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Reservoir computing, a recurrent neural network paradigm in which only the output layer is trained, has demonstrated remarkable performance on tasks such as prediction and control of nonlinear systems. Recently, it was demonstrated that adding time-shifts to the signals generated by a reservoir can provide large improvements in performance accuracy. In this work, we present a technique to choose the time-shifts by maximizing the rank of the reservoir matrix using a rank-revealing QR algorithm. This technique, which is not task dependent, does not require a model of the system and, therefore, is directly applicable to analog hardware reservoir computers. We demonstrate our time-shift selection technique on two types of reservoir computer: an optoelectronic reservoir computer and the traditional recurrent network with a t a n h activation function. We find that our technique provides improved accuracy over random time-shift selection in essentially all cases. [ABSTRACT FROM AUTHOR]

This paper investigates in detail the effects of measurement noise on the performance of reservoir computing. We focus on an application in which reservoir computers are used to learn the relationship between different state variables of a chaotic system. We recognize that noise can affect the training and testing phases differently. We find that the best performance of the reservoir is achieved when the strength of the noise that affects the input signal in the training phase equals the strength of the noise that affects the input signal in the testing phase. For all the cases we examined, we found that a good remedy to noise is to low-pass filter the input and the training/testing signals; this typically preserves the performance of the reservoir, while reducing the undesired effects of noise. [ABSTRACT FROM AUTHOR]

A reservoir computer is a type of dynamical system arranged to do computation. Typically, a reservoir computer is constructed by connecting a large number of nonlinear nodes in a network that includes recurrent connections. In order to achieve accurate results, the reservoir usually contains hundreds to thousands of nodes. This high dimensionality makes it difficult to analyze the reservoir computer using tools from the dynamical systems theory. Additionally, the need to create and connect large numbers of nonlinear nodes makes it difficult to design and build analog reservoir computers that can be faster and consume less power than digital reservoir computers. We demonstrate here that a reservoir computer may be divided into two parts: a small set of nonlinear nodes (the reservoir) and a separate set of time-shifted reservoir output signals. The time-shifted output signals serve to increase the rank and memory of the reservoir computer, and the set of nonlinear nodes may create an embedding of the input dynamical system. We use this time-shifting technique to obtain excellent performance from an opto-electronic delay-based reservoir computer with only a small number of virtual nodes. Because only a few nonlinear nodes are required, construction of a reservoir computer becomes much easier, and delay-based reservoir computers can operate at much higher speeds. [ABSTRACT FROM AUTHOR]

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements—which were never published—illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The identification of resonant objects in radar or sonar, important for object identification, is difficult because existing methods require that the signal have a large signal-to-noise ratio. It is shown in this article that a modified version of the Kaplan–Glass (KG) statistic, a phase space statistic used to determine if a signal is deterministic, is sensitive to the properties of resonant objects. The modified KG statistic can be used to detect the presence of a resonant object even when the radar or sonar signal does not come from a deterministic dynamical system. The use of the modified KG statistic both numerically and in a simple experiment is also demonstrated. [ABSTRACT FROM AUTHOR]

Radiofrequency signals can disrupt the operation of low frequency circuits. A digital inverter circuit would seem to be immune to such disruption, because its output state usually jumps abruptly between 0 and 5 V. Nevertheless, when driven with a high frequency signal, the inverter can have two coexisting stable states (which are not at 0 and 5 V). Slow switching between these states (by changing the rf signal) will produce a low frequency signal. I demonstrate the bistability in a circuit experiment and in a simple model of the circuit. [ABSTRACT FROM AUTHOR]

It is known that stray radio frequency signals can produce nonlinear effects that disrupt the operation of circuits, but the mechanisms by which this disruption occurs are not well known. In this paper, an emitter coupled Schmitt trigger circuit is driven with a high-frequency signal to look for disruptive effects. As the circuit makes a transition between mode locked states (period 2 and period 3, for example), there is a region of chaos in which the largest peak in the power spectrum is in between the mode-locked frequencies, and is not related to the driving frequency by an integer multiple. This chaos resembles the chaos seen during a period adding sequence, except that it contains frequencies ranging over many orders of magnitude, from the driving frequencies on the order of megahertz, down to a few hertz. It is found that only a one-transistor circuit is necessary to produce this extremely broadband chaos, and true quasiperiodicity is not seen in this circuit. The single-transistor circuit is then simulated to confirm the frequency conversion effects. [ABSTRACT FROM AUTHOR]

While added noise can destroy synchronization in synchronized chaotic systems, it was shown that some chaotic systems were not sensitive to added noise. In this paper, the mechanism for this noise resistance is explored. It is seen that part of the chaotic system acts like it is resonant, reducing the noise sensitivity of the system. By comparing to a model of a neuron, it is speculated that similar mechanisms may also be present in biological systems. [ABSTRACT FROM AUTHOR]

In a radar system, it is necessary to measure both range and velocity of a target. The movement of the target causes a Doppler shift of the radar signal, and the size of the Doppler shift is used to measure the velocity of the target. In this work, a chaotic drive-response system is simulated that detects a Doppler shift in a chaotic signal. The response system can detect Doppler shifts in more than one signal at a time. [ABSTRACT FROM AUTHOR]

Examines fundamentals of chaotic synchronization. Recognized potential for communications systems; Subfield of nonlinear dynamics; Geometry of synchronization and stability criteria; Notion of synchronous substitution; Synchronization hyperplanes; Cascaded drive-response synchronization; Cuomo-Oppenheim communications scheme; Control theory approaches.

Introduces multivariate statistics which gauge the continuity and differentiability of functions that map one attractor reconstruction to another. Relation of chaotic time series to a filtered version of that time series; Causes of increases in fractal dimension of attractors; Low-pass dynamical filter called linear, time-invariant filter.

We describe two experiments in which we investigate the synchronization of coupled periodic oscillators. Each experimental system consists of two identical coupled electronic periodic oscillators that display bursts of desynchronization events similar to those observed previously in coupled chaotic systems. We measure the degree of synchronization as a function of coupling strength. In the first experiment, high-quality synchronization is achieved for all coupling strengths above a critical value. In the second experiment, no high-quality synchronization is observed. We compare our results to the predictions of the several proposed criteria for synchronization. We find that none of the criteria accurately predict the range of coupling strengths over which high-quality synchronization is observed. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]