Miguel A. F. Sanjuán

Combinatorial detection of determinism in noisy time series

This paper deals with the distinction between white noise and deterministic chaos in multivariate noisy time series. Our method is combinatorial in the sense that it is based on the properties of topological permutation entropy, and it becomes especially interesting when the noise is so high that the standard denoising techniques fail, so a detection of determinism is the most one can hope for. It proceeds by i) counting the number of the so-called ordinal patterns in independent samples of length L from the data sequence and ii) performing a χ2 test based on the results of i), the null hypothesis being that the data are white noise. Holds the null hypothesis, so should all possible ordinal patterns of a given length be visible and evenly distributed over sufficiently many samples, contrarily to what happens in the case of noisy deterministic data. We present numerical evidence in two dimensions for the efficiency of this method. A brief comparison with two common tests for independence, namely, the calculation of the autocorrelation function and the BDS algorithm, is also performed.

Synchronization and propagation of bursts in networks of coupled map neurons.

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy.

Unpredictable behavior in the Duffing oscillator: Wada basins

This paper describes some numerical experiments giving evidence of Wada basin boundaries for the Duffing oscillator. We suggest some mechanisms by which this fractal property of the boundaries appears and discuss the difficulties that the Wada property presents for predicting the behavior of dynamical systems.

Defining strategies to win in the Internet market

This paper analyzes a model for the competition dynamics of web sites in the Internet, based on the Lotka–Volterra competition equations. This model shows the well known appearance of a winner-take-all characteristic and is based in the nonvalidity of traditional offer and demand equilibrium theory of these kinds of markets. From the stability analysis of the model, we establish a series of rules which are useful for defining strategies in the Internet market. One of the most important results that emerge from this simple model is the appearance of some unexpected phenomena related to the collaboration and competition between sites.

THE EFFECT OF NONLINEAR DAMPING ON THE UNIVERSAL ESCAPE OSCILLATOR

This paper analyzes the role of nonlinear dissipation on the universal escape oscillator. Nonlinear damping terms proportional to the power of the velocity are assumed and an investigation on its effects on the dynamics of the oscillator, such as the threshold of period-doubling bifurcation, fractal basin boundaries and how the basins of attraction are destroyed, is carried out. The results suggest that increasing the power of the nonlinear damping, has similar effects as of decreasing the damping coefficient for a linearly damped case, showing the very importance of the level or amount of energy dissipation.

Basin topology in dissipative chaotic scattering

Chaotic scattering in open Hamiltonian systems under weak dissipation is not only of fundamental interest but also important for problems of current concern such as the advection and transport of inertial particles in fluid flows. Previous work using discrete maps demonstrated that nonhyperbolic chaotic scattering is structurally unstable in the sense that the algebraic decay of scattering particles immediately becomes exponential in the presence of weak dissipation. Here we extend the result to continuous-time Hamiltonian systems by using the Henon-Heiles system as a prototype model. More importantly, we go beyond to investigate the basin structure of scattering dynamics. A surprising finding is that, in the common case where multiple destinations exist for scattering trajectories, Wada basin boundaries are common and they appear to be structurally stable under weak dissipation, even when other characteristics of the nonhyperbolic scattering dynamics are not. We provide numerical evidence and a geometric theory for the structural stability of the complex basin topology.

New developments in classical chaotic scattering

Classical chaotic scattering is a topic of fundamental interest in nonlinear physics due to the numerous existing applications in fields such as celestial mechanics, atomic and nuclear physics and fluid mechanics, among others. Many new advances in chaotic scattering have been achieved in the last few decades. This work provides a current overview of the field, where our attention has been mainly focused on the most important contributions related to the theoretical framework of chaotic scattering, the fractal dimension, the basins boundaries and new applications, among others. Numerical techniques and algorithms, as well as analytical tools used for its analysis, are also included. We also show some of the experimental setups that have been implemented to study diverse manifestations of chaotic scattering. Furthermore, new theoretical aspects such as the study of this phenomenon in time-dependent systems, different transitions and bifurcations to chaotic scattering and a classification of boundaries in different types according to symbolic dynamics are also shown. Finally, some recent progress on chaotic scattering in higher dimensions is also described.

ANALYTICAL ESTIMATES OF THE EFFECT OF NONLINEAR DAMPING IN SOME NONLINEAR OSCILLATORS

This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.

Analysis of vibrational resonance in a quintic oscillator

We consider a damped quintic oscillator with double-well and triple-well potentials driven by both low-frequency force f cos (omega)t and high-frequency force g cos (Omega)t with Omega>>omega and analyze the occurrence of vibrational resonance. The response consists of a slow motion with frequency omega and a fast motion with frequency Omega. We obtain an approximate analytical expression for the response amplitude Q at the low-frequency omega. From the analytical expression of Q, we determine the values of omega and g (denoted as omega(VR) and g(VR)) at which vibrational resonance occurs. The theoretical predictions are found to be in good agreement with numerical results. We show that for fixed values of the parameters of the system, as omega varies, resonance occurs at most one value of omega. When the amplitude g is varied we found two and four resonances in the system with double-well and triple-well cases, respectively. We present examples of resonance (i) without cross-well motion and (ii) with cross-well orbit far before and far after it. omega(VR) depends on the damping strength d while g(VR) is independent of d. Moreover, the effect of d is found to decrease the response amplitude Q.

Escape patterns, magnetic footprints, and homoclinic tangles due to ergodic magnetic limiters

The action of a set of ergodic magnetic limiters in tokamaks is investigated from the Hamiltonian chaotic scattering point of view. Special attention is paid to the influence of invariant sets, such as stable and unstable manifolds, as well as the strange saddle, on the formation of the chaotic layer at the plasma edge. The nonuniform escape process associated to chaotic field lines is also analyzed. It is shown that the ergodic layer produced by the limiters has not only a fractal structure, but it possesses the even more restrictive Wada property.