Invariant Spectra in N-Coupled Standard Maps

This paper examines the evolution of the dynamical spectra of stretching numbers in a system of [Formula: see text]-coupled Standard Maps. It is found that the convergence rate of the spectra to their invariant forms is independent of the dimensionality 2[Formula: see text] and the nonlinearity/coupling parameters. This rate is impressively faster than one could predict on the basis of ergodicity. This effect is probably associated with the manifold of the maximum Lyapunov exponent along which the main stretching occurs. It seems that dynamical spectra depend mainly on the dramatically smaller subspace defined by this unstable manifold.

The Role of Chaos in Barred Galaxies

Ordered orbits in barred galaxies appear along the bar and between the −4/1 and −2/1 resonances of the outer spiral. Chaotic orbits appear mainly near corotation. Such orbits support the bar and the spiral for long times and they are important for self-consistency. There are three main mechanisms for transition from order to chaos: (a) infinite bifurcations, (b) infinite gaps, and (c) infinite spirals. The Lyapunov characteristic number is zero for ordered orbits and positive for chaotic orbits. But much more information is provided by the distribution of the stretching numbers (one-period Lyapunov characteristic numbers). The spectrum of stretching numbers is invariant with respect to initial conditions in a connected chaotic domain. We provide examples of such spectra for 2-D maps, plane galactic orbits, 2-D dissipative systems, 3-D systems (represented by 4-D maps), and systems depending periodically on the time.