Search

Phase control of chaos is a powerful technique but little is known about its physical constraints, relevant for real systems. As a fact, it has not been explored whether this technique can also be applied when the controlling perturbation is not harmonic. Here we apply phase control on a driven double well Duffing oscillator using periodic rectangular pulsed perturbations instead of the classical sinusoidal perturbations. Experimental measurements and numerical simulations show that this kind of perturbation is also able to stabilize the chaotic orbits for an adequate selection of the phase. Furthermore, as the duty cycle of the perturbation (that is, the fraction of the time that the periodically pulsed control is active) is increased, two separate regimes occur. In the first one, the perturbations leading to stabilization of periodic solutions are of constant energy (taken as the product of the duty cycle and the amplitude) and in the second one, a saturation phenomenon occurs, implying that increasing energy values of the perturbations are wasted. Our results unveil the versatility of the pulsed phase control scheme and the importance of energy constraints.

We discuss the phase control technique consisting of an applied square pulsed periodic perturbation. We explore the effect of such perturbations to the different terms of the Duffing oscillator. We find that the effect depends sensitively on how the perturbation is applied, indeed, it is specially effective when it modulates the cubic and the linear term and uneffective when applied to the driving term. Our results highlight the highly nontrivial role of the phase when applying a second periodic perturbation to a chaotic system.

Phase-control techniques of chaos aim to extract periodic behaviors from chaotic systems by applying weak harmonic perturbations with a suitably chosen phase. However, little is known about the best strategy for selecting adequate perturbations to reach desired states. Here we use experimental measures and numerical simulations to assess the benefits of controlling individually the three terms of a Duffing oscillator. Using a real-time analog indicator able to discriminate on-the-fly periodic behaviors from chaos, we reconstruct experimentally the phase versus perturbation strength stability areas when periodic perturbations are applied to different terms governing the oscillator. We verify the system to be more sensitive to perturbations applied to the quadratic term of the double-well Duffing oscillator and to the quartic term of the single-well Duffing oscillator.

The partial control technique allows one to keep the trajectories of a dynamical system inside a region where there is a chaotic saddle and from which nearly all the trajectories diverge. Its main advantage is that this goal is achieved even if the corrections applied to the trajectories are smaller than the action of environmental noise on the dynamics, a counterintuitive result that is obtained by using certain safe sets. Using the Henon map as a paradigm, we show here the deep relationship between the safe sets and the sets of points with different escape times, the escape time sets. Furthermore, we show that it is possible to find certain extended safe sets that can be used instead of the safe sets in the partial control technique. Numerical simulations confirm our findings and show that in some situations, the use of extended safe sets can be more advantageous.

We study, both numerically and experimentally, the synchronization of uncoupled excitable systems due to a common noise. We consider two identical FitzHugh-Nagumo systems, which display both spiking and non-spiking behaviours in chaotic or periodic regimes. An electronic circuit provides a laboratory implementation of these dynamics. Synchronization is tested with both white and coloured noise, showing that coloured noise is more effective in inducing synchronization of the systems. We also study the effects on the synchronization of parameter mismatch and of the presence of intrinsic (not common) noise, and we conclude that the best performance of coloured noise is robust under these distortions.

This Letter shows that a suitable chaotic signal can induce resonant effects analogous to those observed in presence of noise in a bistable system under periodic forcing. By constructing groups of chaotic and random perturbations with similar one-time statistics we show that in some cases chaos and noise induce indistinguishable resonant effects. This reinforces the conjecture by which in some situations where noise is supposed to play a key role maybe chaos is the key ingredient. Here we also show that the presence of a chaotic signal as the perturbation leading to a resonance opens new control perspectives based on our ability to stabilize chaos in different periodic orbits. A discussion of the possible implications of these facts is also presented at the end of the Letter. (C) 2007 Elsevier B.V. All rights reserved.

Basin boundary metamorphoses are characteristic in some kinds of chaotic dynamical systems. They take place when one parameter of the system is varied and it passes through a certain critical value. In this paper we show that a parametric harmonic perturbation can produce basin boundary metamorphoses in periodically driven chaotic dynamical systems even when its amplitude is smaller than the amplitude of the main driving. The physical context of this work is related with the phenomenon of trajectories escaping from a potential well, which is illustrated by using as prototype model the Helmholtz oscillator. One of the main contributions of our research is to analyze the role of the phase difference between the parametric harmonic perturbation and the main driving in the appearance of a basin boundary metamorphosis. We also analyze the variation of the size of the basins and the fractal dimension of the corresponding boundaries when this phenomenon occurs. Finally, Melnikov analysis of this phenomenon is carried out. We expect that this work can be useful for a better understanding of basin boundary metamorphoses phenomena.

We call permutation complexity the kind of dynamical complexity captured by any quantity or functional based on order relations, like ordinal patterns and permutation entropies. These mathematical tools have found interesting applications in time series analysis and abstract dynamical systems. In this letter we propose to extend the study of permutation complexity to spatiotemporal systems, by applying some of its tools to a time series obtained by coarse-graining the dynamics and to state vectors at fixed times, considering the latter as sequences. We show that this approach allows to quantify the complexity and to classify different types of dynamics in cellular automata and in coupled map lattices. Furthermore, we show that our analysis can be used to discriminate between different types of spatiotemporal dynamics registered in magnetoencephalograms. Copyright (C) EPLA, 2010

In this paper, we show that there is a simple and intuitive relation between crisis-induced intermittency and a boundary crisis in unimodal maps and in a family of chaotic flows with strong volume contraction. This point of view allows us to identify a set of 'target points' directly from the time series of the considered system that can be used to tame crisis-induced intermittency. To do this, we use an advantageous adaptation of a control scheme that was typically used in the context of control of transient chaos. We illustrate here these ideas with the quadratic map and Rossler flow.

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 chi(2) 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. Copyright c (c) EPLA, 2008

Here we study how to control the dynamics of excitable systems by using the phase control technique. Excitable systems are relevant in neuronal dynamics and therefore this method might have important applications. We use the periodically driven FitzHugh-Nagumo (FHN) model, which displays both spiking and non-spiking behaviours in chaotic or periodic regimes. The phase control technique consists of applying a harmonic perturbation with a suitable phase phi that we adjust in search of different behaviours of the FHN dynamics. We compare our numerical results with experimental measurements performed on an electronic circuit and find good agreement between them. This method might be useful for a better understanding of excitable systems and different phenomena in neuronal dynamics.

In this letter we discuss some properties of order patterns both in deterministic and random orbit generation. As it turns out, the orbits of one- dimensional maps have always forbidden patterns, i. e., order patterns that cannot occur, in contrast with random time series, in which any order pattern appears with probability one. However, finite random sequences may exhibit "false" forbidden patterns with non- vanishing probability. In this case, forbidden patterns decay with the sequence length, thus unveiling the random nature of the sequence. Last but not least, true forbidden patterns are robust against noise and disintegrate with a rate that depends on the noise level. These properties can be embodied in a simple method to distinguish deterministic,finite time series with very high levels of observational noise, from random ones. We present numerical evidence for white noise.

Autore/i: Enti di appartenenza e/o affiliazione: A well-known method to suppress chaos in a periodically forced chaotic system is to add a harmonic perturbation. The phase control of chaos scheme uses the phase difference between a small added harmonic perturbation and the main driving to suppress chaos, leading the system to different periodic orbits. Using the Duffing oscillator as a paradigm, we present here an in-depth study of this technique. A thorough numerical exploration has been made focused in the important role played by the phase, from which new interesting patterns in parameter space have appeared. On the other hand, our novel experimental implementation of phase control in an electronic circuit confirms both the well-known features of this method and the new ones detected numerically. All this may help in future implementations of phase control of chaos, which is globally confirmed here to be robust and easy to implement experimentally.