Thomas L. Carroll

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. >

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. >

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.

Long-term results of calcium hydroxylapatite for vocal fold augmentation†

Studies have shown excellent results for 12‐month post–injection augmentation data for calcium hydroxylapatite (CaHA) for glottal incompetence; however, the longevity of the material past one year was unknown. Our objective was to report the long‐term effectiveness of CaHA as a vocal fold injectable by assessing data from a cohort of patients who underwent injection for glottal insufficiency.

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.

Communicating with use of filtered, synchronized, chaotic signals

The principles of synchronization of chaotic systems are extended to the case where the drive signal is filtered. A feedback loop in the response system with an identical filter is used to reconstruct the original drive signal, allowing synchronization. A simple parameter switching scheme is used to send information from a drive circuit to a receiver. It is also possible to add a chaotic signal with very similar frequency characteristics and still detect information encoded in the original chaotic carrier (but not the added chaotic signal), demonstrating the possibility of adding and separating multiple chaotic carriers with similar frequency characteristics. >

Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data.

We show that one can use recently introduced statistics for continuity and differentiability to show the effect of filters of infinite extent in time on a chaotic time series. The statistics point to a discontinuous or nondifferentiable function between the unfiltered attractor and the filtered attractor as the origin of attractor dimension increase when the filtering is severe. The density of discontinuities as a function of resolution follows a scaling relation. We present direct visualization of this effect in the filtered Henon attractor where the origin of dimension increase becomes obvious.