Braids of entangled particle trajectories

In many applications, the two-dimensional trajectories of fluid particles are available, but little is known about the underlying flow. Oceanic floats are a clear example. To extract quantitative information from such data, one can measure single-particle dispersion coefficients, but this only uses one trajectory at a time, so much of the information on relative motion is lost. In some circumstances the trajectories happen to remain close long enough to measure finite-time Lyapunov exponents, but this is rare. We propose to use tools from braid theory and the topology of surface mappings to approximate the topological entropy of the underlying flow. The procedure uses all the trajectory data and is inherently global. The topological entropy is a measure of the entanglement of the trajectories, and converges to zero if they are not entangled in a complex manner (for instance, if the trajectories are all in a large vortex). We illustrate the techniques on some simple dynamical systems and on float data from the Labrador sea.Comment: 24 pages, 21 figures. PDFLaTeX with RevTeX4 macros. Matlab code included with source. Fixed an inconsistent convention problem. Final versio

Analysing Lyapunov spectra of chaotic dynamical systems

It is shown that the asymptotic spectra of finite-time Lyapunov exponents of a variety of fully chaotic dynamical systems can be understood in terms of a statistical analysis. Using random matrix theory we derive numerical and in particular analytical results which provide insights into the overall behaviour of the Lyapunov exponents particularly for strange attractors. The corresponding distributions for the unstable periodic orbits are investigated for comparison.Comment: 4 pages, 4 figure

Computational Complexity of Iterated Maps on the Interval (Extended Abstract)

The exact computation of orbits of discrete dynamical systems on the interval is considered. Therefore, a multiple-precision floating point approach based on error analysis is chosen and a general algorithm is presented. The correctness of the algorithm is shown and the computational complexity is analyzed. As a main result, the computational complexity measure considered here is related to the Ljapunow exponent of the dynamical system under consideration

Kolmogorov turbulence, Anderson localization and KAM integrability

The conditions for emergence of Kolmogorov turbulence, and related weak wave turbulence, in finite size systems are analyzed by analytical methods and numerical simulations of simple models. The analogy between Kolmogorov energy flow from large to small spacial scales and conductivity in disordered solid state systems is proposed. It is argued that the Anderson localization can stop such an energy flow. The effects of nonlinear wave interactions on such a localization are analyzed. The results obtained for finite size system models show the existence of an effective chaos border between the Kolmogorov-Arnold-Moser (KAM) integrability at weak nonlinearity, when energy does not flow to small scales, and developed chaos regime emerging above this border with the Kolmogorov turbulent energy flow from large to small scales.Comment: 8 pages, 6 figs, EPJB style