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In any control problem it is desirable to apply the control as infrequently as possible. In this paper we address the problem of how to minimize the frequency of control in presence of external perturbations, that we call disturbances, when the goal is to sustain transient chaos. We show here that the partial control method, that allows to find the minimum control required to sustain transient chaos in presence of disturbances, is the key to find such minimum control frequency. We prove first for the paradigmatic tent map of slope greater than 2 that for a constant value of the disturbances, the control required to sustain transient chaos decreases when the control is applied every k iterates of the map. We show that the combination of this property with the fact that the disturbances grow with k implies that there is a minimum control frequency and we provide a procedure to compute it. Finally we give evidence of the generality of this result showing that the same features are reproduced when considering the Henon map. (C) 2013 Elsevier B. V. All rights reserved.

Detecting determinism in univariate and multivariate time series is difficult if the underlying process is nonlinear, and the noise level is high. In a previous paper, the authors proposed a method based on observable ordinal patterns. This method exploits the robustness of admissible ordinal patterns against observational noise, and the super-exponential growth of forbidden ordinal patterns with the length of the patterns. The new method compared favorably to the Brock–Dechert–Scheinkman independence test when applied to time series projected from the Hénon attractor and contaminated with Gaussian noise of different variances. In this paper, we extend this comparison to higher fractal dimensions by using noisy orbits on the attractors of the Lorenz map, and the time-delayed Hénon map. Finally, we make an analysis that enlightens the robustness of admissible ordinal patterns in the presence of observational noise.

Horseshoe maps have played an important role in the study of nonlinear dynamical systems. Here we study maps associated to two simple physical systems: a four-hill potential and an open billiard. They turn out to be very different from the standard horseshoe maps: one iteration stretches and folds a square in phase space an infinite number of times before placing it across itself. In this paper we explore in further depth these infinite horseshoe maps. We show that the infinite folding action requires that the maps are not defined in part of the rectangle. Our exploration also shows that infinite horseshoes provide valuable information on the complexity of the system. In particular, they provide a visual way to code the wide variety of complex orbits existing in the dynamical systems considered. (C) 2014 Published by Elsevier B. V. ZB 0 Z8 0 ZR 0 ZS 0

Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Particles in such kind of systems can exhibit both bounded or unbounded motions for which escapes from the scattering region can take place. This paper analyzes how to control the escape of the particles from the scattering region in the presence of noise. For that purpose, we apply the partial control technique to the Hénon–Heiles system, which implies that we need to use a control smaller than the noise present in the system. The main finding of our work is the successful control of the particles in the scattering region with a control smaller than noise. We have also analyzed and compared the escapes time of orbits in the scattering region for different situations. Finally, we believe that our results might contribute to a better understanding of both chaotic scattering phenomena and the application of the partial control technique to continuous dynamical systems.

Many discrete-time dynamical systems have a region Q from which all or almost all trajectories leave, or at least they leave in the presence of perturbations that we call disturbances. We partially control systems so that despite disturbances the trajectories of a dynamical system stay in the region Q at least for some initial points in Q. The disturbances can be thought of as either noise or as purposeful, hostile efforts of an enemy to drive the trajectory out of the region. Our goal is to keep trajectories inside Q despite the disturbances and our partial control of chaos method succeeds. Surprisingly this goal can be achieved with a control whose maximum allowable size is smaller than the maximum allowed disturbance. A fundamental step towards this goal is to compute a set called the safe set that had, until now, been found only in certain very special situations. This paper provides a general algorithm for computing safe sets. The algorithm is able to compute the safe sets for a specified region in phase space, the maximum disturbance value, and the maximum allowed control. We call it the Sculpting Algorithm. Its operation is analogous to removing material while sculpting a statue. The algorithm sculpts the safe sets. Our Sculpting Algorithm is independent of the dimension and is fast for one- and two-dimensional dynamical systems. As examples, we apply the algorithm to two paradigmatic nonlinear dynamical systems, namely, the Henon map and the Duffing oscillator. (C) 2012 Elsevier B.V. All rights reserved.

We present an analog circuit implementation of the novel partial control method, that is able to sustain chaotic transient dynamics. The electronic circuit simulates the dynamics of the one-dimensional slope-three tent map, for which the trajectories diverge to infinity for nearly all the initial conditions after behaving chaotically for a while. This is due to the existence of a nonattractive chaotic set: a chaotic saddle. The partial control allows one to keep the trajectories close to the chaotic saddle, even if the control applied is smaller than the effect of the applied noise, introduced into the system. Furthermore, we also show here that similar results can be implemented on a circuit that simulates a horseshoe-like map, which is a simple extension of the previous one. This encouraging result validates the theory and opens new perspectives for the application of this technique to systems with higher dimensions and continuous time dynamics.

In this paper, we study the nonlinear response of the nonlinear mass-spring model with nonsmooth stiffness. For this purpose, we take as prototype model, a system that consists of the double-well smooth potential with an additional spring component acting on the system only for large enough displacement. We focus our study on the analysis of the homoclinic orbits for such nonlinear potential for which we observe the appearance of chaotic motion in the presence of damping effects and an external harmonic force, analyzing the crucial role of the linear spring in the dynamics of our system. The results are shown by using both the Melnikov analysis and numerical simulations. We expect our work to have implications on problems concerning the suspension of vehicles, among others.

We have proposed an adaptive procedure in order to estimate unknown parameters of a numerical model of a chaotic CO 2 laser, which is observed through a time series that represents the output intensity of the laser. To do that, we consider a coupled system with the same functional form and adjustable parameters. The salient feature of the proposed technique is that accurate parameter estimation and identical synchronization can be jointly achieved by adaptively adjusting the desired parameters of the coupled system.

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.

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.