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Symmetries are an essential feature of complex networks as they regulate how the graph collective dynamics organizes into clustered states. We here show how to control network symmetries, and how to enforce patterned states of synchronization with nodes clustered in a desired way. Our approach consists of perturbing the original network connectivity, either by adding new edges or by adding/removing links together with modifying their weights. By solving suitable optimization problems, we furthermore guarantee that changes made on the existing topology are minimal. The conditions for the stability of the enforced pattern are derived for the general case, and the performance of the method is illustrated with paradigmatic examples. Our results are relevant to all the practical situations in which coordination of the networked systems into diverse groups may be desirable, such as for teams of robots, unmanned autonomous vehicles, power grids and central pattern generators.

While there has been considerable work addressing consensus and group consensus in single-layer networks, not much attention has been devoted to consensus in multilayer networks. In this paper, we fill this gap by considering multilayer networks consisting of agents of different types while agents of the same type are arranged in separate layers. The patterns of emerging group consensus are determined by the symmetries of the multilayer network. An analysis of these symmetries reveals a partition of the nodes in each layer into clusters where the nodes in each cluster may achieve group consensus. We show that it is possible for group consensus to arise independently of the particular dynamics of the agents, which may be stable, marginally stable, or unstable. The concept of isolated group consensus where certain clusters of nodes in the multilayer network achieve group consensus while others do not is also introduced.

International audience; 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.

We investigate cluster synchronization in experiments with a multilayer network of electronic Colpitts oscillators, specifically a network with two interaction layers. We observe and analytically characterize the appearance of several cluster states as we change coupling in the layers. In this study, we innovatively combine bifurcation analysis and the computation of transverse Lyapunov exponents. We observe four kinds of synchronized states, from fully synchronous to a clustered quasiperiodic state---the first experimental observation of the latter state. Our work is the first to study fundamentally dissimilar kinds of coupling within an experimental multilayer network.

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 input signal and using the node outputs to fit a training signal. In this work, we set up reservoirs where the edges (or connections) between all the network nodes are either +1 or 0, and proceed to alter the network structure by flipping some of these edges from +1 to -1. We use this simple network because it turns out to be easy to characterize; we may use the fraction of edges flipped as a measure of how much we have altered the network. In some cases, the network can be rearranged in a finite number of ways without changing its structure; these rearrangements are symmetries of the network, and the number of symmetries is also useful for characterizing the network. We find that changing the number of edges flipped in the network changes the rank of the covariance of a matrix consisting of the time series from the different nodes in the network, and speculate that this rank is important for understanding the reservoir computer performance., Comment: accepted for Chaos

We study cluster synchronization in networks and show that the stability of all possible cluster synchronization patterns depends on a small set of Lyapunov exponents. Our approach can be applied to clusters corresponding to both orbital partitions of the network nodes (symmetry-cluster synchronization) and equitable partitions of the network nodes (input-cluster synchronization.) Our results are verified experimentally in networks of coupled opto-electronic oscillators., Comment: 19 pages, 8 figures

Synchronization is a collective phenomenon that appears in many natural and man-made networks of oscillators or dynamical systems such as telecommunication, neuronal and biological networks. An interesting form of synchronization is cluster synchronization where the network becomes partitioned into groups of oscillator nodes which synchronize to each other, but not to other nodes in other groups or clusters. We present a technique and develop methods for the analysis of network dynamics that shows the connection between network symmetries and cluster formation. We also experimentally confirm these approaches in the context of real networks with heterogeneities and noise using an electro-optic network. We find an interesting scenario for the appearance of chimera synchronization in these cases of identical cluster synchronization. We also extend these methods to networks which have Laplacian coupling and show that we can analyze cases where there are synchronization clusters which do not directly arise from symmetries, but can be built from clusters found by a symmetry analysis.

We study cluster synchronization in networks with symmetries in the presence of small generic parametric mismatches of two different types: mismatches affecting the dynamics of the individual uncoupled systems and mismatches affecting the network couplings. We perform a stability analysis of the nearly synchronous cluster synchronization solution and reduce the stability problem to a low-dimensional form. We also show how under certain conditions the low dimensional analysis can be used to predict the overall synchronization error, i.e., how close the individual nearly synchronous trajectories are to each other.

Theory of identical or complete synchronization of identical oscillators in arbitrary networks is introduced. In addition, several graph theory concepts and results that augment the synchronization theory and a tie in closely to random, semirandom, and regular networks are introduced. Combined theories are used to explore and compare three types of semirandom networks for their efficacy in synchronizing oscillators. It is shown that the simplest k-cycle augmented by a few random edges or links are the most efficient network that will guarantee good synchronization.

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.

Recent works by Nichols et al. (Nichols, J.M., Todd, M.D., Seaver, M. and Virgin, L.N. (2003). Use of chaotic excitation and attractor property analysis in structural health monitoring. Phys Rev E, 67(016209)) and Pecora et al. (Todd, M.D., Nichols, J.M., Pecora, L.M. and Virgin, L.N. (2001). Vibration-based damage assessment utilizing state-space geometry changes: Local attractor variance ratio. Smart Materials and Structures, 10, 1000-1008.) have shown that steady-state dynamic analysis of structural health exhibits advantages over transient vibrational analysis. A geometric representation of system dynamics can be used to extract information about a structure’s response to sustained excitation. Analysis of various features of the geometric representation can be used to describe the degree to which the dynamics have been altered by damage. Here, the feature we employ is the ‘‘continuity test,’’ a statistical test first described by Pecora et al. (Pecora, L.M., Carroll, T.L. and Heagy, J.F. (1997). Statistics for continuity and differentiability: an application to attractor reconstruction from time-series. Fields Institute Communications, 11). This test measures the probability that a continuous function exists from one geometric object to another. In this implementation, we formulate a new null hypothesis which serves to make the test less sensitive to noise in the data than the original test. Using experimental data from an excited three-story aluminum frame structure with multiple sensors at the joints, we show that the continuity test can be used not only to detect, but also in some cases to localize damage to particular joints in the frame structure.

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. (c) 2000 American Institute of Physics.

Dynamical systems which can be decomposed into a drive–response pair sometimes display the phenomenon of generalized synchronization wherein the response state y is a function of the drive state x, y = ϕ (x). The questions of detecting when this is true in experimental data, and what are the properties of the function ϕ, require the development of statistics that are based on concepts and definitions from analysis, rather than the standard statistics literature. We show that these statistics can display when such a generalized synchronization situation exists and characterize the nature of the manifold (given by ϕ) of the response over the drive, including when that manifold is fractal.

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.

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 can be tailored to one's choice of stability requirement. This solves, once and for all, the problem of synchronous stability for any linear coupling of that oscillator.

We show that it is possible to synthesize chaotic systems that have improved characteristics for use in communications. These include whiter spectrum, low-time correlation, hyperchaotic behavior, and little or no phase space structure. These systems are based on locally volume-preserving (or expanding) maps. We show how to construct a circuit that produces such characteristics.

The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and "cottage industries" have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution-a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for "secure" communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wavelength bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. (c) 1997 American Institute of Physics.

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.

Chaos has long been recognized to be generally advantageous from the perspective of control. In particular, the infinite number of unstable periodic orbits embedded in a chaotic set and the intrinsically sensitive dependence on initial conditions imply that a chaotic system can be controlled to a desirable state by using small perturbations. Investigation of chaos control, however, was largely limited to nonlinear dynamical systems in the classical realm. In this paper, we show that chaos may be used to modulate or harness quantum mechanical systems. To be concrete, we focus on quantum transport through nanostructures, a problem of considerable interest in nanoscience, where a key feature is conductance fluctuations. We articulate and demonstrate that chaos, more specifically transient chaos, can be effective in modulating the conductance-fluctuation patterns. Experimentally, this can be achieved by applying an external gate voltage in a device of suitable geometry to generate classically inaccessible potential barriers. Adjusting the gate voltage allows the characteristics of the dynamical invariant set responsible for transient chaos to be varied in a desirable manner which, in turn, can induce continuous changes in the statistical characteristics of the quantum conductance-fluctuation pattern. To understand the physical mechanism of our scheme, we develop a theory based on analyzing the spectrum of the generalized non-Hermitian Hamiltonian that includes the effect of leads, or electronic waveguides, as self-energy terms. As the escape rate of the underlying non-attracting chaotic set is increased, the imaginary part of the complex eigenenergy becomes increasingly large so that pointer states are more difficult to form, making smoother the conductance-fluctuation pattern.

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.

Synchronous chaotic behavior is often interrupted by bursts of desynchronized behavior. We investigate the role of unstable periodic orbits in bursting events and show that they serve as sources of local transverse instability within a synchronous chaotic attractor. Analysis of bursts in both model and experimental studies of two coupled R\"ossler-like oscillators reveals the importance of unstable periodic orbits in bursting events.

We report the presence of short wavelength bifurcations from synchronous chaotic states in coupled oscillator systems. The bifurcations immediately excite the shortest spatial wavelength mode present in the system as the coupling between the oscillators is increased beyond a critical value. An associated size instability places an upper bound on the number of oscillators that can support stable synchronous chaotic oscillations; an exact expression is given for the upper bound. Results are demonstrated with numerical simulations and electronic circuits.

Recent numerical experiments of Pecora et al. [Phys. Rev. E 83, 065201 (2011)] have investigated tunneling between two-dimensional symmetric double wells separated by a tunneling barrier. The wells were bounded by hard walls and by the potential barrier which was created by a step increase from the zero potential within a well to a uniform barrier potential within the barrier region, which is a situation potentially realizable in the context of quantum dots. Numerical results for the splitting of energy levels between symmetric and antisymmetric eigenstates were calculated. It was found that the splittings vary erratically from state to state, and the statistics of these variations were studied for different well shapes with the fluctuation levels being much less in chaotic wells than in comparable nonchaotic wells. Here we develop a quantitative theory for the statistics of the energy level splittings for chaotic wells. Our theory is based on the random plane wave hypothesis of Berry. While the fluctuation statistics are very different for chaotic and nonchaotic well dynamics, we show that the mean splittings of differently shaped wells, including integrable and chaotic wells, are the same if their well areas and barrier parameters are the same. We also consider the case of tunneling from a single well into a region with outgoing quantum waves.

We study tunneling in various shaped, closed, two-dimensional, flat-potential, double wells by calculating the energy splitting between symmetric and antisymmetric state pairs. For shapes that have regular or nearly regular classical behavior (e.g. rectangular or circular) the tunneling rates vary greatly over wide ranges often by several orders of magnitude. However, for well shapes that admit more classically chaotic behavior (e.g. the stadium, the Sinai billiard) the range of tunneling rates narrows, often by orders of magnitude. This dramatic narrowing appears to come from destabilization of periodic orbits in the regular wells that produce the largest and smallest tunneling rates and causes the splitting versus energy relation to take on a possibly universal shape. It is in this sense that we say the quantum chaos regularizes the tunneling rates.

We present direct experimental and numerical evidence for riddled basins of attraction for the synchronous chaotic state in a system of coupled chaotic oscillator circuits. Both experiment and computation show scaling typical of basin riddling.

We investigate the synchronization of chaotic oscillations in coupled oscillator systems, both theoretically and in analog electronic circuits. Particular attention is paid to deriving and testing general conditions for the stability of synchronous chaotic behavior in cases where the coupled oscillator array possesses a shift-invariant symmetry. These cases include the well studied cases of nearest-neighbor diffusive coupling and all-to-all or global coupling. An approximate criterion is developed to predict the stability of synchronous chaotic oscillations in the strong coupling limit, when the oscillators are coupled through a single coordinate (scalar coupling). This stability criterion is illustrated numerically in a set of coupled R\"ossler-like oscillators. Synchronization experiments with coupled R\"ossler-like oscillator circuits are also carried out to demonstrate the applicability of the theory to real systems.

We show that chaotically driving identical multiple period systems with a chaotic system can cause them to remain synchronized. That is the basins of attraction merge. We show this for the logistic map and relate it to other work in chaotic and quasiperiodic driving of nonlinear systems.

We have studied the synchronization of an array of three globally pulse coupled relaxation oscillators. We vary the firing rate on one of the oscillators and see how this affects the synchronization of the oscillators for different coupling symmetrics. We find that certain types of coupling make the array more responsive to parameter changes but still robust against noise.

Weakly chaotic or weakly interacting systems have a wide regime where the common random matrix theory modeling does not apply. As an example we consider cold atoms in a nearly integrable optical billiard with displaceable wall ("piston"). The motion is completely chaotic but with small Lyapunov exponent. The Hamiltonian matrix does not look like one taken from a Gaussian ensemble, but rather it is very sparse and textured. This can be characterized by parameters $s$ and $g$ that reflect the percentage of large elements, and their connectivity, respectively. For $g$ we use a resistor network calculation that has a direct relation to the semi-linear response characteristics of the system, hence leading to a novel prediction regarding the rate of heating of cold atoms in optical billiards with vibrating walls., 18 pages, 11 figures, improved PRE accepted version

Quantum tunneling rates through a barrier separating two-dimensional, symmetric, double-well potentials are shown to depend on the classical dynamics of the billiard trajectories in each well and, hence, on the shape of the wells. For shapes that lead to regular (integrable) classical dynamics the tunneling rates fluctuate greatly with eigenenergies of the states sometimes by over two orders of magnitude. Contrarily, shapes that lead to completely chaotic trajectories lead to tunneling rates whose fluctuations are greatly reduced, a phenomenon we call regularization of tunneling rates. We show that a random-plane-wave theory of tunneling accounts for the mean tunneling rates and the small fluctuation variances for the chaotic systems.

Some monitoring programs for ecological resources are developed as components of larger science or management programs and are thus guided by a priori hypotheses. More commonly, ecological monitoring programs are initiated for the purpose of surveillance with no a priori hypotheses in mind. No conceptual framework currently exists to guide the development of surveillance monitoring programs, resulting in substantial debate about program design. We view surveillance monitoring programs as providing information about system dynamics and focus on methods for extracting such information from time series of monitoring data. We briefly describe methods from the general field of nonlinear dynamics that we believe may be useful in extracting information about system dynamics. In looking at the system as a network of locations or components, we emphasize methods for assessing coupling between system components for use in understanding system dynamics and interactions and in detecting changes in system dynamics. More specifically, these methods hold promise for such ecological problems as identifying indicator species, developing informative spatial monitoring designs, detecting ecosystem change and damage, and investigating such topics as population synchrony, species interactions, and environmental drivers. We believe that these ideas and methods provide a useful conceptual framework for surveillance monitoring and can be used with model systems to draw inferences about the design of surveillance monitoring programs. In addition, some of the current methods should be useful with some actual ecological monitoring data, and methodological extensions and modifications should increase the applicability of these approaches to additional sources of actual ecological data.

A cascaded synchronized nonlinear system includes a nonlinear transmitter having stable first and second subparts. The first subpart produces a first transmitter signal for driving the second subpart and the second subpart produces a second transmitter signal for driving the first subpart. The nonlinear transmitter transmits the second transmitter signal to a nonlinear cascaded receiver. The receiver, being for producing an output signal in synchronization with the second transmitter signal, includes a first stage (a duplicate of the first subpart) responsive to the second transmitter signal for producing a first receiver signal. The receiver further includes a second stage (a duplicate of the second subpart) responsive to the first receiver signal for producing the output signal. The cascaded synchronized nonlinear system can be used in an information transfer system. The transmitter, responsive to an information signal produces a drive signal for transmission to the receiver. An error detector compares the drive signal and the output signal produced by the receiver to produce an error signal indicative of the information contained in the information signal.

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. >

Recent work on the synchronizing of chaotic systems raises the possibility of finding ways to apply chaos to real problems. Studying these applications requires that real physical systems be used, as well as mathematical models. There are no general rules for designing or building such physical systems. We present here a circuit that is useful for studying applications of synchronized chaotic systems and discuss some of the considerations that went into designing this circuit. We also show how this circuit is used for studying cascaded synchronized chaotic systems.

Master-stability functions (MSFs) are fundamental to the study of synchronization in complex dynamical systems. For example, for a coupled oscillator network, a necessary condition for synchronization to occur is that the MSF at the corresponding normalized coupling parameters be negative. To understand the typical behaviors of the MSF for various chaotic oscillators is key to predicting the collective dynamics of a network of these oscillators. We address this issue by examining, systematically, MSFs for known chaotic oscillators. Our computations and analysis indicate that it is generic for MSFs being negative in a finite interval of a normalized coupling parameter. A general scheme is proposed to classify the typical behaviors of MSFs into four categories. These results are verified by direct simulations of synchronous dynamics on networks of actual coupled oscillators.

The basins of attraction of period-2 attractors can be subdivided into two domains, one for each of the stable trajectories which are one drive cycle apart. This generalizes to period-n attractors. Periodic drive signals can be replaced by certain chaotic signals which result in the elimination of multiple domains of attraction. The attractors are similar to the original ones, but two systems on the same attractor cannot get out of phase with each other

We generalize the idea of driving a stable system to the situation when the drive signal is chaotic. This leads to the concept of conditional Lyapunov exponents and also generalizes the usual criteria of the linear stability theorem. We show that driving with chaotic signals can be done in a robust fashion, rather insensitive to changes in system parameters. The calculation of the stability criteria leads naturally to an estimate for the convergence of the driven system to its stable state. We focus on a homogeneous driving situation that leads to the construction of synchronized chaotic subsystems. We apply these ideas to the Lorenz and R\"ossler systems, as well as to an electronic circuit and its numerical model.

Certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signals. The criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits.

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We investigate the use of a vibrational approach for the detection of barely visible impact damage in a composite UAV wing. The wing is excited by a shaker according to a predetermined signal, and the response is observed by a system of fiber Bragg grating strain sensors. We use two different driving sequences: a stochastic signal consisting of white noise, and the output from a chaotic Lorenz oscillator. On these data we apply a variety of time series analysis techniques to detect, quantify, and localize the damage incurred from a pendulum impactor, including classical linear analysis (e.g. modal analyses), as well as recently developed nonlinear analysis methods. We compare the performance of these methods, investigate the reproducibility of the results, and find that two nonlinear statistics are able to detect barely visible damage.

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 heuristic and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several long-standing, 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., Comment: 22 pages, revised version as submitted to CHAOS. Manuscript is currently under review. 4 Figures, 31 references

A rigorous analytical approach is developed to test for the existence of a continuous nonlinear functional relationship between systems. We compare the application of this nonlinear local technique to the existing analytical linear global approach in the setting of increasing additive noise. For natural systems with unknown levels of noise and nonlinearity, we propose a general framework for detecting coupling. Lastly, we demonstrate the applicability of this method to detect coupling between simultaneous, experimentally measured, intracellular voltages between neurons within a mammalian neuronal network.

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 detect changes of as little as 2% in the coupling between two oscillators in the circuit. This corresponded to detection of a 2% loss in stiffness to one spring in the modeled system.