Synchronization of Chaotic Optoelectronic Oscillators: Adaptive Techniques and the Design of Optimal Networks

Synchronization in networks of chaotic systems is an interesting phenomenon with potential applications to sensing, parameter estimation and communications. Synchronization of chaos, in addition to being influenced by the dynamical nature of the constituent network units, is critically dependent upon the maintenance of a proper coupling between the systems. In practical situations, however, synchronization in chaotic networks is negatively affected by perturbations in the coupling channels. Here, using a fiber-optic network of chaotic optoelectronic oscillators, we experimentally demonstrate an adaptive algorithm that maintains global network synchrony even when the coupling strengths are unknown and time-varying. Our adaptive algorithm operates by generating real-time estimates of the coupling perturbations which are subsequently used to suitably adjust internal node parameters in order to compensate for external disturbances. In our work, we also examine the influence of network configuration on synchronization. Through measurements of the convergence rate to synchronization in networks of optoelectronic systems, we show that having more network links does not necessarily imply faster or better synchronization as is generally thought. We find that the convergence rate is maximized for certain network configurations, called optimal networks, which are identified based on the eigenvalues of the coupling matrix. Further, based on an analysis of the eigenvectors of the coupling matrix, we introduce a classification system that categorizes networks according to their sensitivity to coupling perturbations as sensitive and nonsensitive configurations. Though our experiments are performed on networks consisting of specific nonlinear optoelectronic oscillators, the theoretical basis of our studies is general and consequently many of our results are applicable to networks of arbitrary dynamical oscillators

Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators

We describe a flexible and modular delayed-feedback nonlinear oscillator that is capable of generating a wide range of dynamical behaviours, from periodic oscillations to high-dimensional chaos. The oscillator uses electrooptic modulation and fibre-optic transmission, with feedback and filtering implemented through real-time digital-signal processing. We consider two such oscillators that are coupled to one another, and we identify the conditions under which they will synchronize. By examining the rates of divergence or convergence between two coupled oscillators, we quantify the maximum Lyapunov exponents or transverse Lyapunov exponents of the system, and we present an experimental method to determine these rates that does not require a mathematical model of the system. Finally, we demonstrate a new adaptive control method that keeps two oscillators synchronized even when the coupling between them is changing unpredictably.Comment: 24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special theme issue to accompany 2009 International Workshop on Delayed Complex Systems

Using Synchronization for Prediction of High-Dimensional Chaotic Dynamics

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.Comment: 10 pages, 4 figure