Narrow Results

The standard map x′ = x + y′, y′ = y + (K/2π)sin(2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ΔK, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theoretical results.

We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary "critical" region containing a smooth curve of tangencies between certain canonical "stable" foliations.

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analyzed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator.

The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same.

Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps, the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps, the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.

We apply to bidimensional chaotic maps the numerical method proposed by Ginelli et al. [2007] to approximate the associated Oseledets splitting, i.e. the set of linear subspaces spanned by the so-called covariant Lyapunov vectors (CLV) and corresponding to the Lyapunov spectrum. These subspaces are the analog of linearized invariant manifolds for nonperiodic points, so the angles between them can be used to quantify the degree of hyperbolicity of generic orbits; however, such splitting being noninvariant under smooth transformations of phase space, it is interesting to investigate the properties of transversality when coordinates change, e.g. to study it in distinct dynamical systems. To illustrate this issue on the Chirikov–Taylor standard map, we compare the probability densities of transversality for two different coordinate systems; these are connected by a linear transformation that deforms splitting angles through phase space, changing also the probability density of almost-zero angles although complete tangencies are in fact invariant. This is completely due to the PDF transformation law and strongly suggests that any statistical inference from such distributions must be generally taken with care.

We introduce a new dynamical indicator of stability based on the Extreme Value statistics showing that it provides an insight into the local stability properties of dynamical systems. The indicator performs faster than others based on the iteration of the tangent map since it requires only the evolution of the original systems and, in the chaotic regions, gives further information about the local information dimension of the attractor. A numerical validation of the method is presented through the analysis of the motions in the Standard map.

A new chaos indicator Ultra Fast Lyapunov Indicator (UFLI) is proposed in this paper. UFLI uses second order derivatives and can distinguish chaotic orbits from torus orbits (orbits on a torus) more sensitively than FLI, OFLI and OFLI² in a short iterations for the standard map for δ = 0.9 and the Froeschlé map ε = 0.6 if we chose an initial variational vector properly. For the generalized Boole transformations (Boole transformation) whose analytic formula of Lyapunov exponent is given, the value of UFLI grows more rapidly than FLI except for α = 0.999, which is the weak chaos near to the infinite ergodic point while the difference in growing speed of the indicator values is relatively small compared with the standard map or the Froeschlé map.