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For the case of generic 4D symplectic maps with a mixed phase space we investigate the global organization of regular tori. For this we compute elliptic 1-tori of two coupled standard maps and display them in a 3D phase-space slice. This visualizes how all regular 2-tori are organized around a skeleton of elliptic 1-tori in the 4D phase space. The 1-tori occur in two types of one-parameter families: (a) Lyapunov families emanating from elliptic-elliptic periodic orbits, which are observed to exist even far away from them and beyond major resonance gaps, and (b) families originating from rank-1 resonances. At resonance gaps of both types of families either (i) periodic orbits exist, similar to the Poincare-Birkhoff theorem for 2D maps, or (ii) the family may form large bends. In combination these results allow for describing the hierarchical structure of regular tori in the 4D phase space analogously to the islands-around-islands hierarchy in 2D maps.
In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems
A recent model of Ariel et al. [1] for explaining the observation of Levy walks in swarming bacteria suggests that self-propelled, elongated particles in a periodic array of regular vortices perform a super-diffusion that is consistent with Levy walk
Extending the work of del-Castillo-Negrete, Greene, and Morrison, Physica D {bf 91}, 1 (1996) and {bf 100}, 311 (1997) on the standard nontwist map, the breakup of an invariant torus with winding number equal to the inverse golden mean squared is stu
We investigate the behavior of the Generalized Alignment Index of order $k$ (GALI$_k$) for regular orbits of multidimensional Hamiltonian systems. The GALI$_k$ is an efficient chaos indicator, which asymptotically attains positive values for regular
The spatiotemporal dynamics of Lyapunov vectors (LVs) in spatially extended chaotic systems is studied by means of coupled-map lattices. We determine intrinsic length scales and spatiotemporal correlations of LVs corresponding to the leading unstable