Transport, diffusion, and energy studies in the Arnold-Beltrami-Childress map

Power-law trapping in the volume-preserving Arnold-Beltrami-Childress map

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.

Characteristic times in the standard map

We study and compare three characteristic times of the standard map: the Lyapunov time t_{L}, the Poincaré recurrence time t_{r}, and the stickiness (or escape) time t_{st}. The Lyapunov time is the inverse of the Lyapunov characteristic number (L) and in general is quite small. We find empirical relations for the L as a function of the nonlinearity parameter K and of the chaotic area A. We also find empirical relations for the Poincaré recurrence time t_{r} as a function of the nonlinearity parameter K, of the chaotic area A, and of the size of the box of initial conditions ε. As a consequence, we find relations between t_{r} and L. We compare the distributions of the stickiness time and the Poincaré recurrence time. The stickiness time inside the sticky regions at the boundary of the islands of stability is orders of magnitude smaller than the Poincaré recurrence time t_{r} and this affects the diffusion exponent μ, which converges always to the value μ=1. This is shown in an extreme stickiness case. The diffusion is anomalous (ballistic motion) inside the accelerator mode islands of stability with μ=2 but it is normal everywhere outside the islands with μ=1. In a particular case of extreme stickiness, we find the hierarchy of islands around islands.

Global and local diffusion in the standard map

We study the global and the local transport and diffusion in the case of the standard map, by calculating the diffusion exponent μ. In the global case, we find that the mean diffusion exponent for the whole phase space is either μ=1, denoting normal diffusion, or μ=2 denoting anomalous diffusion (and ballistic motion). The mean diffusion of the whole phase space is normal when no accelerator mode exists and it is anomalous (ballistic) when accelerator mode islands exist even if their area is tiny in the phase space. The local value of the diffusion exponent inside the normal islands of stability is μ=0, while inside the accelerator mode islands it is μ=2. The local value of the diffusion exponent in the chaotic region outside the islands of stability converges always to the value of 1. The time of convergence can be very long, depending on the distance from the accelerator mode islands and the value of the nonlinearity parameter K. For some values of K, the stickiness around the accelerator mode islands is maximum and initial conditions inside the sticky region can be dragged in a ballistic motion for extremely long times of the order of 10^{7} or more but they will finally end up in normal mode diffusion with μ=1. We study, in particular, cases with maximum stickiness and cases where normal and accelerator mode islands coexist. We find general analytical solutions of periodic orbits of accelerator type and we give evidence that they are much more numerous than the normal periodic orbits. Thus, we expect that in every small interval ΔK of the nonlinearity parameter K of the standard map there exist smaller intervals of accelerator mode islands. However, these smaller intervals are in general very small, so that in the majority of the values of K the global diffusion is normal.