

    Citation:

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  • Figure 1.  An illustration of a cylinder chain, consisting of a sequence of tame cylinders with boundary, connected via homoclinic and heteroclinic orbits. This chain also contains a singular cylinder (shown in the center of the figure); see Section 3.3 for details

    Figure 2.  Homoclinic correspondence on the cylinder and the induced homoclinic correspondence on the Poincaré section

    Figure 3.  Splitting arcs

    Figure 4.  A triangular domain associated to a right splitting arc

    Figure 5.  The setting of Proposition 3.2

    Figure 6.  Positively and negatively tilted arcs

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