Louis M. Pecora

Synchronization in small-world systems

We quantify the dynamical implications of the small-world phenomenon by considering the generic synchronization of oscillator networks of arbitrary topology. The linear stability of the synchronous state is linked to an algebraic condition of the Laplacian matrix of the network. Through numerics and analysis, we show how the addition of random shortcuts translates into improved network synchronizability. Applied to networks of low redundancy, the small-world route produces synchronizability more efficiently than standard deterministic graphs, purely random graphs, and ideal constructive schemes. However, the small-world property does not guarantee synchronizability: the synchronization threshold lies within the boundaries, but linked to the end of the small-world region.

Synchronizing chaotic circuits

The authors describe the conditions necessary for synchronizing a subsystem of one chaotic system with a separate chaotic system by sending a signal from the chaotic system to the subsystem. The general scheme for creating synchronizing systems is to take a nonlinear system, duplicate some subsystem of this system, and drive the duplicate and the original subsystem with signals from the unduplicated part. This is a generalization of driving or forcing a system. The process can be visualized with ordinary differential equations. The authors have build a simple circuit based on chaotic circuits described by R. W. Newcomb et al. (1983, 1986), and they use this circuit to demonstrate this chaotic synchronization. >

Early seizure detection.

Summary: For patients with medically intractable epilepsy, there have been few effective alternatives to resective surgery, a destructive, irreversible treatment. A strategy receiving increased attention is using interictal spike patterns and continuous EEG measurements from epileptic patients to predict and ultimately control seizure activity via chemical or electrical control systems. This work compares results of seven linear and nonlinear methods (analysis of power spectra, cross‐correlation, principal components, phase, wavelets, correlation integral, and mutual prediction) in detecting the earliest dynamical changes preceding 12 intracranially‐recorded seizures from 4 patients. A method of counting standard deviations was used to compare across methods, and the earliest departures from thresholds determined from non‐seizure EEG were compared to a neurologists judgement. For these data, the nonlinear methods offered no predictive advantage over the linear methods. All the methods described here were successful in detecting changes leading to a seizure between one and two minutes before the first changes noted by the neurologist, although analysis of phase correlation proved the most robust. The success of phase analysis may be due in part to its complete insensitivity to amplitude, which may provide a significant source of error.

Synchronizing nonautonomous chaotic circuits

Shows that the synchronizing of chaotic circuits may be extended to circuits that are periodically forced. The authors use a phase correction circuit to match the phase in a response circuit to the phase in a drive circuit. These periodically forced synchronized chaotic circuits are much more noise-resistant than autonomous synchronized chaotic circuits, even when the noise is chaos with large components in the same frequency band as the synchronizing signal. >

Cluster synchronization and isolated desynchronization in complex networks with symmetries

Synchronization is of central importance in power distribution, telecommunication, neuronal and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form. Here we present a new framework and develop techniques for the analysis of network dynamics that shows the connection between network symmetries and cluster formation. The connection between symmetries and cluster synchronization is experimentally confirmed in the context of real networks with heterogeneities and noise using an electro-optic network. We experimentally observe and theoretically predict a surprising phenomenon in which some clusters lose synchrony without disturbing the others. Our analysis shows that such behaviour will occur in a wide variety of networks and node dynamics. The results could guide the design of new power grid systems or lead to new understanding of the dynamical behaviour of networks ranging from neural to social.

MASTER STABILITY FUNCTIONS FOR SYNCHRONIZED COUPLED SYSTEMS

We show that many coupled oscillator array configurations considered in the literature can be put into a simple form so that determining the stability of the synchronous state can be done by a master stability function which solves, once and for all, the problem of synchronous stability for many couplings of that oscillator.

Synchronization of chaotic systems.

We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.

SYNCHRONIZATION STABILITY IN COUPLED OSCILLATOR ARRAYS: SOLUTION FOR ARBITRARY CONFIGURATIONS

The stability of the state of motion in which a collection of coupled oscillators are in identical synchrony is often a primary and crucial issue. When synchronization stability is needed for many different configurations of the oscillators the problem can become computationally intense. In addition, there is often no general guidance on how to change a configuration to enhance or diminsh stability, depending on the requirements. We have recently introduced a concept called the Master Stability Function that is designed to accomplish two goals: (1) decrease the numerical load in calculating synchronization stability and (2) provide guidance in designing coupling configurations that conform to the stability required. In doing this, we develop a very general formulation of the identical synchronization problem, show that asymptotic results can be derived for very general cases, and demonstrate that simple oscillator configurations can probe the Master Stability Function.

A Unified Approach to Attractor Reconstruction

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several longstanding, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction.

Synchronizing chaotic systems

We have shown that one may use chaotic signals to drive dynamical systems. When the driven system is stable to the driving signal and the driven system matches the system that produced the chaotic driving signal, the driven system will produce chaotic signals that are synchronized to the driving system. This may be seen in both autonomous and nonautonomous chaotic systems. This synchronized chaos may be useful in spread spectrum communications applications.