Abstract
Understanding stickiness and power-law behavior of Poincaré recurrence statistics is an open problem for higher-dimensional systems, in contrast to the well-understood case of systems with two degrees of freedom. We study such intermittent behavior of chaotic orbits in three-dimensional volume-preserving maps using the example of the Arnold-Beltrami-Childress map. The map has a mixed phase space with a cylindrical regular region surrounded by a chaotic sea for the considered parameters. We observe a characteristic overall power-law decay of the cumulative Poincaré recurrence statistics with significant oscillations superimposed. This slow decay is caused by orbits which spend long times close to the surface of the regular region. Representing such long-trapped orbits in frequency space shows clear signatures of partial barriers and reveals that coupled resonances play an essential role. Using a small number of the most relevant resonances allows for classifying long-trapped orbits. From this the Poincaré recurrence statistics can be divided into different exponentially decaying contributions, which very accurately explains the overall power-law behavior including the oscillations.
Highlights
Hamiltonian systems generically have a mixed phase space in which regular and chaotic motion coexist
The 3D phase space of the map is mixed with two disjoint regular regions embedded in a chaotic sea
For the uncoupled case regular regions of the 3D map are in the form of concentric cylinders organized around 1D elliptic tori corresponding to the fixed points of the 2D subsystem
Summary
Hamiltonian systems generically have a mixed phase space in which regular and chaotic motion coexist. A typical chaotic trajectory is often trapped intermittently close to regular regions of phase space for arbitrary long but finite times. We investigate power-law trapping in threedimensional volume-preserving maps using the specific example of the Arnold-Beltrami-Childress (ABC) map. One long-trapped orbit is displayed in the 3D phase space, on a 2D Poincaré section, and in frequency-time representation in Sec. III B. III D we use the frequency analysis to identify a small number of the most relevant resonances This allows for classifying long-trapped orbits and by this to divide the Poincaré recurrence statistics into different exponentially decaying contributions. This very accurately explains the overall power-law behavior including the oscillations.
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