Tool to recover scalar time-delay systems from experimental time series

Recovery of couplings and parameters of elements in networks of time-delay systems from time series

We propose a method for the recovery of coupling architecture and the parameters of elements in networks consisting of coupled oscillators described by delay-differential equations. For each oscillator in the network, we introduce an objective function characterizing the distance between the points of the reconstructed nonlinear function. The proposed method is based on the minimization of this objective function and the separation of the recovered coupling coefficients into significant and insignificant coefficients. The efficiency of the method is shown for chaotic time series generated by model equations of diffusively coupled time-delay systems and for experimental chaotic time series gained from coupled electronic oscillators with time-delayed feedback.

Determining the sub-Lyapunov exponent of delay systems from time series

For delay systems the sign of the sub-Lyapunov exponent (sub-LE) determines key dynamical properties. This includes the properties of strong and weak chaos and of consistency. Here we present a robust algorithm based on reconstruction of the local linearized equations of motion, which allows for calculating the sub-LE from time series. The algorithm is inspired by a method introduced by Pyragas for a nondelayed drive-response scheme [K. Pyragas, Phys. Rev. E 56, 5183 (1997)]. In the presented extension to delay systems, the delayed feedback takes over the role of the drive, whereas the response of the low-dimensional node leads to the sub-Lyapunov exponent. Our method is based on a low-dimensional representation of the delay system. We introduce the basic algorithm for a discrete scalar map, extend the concept to scalar continuous delay systems, and give an outlook to the case of a full vector-state system, from which only a scalar observable is recorded.

Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system

In this paper, evolution of time delay (TD) signature of chaos generated in a mutual delay-coupled semiconductor lasers (MDC-SLs) system is investigated experimentally and theoretically. Two statistical methods, including self-correlation function (SF) and permutation entropy (PE), are used to estimate the TD signature of chaotic time series. Through extracting the characteristic peak from the SF curve, a series of TD signature evolution maps are firstly obtained in the parameter space of coupled strength and frequency detuning. Meantime, the influences of injection current on the evolution maps of TD signature have been discussed, and the optimum scope of TD signature suppression is also specified. An overall qualitative agreement between our theoretical and experimental results is obtained.

Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems

Existence of a new type of oscillating synchronization that oscillates between three different types of synchronizations (anticipatory, complete, and lag synchronizations) is identified in unidirectionally coupled nonlinear time-delay systems having two different time-delays, that is feedback delay with a periodic delay time modulation and a constant coupling delay. Intermittent anticipatory, intermittent lag, and complete synchronizations are shown to exist in the same system with identical delay time modulations in both the delays. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay with suitable stability condition is discussed. The intermittent anticipatory and lag synchronizations are characterized by the minimum of the similarity functions and the intermittent behavior is characterized by a universal asymptotic -32 power law distribution. It is also shown that the delay time carved out of the trajectories of the time-delay system with periodic delay time modulation cannot be estimated using conventional methods, thereby reducing the possibility of decoding the message by phase space reconstruction.

Delay Fokker-Planck equations, Novikov's theorem, and Boltzmann distributions as small delay approximations

We study time-delayed stochastic systems that can be described by means of so-called delay Fokker-Planck equations. Using Novikov's theorem, we first show that the theory of delay Fokker-Planck equations is on an equal footing with the theory of ordinary Fokker-Planck equations. Subsequently, we derive stationary distributions in the case of small time delays. In the case of additive noise systems, these distributions can be cast into the form of Boltzmann distributions involving effective potential functions.

Estimation of coupling between time-delay systems from time series

We propose a method for estimation of coupling between the systems governed by scalar time-delay differential equations of the Mackey-Glass type from the observed time series data. The method allows one to detect the presence of certain types of linear coupling between two time-delay systems, to define the type, strength, and direction of coupling, and to recover the model equations of coupled time-delay systems from chaotic time series corrupted by noise. We verify our method using both numerical and experimental data.

Extracting information masked by the chaotic signal of a time-delay system

We further develop the method proposed by Bezruchko et al. [Phys. Rev. E 64, 056216 (2001)] for the estimation of the parameters of time-delay systems from time series. Using this method we demonstrate a possibility of message extraction for a communication system with nonlinear mixing of information signal and chaotic signal of the time-delay system. The message extraction procedure is illustrated using both numerical and experimental data and different kinds of information signals.

Reconstruction of time-delay systems from chaotic time series

We propose a method that allows one to estimate the parameters of model scalar time-delay differential equations from time series. The method is based on a statistical analysis of time intervals between extrema in the time series. We verify our method by using it for the reconstruction of time-delay differential equations from their chaotic solutions and for modeling experimental systems with delay-induced dynamics from their chaotic time series.

The control of high-dimensional chaos in time-delay systems to an arbitrary goal dynamics

We present the control of high-dimensional chaos, with possibly a large number of positive Lyapunov exponents, of unknown time-delay systems to an arbitrary goal dynamics. We give an existence-and-uniqueness theorem for the control force. In the case of an unknown system, a formula to compute a model-based control force is derived. We give an example by demonstrating the control of the Mackey-Glass system toward a fixed point and a Rossler dynamics. (c) 1999 American Institute of Physics.