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## Publications

Publications (190)

It is common for researchers to record long, multiple time series from experiments or calculations. But sometimes there are no good models for the systems or no applicable mathematical theorems that can tell us when there are basic relationships between subsets of the time series data such as continuity, differentiability, embeddings, etc. The data...

There are several algorithms for estimating the embedding dimension from a time series, but they rely on setting an arbitrary threshold. These algorithms also do not give any indication of how likely the results are to be valid. We have published a method for estimating embedding dimension based on the geometry of a point cloud in the embedding spa...

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...

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 t...

Due to the increased spectrum congestion, there has been a tremendous interest in radar and communication devices operating on a shared platform. Therefore, the waveform design for joint radar communication (JRC) systems has recently gained significant attention. Despite several efforts, most state-of-the-art waveforms are far from achieving ideal...

This paper investigates in detail the effects of 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 differently the training and testing phases. We find that the best perf...

Reservoir computing is a recurrent neural network paradigm in which only the output layer is trained. 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 optimal time shifts. Our technique maximizes th...

While there have been many publications on potential applications of chaos to fields such as communications, radar, sonar, random signal generation, channel equalization and others, designing continuous chaotic systems is still an unsolved problem. There are a number of well known chaotic systems used for applications, but if any application is to...

While there have been many publications on potential applications of chaos to fields such as communications, radar, sonar, random signal generation, channel equalization and others, designing continuous chaotic systems is still an unsolved problem. There are a number of well known chaotic systems used for applications , but if any application is to...

While there have been many publications on potential applications of chaos to fields such as communications, radar, sonar, random signal generation, channel equalization and others, designing continuous chaotic systems is still an unsolved problem. There are a number of well known chaotic systems used for applications, but if any application is to...

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...

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...

A reservoir computer is a way of using a high dimensional dynamical system for computation. One way to construct a reservoir computer is by connecting a set of nonlinear nodes into a network. Because the network creates feedback between nodes, the reservoir computer has memory. If the reservoir computer is to respond to an input signal in a consist...

A reservoir computer is a way of using a high dimensional dynamical system for computation. One way to construct a reservoir computer is by connecting a set of nonlinear nodes into a network. Because the network creates feedback between nodes, the reservoir computer has memory. If the reservoir computer is to respond to an input signal in a consist...

The authors propose a novel signal design for generating wideband quasi-Frequency
modulated (FM) waveforms using chaotic systems. The receiver is based on a self-synchronizing chaotic system, making for fast synchronization that is robust to timing errors or Doppler shifts. The chaotic oscillator has fast and slow time scales, and the slow oscilla...

Reservoir computers are a type of recurrent neural network for which the network connections are not changed. To train the reservoir computer, a set of output signals from the network are fit to a training signal by a linear fit. As a result, training of a reservoir computer is fast, and reservoir computers may be built from analog hardware, result...

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...

A reservoir computer is a complex dynamical system, often created by coupling nonlinear nodes in a network. The nodes are all driven by a common driving signal. Reservoir computers can contain hundreds to thousands of nodes, resulting in a high dimensional dynamical system, but the reservoir computer variables evolve on a lower dimensional manifold...

In this paper, we develop a joint radar-communication waveform based on the unstable basis function. Basis functions of a chaotic waveform are uncertain, vary continuously and hence the use of a traditional matched filter is not practical. Corron et al. [1,2] demonstrated a chaotic circuit for which an analytic matched filter could be constructed,...

The topology of a network associated with a reservoir computer is often taken so that the connectivity and the weights are chosen randomly. Optimization is hardly considered as the parameter space is typically too large. Here we investigate this problem for a class of reservoir computers for which we obtain a decomposition of the reservoir dynamics...

It has been demonstrated that cellular automata had the highest computational capacity at the edge of chaos [N. H. Packard, in Dynamic Patterns in Complex Systems, edited by J. A. S. Kelso, A. J. Mandell, and M. F. Shlesinger (World Scientific, Singapore, 1988), pp. 293–301; C. G. Langton, Physica D 42(1), 12–37 (1990); J. P. Crutchfield and K. You...

It has been demonstrated that cellular automata had the highest computational capacity at the edge of chaos, the parameter at which their behavior transitioned from ordered to chaotic. This same concept has been applied to reservoir computers; a number of researchers have stated that the highest computational capacity for a reservoir computer is at...

Reservoir computers are a type of neuromorphic computer that may be built with analog hardware, potentially creating powerful computers that are small, light and consume little power. Typically a reservoir computer is build by connecting together a set of nonlinear nodes into a network; connecting the nonlinear nodes may be difficult or expensive,...

Reservoir computers are a type of neuromorphic computer that may be built a an analog system, potentially creating powerful computers that are small, light and consume little power. Typically a reservoir computer is build by connecting together a set of nonlinear nodes into a network; connecting the nonlinear nodes may be difficult or expensive, ho...

Because reservoir computers are high dimensional dynamical systems, designing a good reservoir computer is difficult. In many cases, the designer must search a large nonlinear parameter space, and each step of the search requires simulating the full reservoir computer. In this work, I show that a simple statistic based on the mean path length betwe...

With the increase in demand for spectral resources and bandwidth constraints, efficient solutions for the coexistence of radar and communication systems are needed. Simultaneously, the development of multifunctional radio frequency (RF) systems with less hardware have received considerable attention. Most previous work on coexistence has focused on...

A reservoir computer is a complex dynamical system, often created by coupling nonlinear nodes in a network. The nodes are all driven by a common driving signal. In this work, three dimension estimation methods, false nearest neighbor, covariance dimension, and Kaplan-Yorke dimension, are used to estimate the dimension of the reservoir dynamical sys...

A reservoir computer is a complex dynamical system, often created by coupling nonlinear nodes in a network. The nodes are all driven by a common driving signal. In this work, three dimension estimation methods, false nearest neighbor, covariance and Kaplan-Yorke dimensions, are used to estimate the dimension of the reservoir dynamical system. It is...

A reservoir computer is a complex nonlinear dynamical system that has been shown to be useful for solving certain problems, such as prediction of chaotic signals, speech recognition, or control of robotic systems. Typically, a reservoir computer is constructed by connecting a large number of nonlinear nodes in a network, driving the nodes with an i...

The edge of chaos is not always the optimum parameter to operate a reservoir computer

A reservoir computer is a dynamical system that may be used to perform computations. A reservoir computer usually consists of a set of nonlinear nodes coupled together in a network so that there are feedback paths. Training the reservoir computer consists of inputing a signal of interest and fitting the time series signals of the reservoir computer...

A reservoir computer is a complex nonlinear dynamical system that has been shown to be useful for solving certain problems, such as prediction of chaotic signals, speech recognition or control of robotic systems. Typically a reservoir computer is constructed by connecting a large number of nonlinear nodes in a network, driving the nodes with an inp...

We describe a method to estimate embedding dimension from a time series. This method includes an estimate of the probability that the dimension estimate is valid. Such validity estimates are not common in algorithms for calculating the properties of dynamical systems. The algorithm described here compares the eigenvalues of covariance matrices crea...

Several recent papers have shown that reservoir computers are useful for analyzing and predicting dynamical systems. Reservoir computers have also been shown to be useful for various classification problems. In this work, a reservoir computer is used to identify one out of the 19 different Sprott systems. An advantage of reservoir computers for thi...

Analyzing data from dynamical systems often begins with creating a reconstruction of the trajectory based on one or more variables, but not all variables are suitable for reconstructing the trajectory. The concept of nonlinear observability has been investigated as a way to determine if a dynamical system can be reconstructed from one signal or a c...

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing mo...

Several recent papers have shown that reservoir computers are useful for analyzing and predicting dynamical systems. Reservoir computers have also been shown to be useful for various classification problems. In this work, a reservoir computer is used to identify one out of the 19 different Sprott systems. An advantage of reservoir computers for thi...

Analyzing data from dynamical systems often begins with creating a reconstruction of the trajectory based on one or more variables, but not all variables are suitable for reconstructing the trajectory. The concept of nonlinear observability has been investigated as a way to determine if a dynamical system can be reconstructed from one signal or a c...

The same broad-band properties that make chaos interesting as a potential communications carrier also cause trouble when chaotic signals are transmitted through real communications channels. Phase and amplitude distortion of a chaotic signal can ruin the ability of a chaotic receiver to synchronize to an incoming chaotic signal. We show that it is...

Work by Corron et al. [1,2] represented a chaotic signal as a set of basis functions, and built a matched filter for the resulting signal. This paper makes use of basis functions without an underlying chaotic system. Matched filtering is still possible, allowing communication in noisy environments, but the resulting signals can be broad band, which...

1. Abstract Many methods for using chaos as a communications signal have been suggested , but all suffer from the difficulty in synchronizing the transmitter and receiver. Chaotic signals are broad band and unpredictable, making them potentially useful when the goal is low interference communications or even low probability of detection (LPD) commu...

Stationary dynamical systems have invariant measures (or densities) that are characteristic of the particular dynamical system. We develop a method to characterize this density by partitioning the attractor into the smallest regions in phase space that contain information about the structure of the attractor. To accomplish this, we develop a statis...

Stationary dynamical systems have invariant measures (or densities) that are characteristic of the particular dynamical system. We develop a method to characterize this density by partitioning the attractor into the smallest regions in phase space that contain information about the structure of the attractor. To accomplish this, we develop a statis...

We describe a method to estimate embedding dimension from a time series. This method includes an estimate of the probability that the dimension estimate is valid. Such validity estimates are not common in algorithms for calculating the properties of dynamical systems. The algorithm described here compares the eigenvalues of covariance matrices crea...

There have been many attempts to apply chaotic signals to communications or radar, but one obstacle has been that there is no effective way to recover chaotic signals from noise larger than the signal. In this work, we create “pseudo-chaotic” signals by concatenating dictionary sequences generated from a chaotic attractor. Because the number of dic...

Stationary dynamical systems have invariant measures (or densities) that are characteristic of the particular dynamical system. We develop a method to characterize this density by partitioning the attractor into the smallest regions in phase space that contain information about the structure of the attractor. To accomplish this, we develop a statis...

Recognizing a chaotic attractor can be seen as a problem in pattern recognition. Some feature vector must be extracted from the attractor and used to compare to other attractors. The field of machine learning has many methods for extracting feature vectors, including clustering methods, decision trees, support vector machines, and many others. In t...

Currently radar target identification is effected by finding the downrange distribution of scatterers on a platform using high definition radar and referencing this distribution to a look-up table of measured radar returns. This method is difficult due to the rapid scintillation of the radar signal of a flying target at short wavelengths. We invest...

The response of a radar or sonar target to a signal may be described by an impulse response function, which means that the target may be considered as a filter acting on a signal. It is known that filters are not exactly invertible, and this lack of invertibility may be used to identify the particular target that reflected a signal. We apply techni...

A signal reflected from a radar target is modified according to the impulse response function of the target. This interaction may be thought of as a linear filter acting on the incident signal. Filters are not exactly invertible, which means that there is no continuous function between the filter output and its input. Likewise, there is no invertib...

The authors use an acoustic experiment to test a method for identifying radar or sonar targets based on information from the target's impulse response function. The new algorithm uses a long modulated pulse of relatively low resolution. Using concepts from the field of dynamics, the algorithm measures the similarity between a signal reflected from...

Reflecting signals off of targets is a method widely used to locate objects, but the reflected signal also contains information that can be used to identify the object. In radar or sonar, the signal amplitudes used are small enough that only linear effects are present, so we can consider the effect of the target on the signal as a linear filter. Us...

We have developed a method for radar/sonar target discrimination
employing techniques from non-linear dynamics. We demonstrate our method
by simulating radar scattering from four similar targets where the radar
wavelength and bandwidth resolution is on the order of the target size.
We find that this method results in a high probability of target
di...

One may describe the effect of a radar or sonar target on an incoming signal as a filter that produces a scattered signal. Chaotic signals are very sensitive to the effect of filters, and so a radar or sonar target imposes a unique signature on a scattered chaotic signal. In this study the authors describe a method that uses the concept of phase sp...

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing mo...

One may describe the effect of a radar or sonar target on an incoming signal as a filter which produces a scattered signal. Chaotic signals are very sensitive to the effect of filters, so a radar or sonar target imposes a unique signature on a scattered chaotic signal. In this paper we describe a method that uses the concept of phase space dimensio...

Filtering a chaotic signal through a recursive [or infinite impulse response (IIR)] filter has been shown to increase the dimension of chaos under certain conditions. Filtering with a nonrecursive [or finite impulse response (FIR)] filter should not increase dimension, but it has been shown that if the FIR filter has a long tail, measurements of ac...

We present a low resolution method for radar target identification. The method employs optimized waveforms in cross correlation with the return from a low frequency chirp to distinguish between similar candidate targets. Target signature waveforms were generated every degree in azimuth for generic wing-body-tail targets using FDTD simulation. The t...

The standard method used for detecting signals in radar or sonar is cross correlation. The accuracy of the detection with cross correlation is limited by the bandwidth of the signals. We show that by calculating the cross correlation based on points that are nearby in phase space rather than points that are simultaneous in time, the detection accur...

We seek in this paper to differentiate driven nonlinear systems using only a single output signal from the driven system. We do not have access to the driving signal. We demonstrate the phase space identification techniques with an experimental model of a radio transmitter. We restrict the driving signals to nearly periodic signals, because these t...

There has been interest in the use of chaotic signals for radar, but most researchers consider only a few chaotic systems and how these signals perform for the detection of point targets. The range of possible chaotic signals is far greater than what most of these researchers consider, so to demonstrate this, I use a chaotic map whose parameters ma...

When a radio frequency signal is radiated by a transmitter, the properties of the transmitter itself affect the properties of the signal. These transmitter-induced changes are known as unintentional modulation, to differentiate them from intentional modulation used to add information to the signal. The unintentional modulation can be used to identi...

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 approac...

IntroductionMethods of Detecting Coupling Cross-CorrelationMutual InformationMutual Information in Two DimensionsPhase CorrelationContinuity MeasureLinear and Nonlinear Systems Gaussian Distributed White NoiseAutoregressive ModelHénon MapRössler AttractorCircuit DataUncoupled Systems Correlation Between Gaussian Distributed Random Data SetsCorrelat...

Given a sampling of pairs of points from a function’s domain and range, what can we say about the mathematical properties, if any, of that function? For example, were the point pairs related by a function that is continuous? Or differentiable? This is the same situation we are faced with when we apply standard statistical tools, like linear correla...

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 deterministi...

Introduction and MotivationThe Geometry of Synchronization Simple ExamplesSome Generalizations and a Definition of Identical SynchronizationThe Dynamics of Synchronization Stability and the Transverse ManifoldSynchronizing Chaotic Systems, Variations on ThemesSynchronous Circuits and ApplicationsStability and Bifurcations of Synchronized, Mutually...

We recently showed that the seemingly separate problems of finding a proper time delay and then finding a proper embedding dimension for attractor reconstruction are really the same problem which can be solved with a mathematical statistic faithful to the Takens reconstruction theorem. This approach also deals well with disparate time scales in dat...

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)....

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...

It was shown previously in an experiment that when high frequency signals (on the order of 1 MHz) were injected into this low frequency amplifier, the nonlinearities of the pn junctions caused period doubling, chaos, and very low frequency oscillations (on the order of 1 Hz). In this paper we present theory and simulations to explain the existence...

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 re...

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 t...

We describe an experiment using a chaotically driven metal plate with incremental damage. Damage in the plate is manifested as a local change in the plate's response (loss of stiffness). We develop a statistical test for Holder continuity and demonstrate its use by examining the map between responses of the undamaged plate and responses of the dama...

Research into applications of synchronized chaotic systems assumes that it will be necessary to build many different drive-response pairs, but little is known in general about designing higher dimensional chaotic flows. In this paper, I do not add any design techniques, but I show that it is possible to create multiple drive-response pairs from one...

We employ chaotic forcing to vibrate linear structures (e.g. beams) to
determine when parameters of the structure change, i.e. damage has
ocurred, during the lifetime of the structure. We explain the procedure
and show proof of concept first using a circuit simulation of a
structure. The circuit simulation provides a realistic test of our
attractor...

We employ chaotic interrogation of a circuit simulation of a structure in order to test for damage to the structure. The circuit simulation provides a realistic test of our attractor-based method and permits close control over parameters in the structure. In this circuit, simulating an eight-degree-of-freedom spring-mass system, we were able to det...

Long‐term, high‐level performance demands on a variety of structures and equipment have stimulated significant research in the field of structural health monitoring. The primary goals of this field are to provide information regarding structural performance capability, damage assessment, and even structural prognosis, all of which may potentially r...

Analysis of data from experiments on dynamical systems often centers on the embedding of time series data to reconstruct an atttractor. In our system, we consider output expressed as multiple time series from a circuit designed to simulate a spring-mass system in both an undamaged and a damaged state. In order to analyze differences in the reconstr...

Many experiments have the ability to record more than one time series of data simultaneously. We explore two issues that are present when multiple time series are used to reconstruct attractors which are not present in the case of one time series. First, we show that there is an algorithmic approach to false nearest neighbors that naturally extends...

It is known from extensive work with the diode resonator that the nonlinear properties of a P-N junction can lead to period doubling, chaos, and other complicated behaviors in a driven circuit. There has been very little work on what happens when more than one P-N junction is present. In this work, the first step towards multiple P-N junction circu...

Work on self-synchronizing systems for communications has had limited practicality because the chaotic signals were not as easy to detect in the presence of noise as conventional spread-spectrum signals. This difficulty may actually be an advantage in some cases, where one wants to conceal the existence of the communications signal. Conventional co...

One of the classic problems in the study of nonlinear dynamics has been the diode resonator. Previous work with the diode resonator sought to explain the causes of period doubling and chaos, and often used simplified models. This paper instead seeks to link the onset of nonlinear dynamical effects to measurable parameters by comparing experiments a...

Synchronized chaotic systems are highly vulnerable to noise added to the synchronizing signal. It was previously shown that chaotic circuits could be built that were less sensitive to this type of noise. In this work, simple chaotic maps are demonstrated that are also less sensitive to added noise. These maps are based on coupling a shift map to a...

The synchronization of chaotic circuits is greatly degraded by noise, so that for noise as large or larger than the synchronizing signal, no useful synchronization is possible. I demonstrate here a set of synchronized chaotic circuits that is robust to additive white noise. For any noise level, it is possible to achieve arbitrarily small synchroniz...

A method for reconstructing dimensions of subspaces for weakly coupled dynamical systems is offered. The tool is able to extrapolate the subspace dimensions from the zero coupling limit, where the division of dimensions as per the algorithm is exact. Implementation of the proposed technique to multivariate data demonstrates its effectiveness in dis...

Cyclostationary signals have an expectation value which varies
periodically in time. Chaotic signals that have large components at some
discrete frequencies in their power spectra can be cyclostationary. The
cyclostationarity persists even if the discrete frequency components are
removed from the chaotic signal, leaving a signal with a purely broad...

Until now, research on the applications of self-synchronized chaotic circuits to communications has been hindered by the great sensitivity of self-synchronized chaotic systems to additive noise. In this paper, the author demonstrates a self-synchronized chaotic system that synchronizes even in the presence of noise much larger than the signal. This...

We describe a method to characterize the predictability and functionality between two simultaneously generated time series. This nonlinear method requires minimal assumptions and can be applied to data measured either from coupled systems or from different positions on a spatially extended system. This analysis generates a function statistic, Theta...

The practical applications of self-synchronizing chaotic systems are greatly limited by their sensitivity to noise. Even small amounts of noise added to the synchronizing signal can degrade synchronization to the point where information encoded on the chaotic signal can't be recovered. In this paper, I show that it is possible to build chaotic circ...