Using Synchronization for Prediction of High-Dimensional Chaotic Dynamics

Abstract

We experimentally observe the nonlinear dynamics of an optoelectronic time-delayed feedback loop designed for chaotic communication using commercial fiber optic links, and we simulate the system using delay differential equations. We show that synchronization of a numerical model to experimental measurements provides a new way to assimilate data and forecast the future of this time-delayed high-dimensional system. For this system, which has a feedback time delay of 22 ns, we show that one can predict the time series for up to several delay periods, when the dynamics is about 15 dimensional. The question of how to predict the future of a dynamical system with time delay is of interest in many applications [1][2]. In chaotic encrypted communication systems, the predictability is related to how difficult it would be for an eavesdropper to intercept a message. A better understanding of predictability in these systems could guide the development of new strategies to improve security, such as periodically changing the system parameters or protocols to avoid interception [3]. Prediction and anticipation are thought to underlie the process of image recognition and motion tracking in the retina [4]. In networked sensor arrays designed to detect spatiotemporal disturbances, prediction methods could enable efficient acquisition and incorporation of data from multiple sensors [5]. In biomedical treatment, prediction models could lead to improved strategies for adjusting drug dosage and delivery or physiological control [6].Recent studies on prediction address system identification and model development [7], as well as the use of anticipated synchronization between coupled identical systems [8]. In this work, we demonstrate that synchronization of a numerical model to an experimentally measured waveform allows us to both forecast the future dynamics of a high-dimensional system and estimate the local maximum Lyapunov exponent and its distribution. The inverse of the maximum Lyapunov exponent defines the prediction horizon -the time over which the system behavior can be forecast.The optoelectronic system studied is shown in Figure 1. A similar system was used by Argyris et al. as a transmitter and receiver for a high-speed chaotic communication 1

Complex dynamics and synchronization of delayed-feedback nonlinear oscillators

“…Conversely, if the two systems are initially synchronized, the coupling can be suddenly turned off, allowing the trajectories to exponentially diverge. By measuring the rate of exponential divergence, we find the maximal Lyapunov exponent of the system (Cohen et al 2008). Unlike conventional methods, which require numerical solution of a linearized system of equations, this approach can be applied even in cases when an exact model of the physical system is unavailable or impractical.…”

“…Table 1 lists all of the parameter values used in the experiments and simulations. To simplify the experimental implementation, we consider here a low-frequency system that operates at audio frequencies, but this system can also be scaled to radio or microwave frequencies (Goedgebuer et al 2002;Kouomou et al 2005;Cohen et al 2008). In practice, the round-trip gain (β) and time delay (τ ) were measured experimentally by interrupting the feedback loop at the input to the modulator and measuring the round-trip small signal gain and group delay using a vector network analyser.…”

“…One contribution of 13 to a Theme Issue 'Delayed complex systems '. This journal is © 2010 The Royal Society mechanical scenarios on nano-to macroscopic spatial scales (Argyris et al 2005(Argyris et al , 2008Kouomou et al 2005;Uchida et al 2005Uchida et al , 2008Illing et al 2007;Yousefi et al 2007;Cohen et al 2008;Sorrentino & Ott 2008, 2009aReidler et al 2009). …”

“…In §2, we introduce the basic opto-electronic system and the delay differential equations used for a continuous-time description of the dynamics (Kouomou et al 2005;Cohen et al 2008). The intrinsic nonlinearity of the system arises from the integrated optical Mach-Zehnder modulator, which changes the intensity of light transmitted depending on the cosine squared of a modulation voltage applied to its electrodes.…”

“…The time scales for these transients set the limits on communication rates and detection of environmental perturbations. One may determine the maximum Lyapunov exponent for a dynamical system by measurement of transients away from synchrony (Cohen et al 2008). When a mathematical model is available, one may predict the dynamics of an experimental system for several delay times by performing assimilation of experimental data using synchronization of the mathematical model to the data (So et al 1994;Cohen et al 2008;Mariño et al 2009;Quinn et al 2009;Sorrentino & Ott 2009b).…”