Time-dependent Control Parameters Research Articles

Analytical investigation of modulated time-delayed feedback control

The influence of time-dependent control parameters on time-delayed feedback schemes for control of chaos is investigated by analytical means. For the logistic map the linear stability of the period-two orbit subjected to a modulated time-delayed feedback loop is calculated. We find enhanced control performance due to phase lags between the periodic orbit and the controller.

Stochastic resonance in spatiotemporal on–off intermittency

Stochastic resonance is investigated in a generic system with spatiotemporal on–off intermittency: a chain of coupled logistic maps with a time-dependent control parameter, driven by a spatiotemporal periodic signal. Spatiotemporal correlation function between the periodic signal and the output signal, reflecting the occurrence of laminar phases and chaotic bursts, has a maximum as a function of the mean value of the control parameter. For a given period and length of the periodic signal the height of this maximum can be increased by choosing an optimum coupling strength between maps. It is argued that the obtained result can be interpreted as an example of noise-free (dynamical) stochastic resonance in a system with spatiotemporal intermittency.

Moment evolution and level-crossing statistics in dichotomous and multilevel flows with time-dependent control parameters.

We study the dynamics of the first two moments and of threshold crossings by the stochastic trajectory in dichotomous diffusion x=xi(t), where xi(t) is a dichotomous Markov process. The transition rate of the latter is regarded as a control parameter and allowed to have specified time variations. The stabilizing or destabilizing effect of this variation is demonstrated, and qualitative changes in the statistical properties of the system are shown to occur. The analysis is then extended to linear dichotomous flow, and to a generalization of dichotomous diffusion in which x is driven by a multilevel Markov noise.

Symbolic dynamics of event-related brain potentials

We apply symbolic dynamics techniques such as word statistics and measures of complexity to nonstationary and noisy multivariate time series of electroencephalograms (EEG) in order to estimate event-related brain potentials (ERP). Their significance against surrogate data as well as between different experimental conditions is tested. These methods are validated by simulations using stochastic dynamical systems with time-dependent control parameters and compared with traditional ERP-analysis techniques. Continuous EEG data are cut into epochs according to stimuli events presented to the subjects. These ensembles of time series can be considered as ensembles of trajectories given by some dynamical systems. We employ a statistical mechanics approach motivated by the Frobenius-Perron equation and apply it to coarse-grained symbolic descriptions of the dynamics. We develop time-dependent measures of complexity founded on running cylinder sets and show that these quantities are able to distinguish simulated data obtained by different control parameters as well as experimental data between different experimental conditions. As a first finding, our approach restores the well-known ERP components and it reveals additionally qualitative changes in the EEG that cannot be detected by means of the traditional techniques. We criticize the prerequisites of the traditional approach to ERP analysis and propose to consider ERP instead in terms of dynamical system theory and information theory.

Global reconstruction in application to multichannel communication

We propose a method for restoration of time-dependent control parameters of dynamical system using the technique of global reconstruction. The technique presented is applied to multichannel confidential communication by means of parameter modulation.

Slow passage through a supercritical Hopf bifurcation: Time-delayed response in the Belousov–Zhabotinsky reaction in a batch reactor

When the control parameter of a dynamical system varies continuously, bifurcation is delayed due to the inertia in the system’s response. We study experimentally the time delay of the supercritical Hopf bifurcation that arises from the parametric drift of the oscillating Belousov–Zhabotinsky reaction under batch conditions. The time-dependent oscillation amplitude and period are analyzed using the normal form of the supercritical Hopf bifurcation with a time-dependent control parameter. We show that this approach describes the time evolution of the entire oscillatory domain from high amplitude to vanishing oscillations.

On the transient stochastic dynamics driven by Gaussian colored noise of systems with time-dependent control parameters. The effect of initial conditions

On a recent work we studied the transient stochastic dynamics driven by Gaussian white noise (GWN), of linear systems with time-dependent control parameters, by means of the connection between the nonlinear relaxation times (NLRT) and the quasideterministic (QD) approach. In the present study we make an extension of the analysis to the Gaussian colored noise (GCN) problem. Here, we first calculate the characteristic time associated with the decay of the unstable state of such linear systems, when the control parameter is a linear function of time of the form a( t) = bt − a 0, with b > a 0 > 0, which is continuously swept from below to above a threshold t ̄ = (a 0/b) . This type of linear modulation is known as the ramp model. Then, going further, we consider a general case where the control parameter is modulated by a family of functions a( t) = bt δ ∮ a 0, with δ > 0. The effects of the coupling between the initial state, at time t = 0, of the system with the noise are specially emphasized. The results of NLRT for the ramp model and the general case are compared.

The characteristic times of the transient stochastic dynamics with time-dependent control parameters distributed initial conditions

A systematic method is developed for the calculation of characteristic times, called nonlinear relaxation times (NLRT), to describe the dynamical relaxation of the linear transient stochastic systems whose control parameters are time-dependent functions. For those control parameters, which are modulated by a family of functions of the form a( t) = bt δ − a 0, with δ > 0, the method is applied to calculate the NLRT associated with the decay of unstable states of the linear stochastic systems when these parameters are continuously swept from below to above threshold ( t - = ( a 0 b ) 1 δ ) . The ramp modulation is a model for which δ = 1, it is studied and formulated in terms of the time differences s = t − t _ with t - = ( a 0 b ) . The time scales of both models are compared under certain requirements of the involved parameters.

On evolution of open dynamical systems: Some algebraic methods

Open dynamical systems which are governed by a finite number of ordinary differential equations with controls (time-dependent control parameters) constitute a large and important class of models for practical purposes. In the last few years, there has been considerable interest and progress in algebraic methods for solving the equations of the form $$\dot x\left( t \right) = L_0 x\left( t \right) + \sum\limits_{j = 1}^r {u\left( t \right)L_i x\left( t \right)} ,$$ i.e. bilinear models. In this paper, intended as an expository introduction to the main results of system-theoretic approach to the modelling of open systems, a new polynomial representation of solutions to (*) is discussed.

Dynamic behavior of bistability in a laser with a saturable absorber

Optical bistability in a CO2 infrared laser containing SF6 as a saturable intracavity absorber has been studied under dynamic conditions as a function of a time-dependent laser control parameter. The width of the optical bistability region, the fluctuations in the bistability switching values, and the dependence on the initial conditions have been investigated in this transient optical bistability. A numerical solution of the laser rate equations provides a good interpretation of the experimental results, but some experimental observations cannot be completely explained.

Imperfect Bifurcation with a Slowly-Varying Control Parameter

We consider a general class of imperfect bifurcation problems described by the following first order nonlinear differential equation:\[ y_i = ky^p + \lambda (t)y + \delta , \] where $k = 1$ or $-1$ and $p = 2$ or 3 are fixed quantities. The solution depends on the values of the “imperfection” parameter $\delta (0 < \delta \ll 1)$ and the time-dependent control parameter $\lambda (t) = \lambda _0 + \varepsilon t$ and $0 < \varepsilon \ll 1$). If $\delta = \varepsilon = 0$, this equation admits at $\lambda = 0$ a bifurcation from the basic state $y = 0$ to nonzero steady states. In the first part of the paper, we analyze the perturbation of the bifurcation solutions produced both by the small imperfection $(\delta \ne 0)$ and the slow variation of $\lambda (\varepsilon \ne 0)$. We show that $\lambda = 0$ does not correspond to the transition between the two branches of slowly-varying steady states. This transition appears at a larger value of $\lambda = \lambda _1 $. Provided that $\delta $ is sufficiently ...

Branch selection in the presence of coloured noise

We have experimentally studied the effects of exponentially correlated, gaussian, additive noise on an electronic bistable system whose instability is driven by a time dependent external control parameter. The results, which indicate that the noise correlation time has little influence on the statistics of branch selection, are in substantial agreement with the predictions of a simple, physically motivated theory outlined herein.

Dynamic effects in optical bistability

In the first part we give a brief review of some theoretical problems that still hinder the practical use of bistable optical devices in logic circuits. In the second part we study the influence of a time-dependent control parameter on the solution of differential equations. We prove that even for small sweeping rates large delays can be expected, resulting in the dynamical stabilization of unstable solutions. Examples are given for dynamically induced optical bistability in a laser and dynamical stabilization in a laser with saturable absorber.

Determination of an Unknown Heat Source from Overspecified Boundary Data

Determination of an Unknown Heat Source from Overspecified Boundary Data