Chaos with a high-dimensional torus

Abstract

Transition from quasiperiodicity with many frequencies (i.e., a high-dimensional torus) to chaos is studied by using N -dimensional globally coupled circle maps. First, the existence of N -dimensional tori with N ≥ 2 is confirmed while they become exponentially rare with N . Besides, chaos exists even when the map is invertible, and such chaos has more null Lyapunov exponents as N increases. This unusual form of "chaos on a torus," termed toric chaos, exhibits delocalization and slow dynamics of the first Lyapunov vector. Fractalization of tori at the transition to chaos is also suggested. The relevance of toric chaos to neural dynamics and turbulence is discussed in relation to chaotic itinerancy.

Statistical Features of High-Dimensional Hamiltonian Systems

“…It is clear that as soon as N ≥ 3 the motion lives in dimension 2N − 1 while the KAM tori have dimension N , so that they are not able to separate the phase space in disjoint components and, as we have said, ergodicity is guaranteed asymptotically from the point of view of dynamics. A trajectory initially closed to a KAM torus may visit any region of the energy hypersurface, but, since the compenetration of chaotic regions and invariant manifolds typically follows the pattern of a fractal geometry, the diffusion across the isoenergetic hypersurface is usually very slow; as we said, such a phenomenon is called Arnol'd diffusion [13,15]. The important and difficult problem is therefore to understand the "speed" of the Arnold diffusion, i.e.…”

Three-dimensional quasiperiodic torsional flows in rotating spherical fluids at very low Prandtl numbers

“…However, the ratios f 1 /f 3 and f 2 /f 3 have a strong variation among the solutions due to the fast increase of f 3 . These ratios determine the type of quasiperiodicity on the T 3[36,37]. Despite the spectra of Figs.14(a) and (b) have the same high peaks, at Ra = 7700, f 3 = 84.5 and at Ra = 8450 it is already 107.1, and the PS look very different, mainly when they are RS 3 (or near) as those of Figs.…”

“…and Kaneko [37] suggest the fractalization of the three-torus at the transition to chaos. No hint of this behavior was found here.…”

“…On the other hand the results have been compared with those found in Refs. [32,[36][37][38], which study the dynamics in three-tori of low-dimensional systems, and the transition from highdimensional tori to torus chaos or more complex chaotic attractors. These models allow to explore deeply the range of parameters, and to compute the Lyapunov exponents to determine when and how the dynamics becomes chaotic.…”

Synchronization–desynchronization transitions in networks of circle maps with sinusoidal coupling

“…[30,31] As a population of sine-circle maps are coupled to each other, the intermingled influences of coupling and mapping nonlinearities will make a clear analysis very difficult. [32][33][34][35][36][37][38][39][40][41] On the other hand, even for an unperturbed circle map, very complicated synchronization-desynchronization dynamics can be found as these simple oscillators are coupled to each other. In this paper, it is our mission to extensively investigate the synchronization dynamics of coupled circle maps.…”