Delay-induced resonance suppresses damping-induced unpredictability.

Combined effects of the damping and forcing in the underdamped time-delayed Duffing oscillator are considered in this paper. We analyse the generation of a certain damping-induced unpredictability due to the gradual suppression of interwell oscillations. We find the minimal amount of the forcing amplitude and the right forcing frequency to revert the effect of the dissipation, so that the interwell oscillations can be restored, for different time delay values. This is achieved by using the delay-induced resonance, in which the time delay replaces one of the two periodic forcings present in the vibrational resonance. A discussion in terms of the time delay of the critical values of the forcing for which the delay-induced resonance can tame the dissipation effect is finally carried out. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.

Transient chaos under coordinate transformations in relativistic systems.

We use the Hénon-Heiles system as a paradigmatic model for chaotic scattering to study the Lorentz factor effects on its transient chaotic dynamics. In particular, we focus on how time dilation occurs within the scattering region by measuring the time with a clock attached to the particle. We observe that the several events of time dilation that the particle undergoes exhibit sensitivity to the initial conditions. However, the structure of the singularities appearing in the escape time function remains invariant under coordinate transformations. This occurs because the singularities are closely related to the chaotic saddle. We then demonstrate using a Cantor-like set approach that the fractal dimension of the escape time function is relativistic invariant. In order to verify this result, we compute by means of the uncertainty dimension algorithm the fractal dimensions of the escape time functions as measured with an inertial frame and a frame comoving with the particle. We conclude that, from a mathematical point of view, chaotic transient phenomena are equally predictable in any reference frame and that transient chaos is coordinate invariant.

Effect of multiple time-delay on vibrational resonance.

We report our investigation on the effect of multiple time-delay on vibrational resonance in a single Duffing oscillator and in a system of n Duffing oscillators coupled unidirectionally and driven by both a low- and a high-frequency periodic force. For the single oscillator, we obtain analytical expressions for the response amplitude Q and the amplitude g of the high-frequency force at which resonance occurs. The regions in parameter space of enhanced Q at resonance, as compared to the case in absence of time-delay, show a bands-like structure. For the two-coupled oscillators, we explain all the features of variation of Q with the control parameter g. For the system of n-coupled oscillators with a single time-delay coupling, the response amplitudes of the oscillators are shown to be independent of the time-delay. In the case of a multi time-delayed coupling, undamped signal propagation takes place for coupling strength (δ) above a certain critical value (denoted as δu). Moreover, the response amplitude approaches a limiting value QL with the oscillator number i. We obtain analytical expressions for both δu and QL.

Novel vibrational resonance in multistable systems.

We investigate the role of multistable states on the occurrence of vibrational resonance in a periodic potential system driven by both a low-frequency and a high-frequency periodic force in both underdamped and overdamped limits. In both cases, when the amplitude of the high-frequency force is varied, the response amplitude at the low-frequency exhibits a series of resonance peaks and approaches a limiting value. Using a theoretical approach, we analyse the mechanism of multiresonance in terms of the resonant frequency and the stability of the equilibrium points of the equation of motion of the slow variable. In the overdamped system, the response amplitude is always higher than in the absence of the high-frequency force. However, in the underdamped system, this happens only if the low-frequency is less than 1. In the underdamped system, the response amplitude is maximum when the equilibrium point around which slow oscillations take place is maximally stable and minimum at the transcritical bifurcation. And in the overdamped system, it is maximum at the transcritical bifurcation and minimum when the associated equilibrium point is maximally stable. When the periodicity of the potential is truncated, the system displays only a few resonance peaks.

Theory and numerics of vibrational resonance in Duffing oscillators with time-delayed feedback.

The influence of linear time-delayed feedback on vibrational resonance is investigated in underdamped and overdamped Duffing oscillators with double-well and single-well potentials driven by both low frequency and high frequency periodic forces. This task is performed through both theoretical approach and numerical simulation. Theoretically determined values of the amplitude of the high frequency force and the delay time at which resonance occurs are in very good agreement with the numerical simulation. A major consequence of time-delayed feedback is that it gives rise to a periodic or quasiperiodic pattern of vibrational resonance profile with respect to the time-delayed parameter. An appropriate time delay is shown to induce a resonance in an overdamped single-well system which is otherwise not possible. For a range of values of the time-delayed parameters, the response amplitude is found to be larger than in delay-time feedback-free systems.

Fractal structures in nonlinear plasma physics.

Fractal structures appear in many situations related to the dynamics of conservative as well as dissipative dynamical systems, being a manifestation of chaotic behaviour. In open area-preserving discrete dynamical systems we can find fractal structures in the form of fractal boundaries, associated to escape basins, and even possessing the more general property of Wada. Such systems appear in certain applications in plasma physics, like the magnetic field line behaviour in tokamaks with ergodic limiters. The main purpose of this paper is to show how such fractal structures have observable consequences in terms of the transport properties in the plasma edge of tokamaks, some of which have been experimentally verified. We emphasize the role of the fractal structures in the understanding of mesoscale phenomena in plasmas, such as electromagnetic turbulence.

Single and multiple vibrational resonance in a quintic oscillator with monostable potentials.

We analyze the occurrence of vibrational resonance in a damped quintic oscillator with three cases of single well of the potential V(x)=1/2omega(0)(2)x(2)+1/4betax(4)+1/6gammax(6) driven by both low-frequency force f cos omegat and high-frequency force g cos Omegat with Omega >> omega. We restrict our analysis to the parametric choices (i) omega(0)(2), beta, gamma > 0 (single well), (ii) omega(0)(2), gamma > 0, beta < 0, beta(2) < 4omega(0)(2)gamma (single well), and (iii) omega(0)(2) > 0, beta arbitrary, gamma < 0 (double-hump single well). From the approximate theoretical expression of response amplitude Q at the low-frequency omega we determine the values of omega and g (denoted as omega(VR) and g(VR)) at which vibrational resonance occurs. We show that for fixed values of the parameters of the system when omega is varied either resonance does not occur or it occurs only once. When the amplitude g is varied for the case of the potential with the parametric choice (i) at most one resonance occur while for the other two choices (ii) and (iii) multiple resonance occur. Further, g(VR) is found to be independent of the damping strength d while omega(VR) depends on d. The theoretical predictions are found to be in good agreement with the numerical result. We illustrate that the vibrational resonance can be characterized in terms of width of the orbit also.

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.

Sparse repulsive coupling enhances synchronization in complex networks.

Through the last years, different strategies to enhance synchronization in complex networks have been proposed. In this work, we show that synchronization of nonidentical dynamical units that are attractively coupled in a small-world network is strongly improved by just making phase-repulsive a tiny fraction of the couplings. By a purely topological analysis that does not depend on the dynamical model, we link the emerging dynamical behavior with the structural properties of the sparsely coupled repulsive network.

Crisis-induced intermittency in two coupled chaotic maps: towards understanding chaotic itinerancy.

The present paper considers crisis-induced intermittency in a system composed of two coupled logistic maps. Its purpose is to clarify a bifurcation scenario generating such intermittent behaviors that can be regarded as a simple example of chaotic itinerancy. The intermittent dynamics appears immediately after an attractor-merging crisis of two off-diagonal chaotic attractors in a symmetrically coupled system. The scenario for the crisis is investigated through analyses of sequential bifurcations leading to the two chaotic attractors and successive changes in basin structures with variation of a system parameter. The successive changes of the basins are also characterized by variation of a dimension of a fractal basin boundary. A numerical analysis shows that simultaneous contacts between the attractors and the fractal basin boundary bring about the crisis and a snap-back repeller generated at the crisis produces the intermittent transitions. Furthermore, a modified scenario for intermittent behaviors in an asymmetrically coupled system is also discussed.

Coupling scheme for complete synchronization of periodically forced chaotic CO2 lasers.

We present a way of coupling two nonautonomous, periodically forced, chaotic C O2 lasers in a master-slave configuration in order to achieve complete synchronization. The method consists of modulating the forcing of the slave laser by means of the difference between the intensities of the two lasers, and lends itself to a simple physical implementation. Experimental evidence of complete synchronization induced by a suitable coupling strength is shown, and a numerical model is used to achieve further insight of the synchronization phenomena. Finally, we describe a possible application of the investigated technique to the design of a digital communication system.

Experimental evidence, numerics, and theory of vibrational resonance in bistable systems.

We consider an overdamped bistable oscillator subject to the action of a biharmonic force with very different frequencies, and study the response of the system when the parameters of the high-frequency force are varied. A resonantlike behavior is obtained when the amplitude or the frequency of this force is modified in an experiment performed by means of an analog circuit. This behavior, confirmed by numerical simulations, is explained on the basis of a theoretical approach.

Vibrational resonance in a noise-induced structure.

We report on the effect of vibrational resonance in a spatially extended system of coupled noisy oscillators under the action of two periodic forces, a low-frequency one (signal) and a high-frequency one (carrier). Vibrational resonance manifests itself in the fact that for optimally selected values of high-frequency force amplitude, the response of the system to a low-frequency signal is optimal. This phenomenon is a synthesis of two effects, a noise-induced phase transition leading to bistability, and a conventional vibrational resonance, resulting in the optimization of signal processing. Numerical simulations, which demonstrate this effect for an extended system, can be understood by means of a zero-dimensional "effective" model. The behavior of this "effective" model is also confirmed by an experimental realization of an electronic circuit.