Accelerator modes and anomalous diffusion in 3D volume-preserving maps

Abstract

Angle-action maps that are periodic in the action direction can have accelerator modes: orbits that are periodic when projected onto the torus, but that lift to unbounded orbits in an action variable. In this paper we construct a volume-preserving family of maps, with two angles and one action, that have accelerator modes created at Hopf-one (or saddle-center-Hopf) bifurcations. Near such a bifurcation we show that there is often a bubble of invariant tori. Computations of chaotic orbits near such a bubble show that the trapping times have an algebraic decay similar to that seen around stability islands in area-preserving maps. As in the 2D case, this gives rise to anomalous diffusive properties of the action in our 3D map.

Accelerator-mode islands and superdiffusion in double-kicked rotor

“…The superdiffusion originates from the existence of stable accelerator modes island (AM island). [12,[34][35][36] The AMs are generalized periodic orbits of maps having translational symmetry in phase space. The cylindrical phase space of DKR allows one to define consistently periodic orbits in a generalized version [23]…”

Bifurcations in Volume-Preserving Systems

“…-Lomelí and Ramírez-Ros [30] concentrate on the splitting of separatrices as the stable and unstable manifold forming the above spherical structure (for which they use the terms spheromak and Hill's spherical vortices) cease to coincide due to a volume-preserving perturbation. -Meiss et al [33] also consider 3-dimensional diffeomorphisms, where chaotic orbits are studied near the spherical structure described above. In particular it is found that trapping times exhibit an algebraic decay.…”

“…The Cantor family is confined to what is left of T × S 2 after separatrix splitting, again see [30]. The construction in [33] relates such vortex bubbles resulting from elliptic periodic volume-preserving Hopf bifurcations to unbounded chaotic motion. The frequency vector (ω, α, η) of the normally elliptic invariant 2-tori can also be effectively controlled by the single parameter β.…”

Dynamics of the Chaplygin ball on a rotating plane

“…The problem of finite motion of the contact point is also unresolved. It is possible that there exist unbounded trajectories due to the phenomenon of diffusion described for three-dimensional maps in the recent paper [33].…”