Showing papers in "Chaos in 2003"
TL;DR: Several phenomenological approaches to applying information theoretic measures of randomness and memory to stochastic and deterministic processes are synthesized by using successive derivatives of the Shannon entropy growth curve to look at the relationships between a process's entropy convergence behavior and its underlying computational structure.
Abstract: We study how the Shannon entropy of sequences produced by an information source converges to the source’s entropy rate. We synthesize several phenomenological approaches to applying information theoretic measures of randomness and memory to stochastic and deterministic processes by using successive derivatives of the Shannon entropy growth curve. This leads, in turn, to natural measures of apparent memory stored in a source and the amounts of information that must be extracted from observations of a source in order for it to be optimally predicted and for an observer to synchronize to it. To measure the difficulty of synchronization, we define the transient information and prove that, for Markov processes, it is related to the total uncertainty experienced while synchronizing to a process. One consequence of ignoring a process’s structural properties is that the missed regularities are converted to apparent randomness. We demonstrate that this problem arises particularly for settings where one has access only to short measurement sequences. Numerically and analytically, we determine the Shannon entropy growth curve, and related quantities, for a range of stochastic and deterministic processes. We conclude by looking at the relationships between a process’s entropy convergence behavior and its underlying computational structure.
407 citations
TL;DR: It is demonstrated that identical neurons which are not coupled but subjected to a common fluctuating input (Gaussian noise) can achieve complete synchronization when the noise amplitude is larger than a threshold.
Abstract: We study nontrivial effects of noise on synchronization and coherence of a chaotic Hodgkin-Huxley model of thermally sensitive neurons. We demonstrate that identical neurons which are not coupled but subjected to a common fluctuating input (Gaussian noise) can achieve complete synchronization when the noise amplitude is larger than a threshold. For nonidentical neurons, noise can induce phase synchronization. Noise enhances synchronization of weakly coupled neurons. We also find that noise enhances the coherence of the spike trains. A saddle point embedded in the chaotic attractor is responsible for these nontrivial noise-induced effects. Relevance of our results to biological information processing is discussed.
171 citations
TL;DR: Analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters, and they are stable and robust against up to 10%-15% parameter mismatch and against small noise.
Abstract: Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%–15% parameter mismatch and against small noise.
120 citations
TL;DR: It is believed that the widespread synchrony of the jumps in analytic phase manifest a metastable cortical state in accord with the theory of self-organized criticality, and appear to be subcritical bifurcations.
Abstract: My objective of this study was to find evidence of chaotic itinerancy in human brains by means of noninvasive recording of the electroencephalogram (EEG) from the scalp of normal subjects. My premise was that chaotic itinerancy occurs in sequences of cortical states marked by state transitions that appear as temporal discontinuities in neural activity patterns. I based my study on unprecedented advances in spatial and temporal resolution of the phase of oscillations in scalp EEG. The spatial resolution was enhanced by use of a high-density curvilinear array of 64 electrodes, 189 mm in length, with 3 mm spacing. The temporal resolution was advanced to the limit provided by the digitizing step, here 5 ms, by use of the Hilbert transform. The numerical derivative of the analytic phase revealed plateaus in phase that lasted on the order of 0.1 s and repeated at rates in the theta (3–7 Hz) or alpha (7–12 Hz) ranges. The plateaus were bracketed by sudden jumps in phase that usually took place within 1 to 2 digitizing steps. The jumps were commonly synchronized in each cerebral hemisphere over distances of up to 189 mm, irrespective of the orientation of the array. The jumps were usually not synchronized across the midline separating the hemisphere or across the sulcus between the frontal and parietal lobes. I believe that the widespread synchrony of the jumps in analytic phase manifest a metastable cortical state in accord with the theory of self-organized criticality. The jumps appear to be subcritical bifurcations. They reflect the aperiodic evolution of brain states through sequences of attractors that on access support the experience of remembering.
96 citations
TL;DR: The relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator is discussed.
Abstract: The phenomenon of frequency and phase synchronization in stochastic systems requires a revision of concepts originally phrased in the context of purely deterministic systems. Various definitions of an instantaneous phase are presented and compared with each other with special attention paid to their robustness with respect to noise. We review the results of an analytic approach describing noise-induced phase synchronization in a thermal two-state system. In this context exact expressions for the mean frequency and the phase diffusivity are obtained that together determine the average length of locking episodes. A recently proposed method to quantify frequency synchronization in noisy potential systems is presented and exemplified by applying it to the periodically driven noisy harmonic oscillator. Since this method is based on a threshold crossing rate pioneered by Rice the related phase velocity is termed the Rice frequency. Finally, we discuss the relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator.
96 citations
TL;DR: The results suggest that the AMRA will be able to simulate the 3D electrical dynamics of canine ventricles quantitatively for 1 s using 32 1-GHz Alpha processors in approximately 9 h.
Abstract: A recently developed space–time adaptive mesh refinement algorithm (AMRA) for simulating isotropic one- and two-dimensional excitable media is generalized to simulate three-dimensional anisotropic media. The accuracy and efficiency of the algorithm is investigated for anisotropic and inhomogeneous 2D and 3D domains using the Luo–Rudy 1 (LR1) and FitzHugh–Nagumo models. For a propagating wave in a 3D slab of tissue with LR1 membrane kinetics and rotational anisotropy comparable to that found in the human heart, factors of 50 and 30 are found, respectively, for the speedup and for the savings in memory compared to an algorithm using a uniform space–time mesh at the finest resolution of the AMRA method. For anisotropic 2D and 3D media, we find no reduction in accuracy compared to a uniform space–time mesh. These results suggest that the AMRA will be able to simulate the 3D electrical dynamics of canine ventricles quantitatively for 1 s using 32 1-GHz Alpha processors in approximately 9 h.
93 citations
TL;DR: Intrinsic localized modes have been observed in micromechanical cantilever arrays, and their creation, locking, interaction, and relaxation dynamics in the presence of a driver have been studied.
Abstract: Intrinsic localized modes (ILMs) have been observed in micromechanical cantilever arrays, and their creation, locking, interaction, and relaxation dynamics in the presence of a driver have been studied. The micromechanical array is fabricated in a 300 nm thick silicon–nitride film on a silicon substrate, and consists of up to 248 cantilevers of two alternating lengths. To observe the ILMs in this experimental system a line-shaped laser beam is focused on the 1D cantilever array, and the reflected beam is captured with a fast charge coupled device camera. The array is driven near its highest frequency mode with a piezoelectric transducer. Numerical simulations of the nonlinear Klein–Gordon lattice have been carried out to assist with the detailed interpretation of the experimental results. These include pinning and locking of the ILMs when the driver is on, collisions between ILMs, low frequency excitation modes of the locked ILMs and their relaxation behavior after the driver is turned off.
91 citations
TL;DR: The incorporation of the alphabeta filter to DFA is presented to determine patterns in the power-law behavior that can be found in persistent long-range correlations in healthy heart rate variability data, suggesting the existence of evolving patterns that cannot be described by scaling exponents estimated using a linear procedure over two predefined ranges.
Abstract: Detrended fluctuation analysis (DFA), suitable for the analysis of nonstationary time series, has confirmed the existence of persistent long-range correlations in healthy heart rate variability data. In this paper, we present the incorporation of the αβ filter to DFA to determine patterns in the power-law behavior that can be found in these correlations. Well-known simulated scenarios and real data involving normal and pathological circumstances were used to evaluate this process. The results presented here suggest the existence of evolving patterns, not always following a uniform power-law behavior, that cannot be described by scaling exponents estimated using a linear procedure over two predefined ranges. Instead, the power law is observed to have a continuous variation with segment length. We also show that the study of these patterns, avoiding initial assumptions about the nature of the data, may confer advantages to DFA by revealing more clearly abnormal physiological conditions detected in congestive heart failure patients related to the existence of dominant characteristic scales.
90 citations
TL;DR: Numerical evidence and theoretical reasoning are presented which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous in the Hindmarsh-Rose model of a neuron.
Abstract: Interior crises are understood as discontinuous changes of the size of a chaotic attractor that occur when an unstable periodic orbit collides with the chaotic attractor. We present here numerical evidence and theoretical reasoning which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous. This occurs in the Hindmarsh–Rose model of a neuron, at the transition point between the bursting and spiking dynamics, which are two different dynamic behaviors that this system is able to present. Moreover, besides the change in attractor size, other significant properties of the system undergoing the transitions do change in a relevant qualitative way. The mechanism for such transition is understood in terms of a simple one-dimensional map whose dynamics undergoes a crossover between two different universal behaviors.
79 citations
TL;DR: In this article, the influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network.
Abstract: The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching are found, depending on the details of the underlying deterministic dynamics: random switching between the heteroclinic cycles determined by the linear dynamics near one of the saddle points, noise induced stability of a cycle, and intermittent switching between cycles. All three responses are explained by examining the size of the stable and unstable eigenvalues at the equilibria.
76 citations
TL;DR: This work develops a digital secure communication scheme that utilizes the instantaneous phase as the signal transmitted from the drive to the response subsystems and operates with a weak positive conditional Lyapunov exponent in theresponse subsystem.
Abstract: Phase synchronization of chaotic systems with both weak and strong couplings has recently been investigated extensively. Similar to complete synchronization, this type of synchronization can also be applied in secure communications. We develop a digital secure communication scheme that utilizes the instantaneous phase as the signal transmitted from the drive to the response subsystems. Simulation results show that the scheme is difficult to be broken by some traditional attacks. Moreover, it operates with a weak positive conditional Lyapunov exponent in the response subsystem.
TL;DR: It is found that a special type of unstable steady or time-periodic solutions called scattors and their stable and unstable manifolds direct the traffic flow of orbits.
Abstract: Scattering of particle-like patterns in dissipative systems is studied, especially we focus on the issue how the input–output relation is controlled at a head-on collision where traveling pulses or spots interact strongly. It remains an open problem due to the large deformation of patterns at a colliding point. We found that a special type of unstable steady or time-periodic solutions called scattors and their stable and unstable manifolds direct the traffic flow of orbits. Such scattors are in general highly unstable even in the one-dimensional case which causes a variety of input–output relations through the scattering process. We illustrate the ubiquity of scattors by using the complex Ginzburg–Landau equation, the Gray–Scott model, and a three-component reaction diffusion model arising in gas-discharge phenomena.
TL;DR: A dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally, from random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network.
Abstract: Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.
TL;DR: This issue focuses on Control and Synchronization in Chaotic Dynamical Systems and Elements and concepts from the theory of systems con and the theoryof communication have been brought in, making this area well-established within nonlinear science.
Abstract: exopl athe unof me ics r nd ired n a its onby esobded rian n tal of a The last ten years have seen remarkable developmen the research of chaotic dynamics, particularly with respec the interaction of chaotic dynamics with other fields of r search and with applications. There is now a developed ence of chaos that has as an essential underpinning the s interaction of theory and experiment. This is a depart from earlier times in which theoretical work existed large in the absence of substantial experimental realizations. Al with this new orientation has come increased apprecia and concern for the implications of chaotic dynamics in pr tical applications. Issues in topics such as the active con of chaotic systems in a broad variety of situations, the us chaos for communication, and the synchronization of cha dynamics for various purposes, are at the forefront of rec application topics in nonlinear science. The common thr through those topics is the marriage between knowledg the basic mathematical properties of chaos and specific p tical considerations of various applications. This Focus Issue resulted from a six-week event at Max Planck Institute for Physics of Complex Systems Dresden in the Fall of 2001. During that Worshop/Semin especially interesting and challenging topics on control a synchronization were addressed. We believe that success the research work coming out from that program will ha far-reaching technological and economical impact for broad area of important practical systems ranging from sers, via engineering to neuroscience and medicine. This issue focuses onControl and Synchronization in Chaotic Dynamical Systems . The fundamentals and the ma jor concepts involved in this area were reviewed in Chaosin a Focus Issue in December 1997 @Chaos7 ~4!#. Since that time, the then novel topics and applications have matu making this area well-established within nonlinear scien Elements and concepts from the theory of systems con and the theory of communication have been brought in, g ing the whole topic a firmer foundation. Therefore, the p
TL;DR: A noisy nonlinear process is presented here as a candidate neural mechanism to explain the majority of reported phenomenology and provide specific quantitative predictions, and shows remarkable quantitative agreement not dependent on adjustable parameters.
Abstract: How the brain estimates the pitch of a complex sound remains unsolved. Complex sounds are composed of more than one tone. When two tones occur together, a third lower pitched tone is often heard. This is referred to as the “missing fundamental illusion” because the perceived pitch is a frequency (fundamental) for which there is no actual source vibration. This phenomenon exemplifies a larger variety of problems related to how pitch is extracted from complex tones, music and speech, and thus has been extensively used to test theories of pitch perception. A noisy nonlinear process is presented here as a candidate neural mechanism to explain the majority of reported phenomenology and provide specific quantitative predictions. The two basic premises of this model are as follows: (I) The individual tones composing the complex tones add linearly producing peaks of constructive interference whose amplitude is always insufficient to fire the neuron (II): The spike threshold is reached only with noise, which natur...
TL;DR: The state of art of the understanding of the principal features of the quantum discrete breathers is presented and some important properties differentiating localized and extended vibrational modes are outlined.
Abstract: The knowledge about discrete quantum breathers, accumulated during the last two decades, is reviewed. “Prehistory” of the problem is described and some important properties differentiating localized and extended vibrational modes are outlined. The state of art of our understanding of the principal features of the quantum discrete breathers is presented.
TL;DR: This article proves the existence of regular repeated sequences, called "epistrophes," which occur at all levels of resolution within the escape-time plot, and gives this plot a certain self-similarity, called 'epistrophic' self- similarity, which need not imply either strict or asymptotic self-Similarity.
Abstract: We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane We record the number of iterates it takes a trajectory to escape from a bounded region of the plane as a function along the line of initial conditions, forming an “escape-time plot” For a chaotic system, this plot is in general not a smooth function, but rather has many singularities at which the escape time is infinite; these singularities form a complicated fractal set In this article we prove the existence of regular repeated sequences, called “epistrophes,” which occur at all levels of resolution within the escape-time plot (The word “epistrophe” comes from rhetoric and means “a repeated ending following a variable beginning”) The epistrophes give the escape-time plot a certain self-similarity, called “epistrophic” self-similarity, which need not imply either strict or asymptotic self-similarity
TL;DR: In this paper, a generalized fluctuation-response relation is derived, which links the average relaxation toward equilibrium to the invariant measure of the system and points out the relevance of the amplitude of the initial perturbation.
Abstract: We study the response of dynamical systems to finite amplitude perturbation. A generalized fluctuation-response relation is derived, which links the average relaxation toward equilibrium to the invariant measure of the system and points out the relevance of the amplitude of the initial perturbation. Numerical computations on systems with many characteristic times show the relevance of the above-mentioned relation in realistic cases.
TL;DR: It is demonstrated that there exists a subset on which a singular Poincare map generated by the model is conjugate to the shift map on two symbols, leading to the classical mathematical phenomenon of Pontryagin's delay of loss of stability.
Abstract: Chaotic dynamics of a classical prey-predator-superpredator ecological model are considered. Although much is known about the behavior of the model numerically, very few results have been proven analytically. A new analytical result is obtained. It is demonstrated that there exists a subset on which a singular Poincare map generated by the model is conjugate to the shift map on two symbols. The existence of such a Poincare map is due to two conditions: the assumption that each species has its own time scale ranging from fast for the prey to slow for the superpredator, and the existence of transcritical points, leading to the classical mathematical phenomenon of Pontryagin’s delay of loss of stability. This chaos generating mechanism is new, neither suspected in abstract form nor recognized in numerical experiments in the literature.
TL;DR: The results indicate that constrained chaos could be potentially useful in complex functioning and controlling for systems with large but finite degrees of freedom typically observed in biological systems and may be such that working in a delicate balance between converging dynamics and diverging dynamics in high dimensional state space depending on given situation, environment and context to be controlled or to be processed.
Abstract: Complex dynamics including chaos in systems with large but finite degrees of freedom are considered from the viewpoint that they would play important roles in complex functioning and controlling of biological systems including the brain, also in complex structure formations in nature. As an example of them, the computer experiments of complex dynamics occurring in a recurrent neural network model are shown. Instabilities, itinerancies, or localization in state space are investigated by means of numerical analysis, for instance by calculating correlation functions between neurons, basin visiting measures of chaotic dynamics, etc. As an example of functional experiments with use of such complex dynamics, we show the results of executing a memory search task which is set in a typical ill-posed context. We call such useful dynamics “constrained chaos,” which might be called “chaotic itinerancy” as well. These results indicate that constrained chaos could be potentially useful in complex functioning and contro...
TL;DR: The energy relaxation process produced by surface damping on lattices of classical anharmonic oscillators is described and a statistical analysis of the resulting quasistationary state through the distribution of breathers' energies yield information on their effective interactions.
Abstract: We describe the energy relaxation process produced by surface damping on lattices of classical anharmonic oscillators. Spontaneous emergence of localized vibrations dramatically slows down dissipation and gives rise to quasistationary states where energy is trapped in the form of a gas of weakly interacting discrete breathers. In one dimension, strong enough on-site coupling may yield stretched-exponential relaxation which is reminiscent of glassy dynamics. We illustrate the mechanism generating localized structures and discuss the crucial role of the boundary conditions. For two-dimensional lattices, the existence of a gap in the breather spectrum causes the localization process to become activated. A statistical analysis of the resulting quasistationary state through the distribution of breathers’ energies yield information on their effective interactions.
TL;DR: The advection and diffusion of a passive scalar is investigated for a map of the 2-torus, and the limit of almost-uniform stretching is considered, allowing an analytic understanding of the transition from a phase of constant scalar variance to exponential decay.
Abstract: The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered. This allows an analytic understanding of the transition from a phase of constant scalar variance (for short times) to exponential decay (for long times). This transition is embodied in a short superexponential phase of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection–diffusion operator, in which most of the scalar variance is concentrated at small scales, even though a large-scale mode sets the decay rate. The duration of the superexponential phase is proportional to the logarithm of the exponential decay rate; if the decay is slow enough then there is no superexponential phase at all.
TL;DR: This paper reviews the experiments on visualization of discrete breathers (intrinsic localized modes) in nonlinear lattices made of Josephson junctions and the visualization technique based on low temperature laser scanning microscopy (LSM).
Abstract: In this paper I review our experiments on visualization of discrete breathers (intrinsic localized modes) in nonlinear lattices made of Josephson junctions. Properties of Josephson junctions and arrays made of such junctions are discussed in the Introduction. The visualization technique based on low temperature laser scanning microscopy (LSM) is described in detail. Images of discrete breathers in Josephson junction arrays of various geometries are presented. Possible further experiments that can be done using LSM technique are envisioned.
TL;DR: This work investigates retrieval dynamics of multiple active patterns in a network of chaotic model neurons and includes a mechanism for self-organized readout or decoding of memory pattern coherence in terms of short-term potentiation and short- term depression of synaptic weights.
Abstract: Associative memory dynamics in neural networks are generally based on attractors. Retrieval based on fixed-point attractors works if only one memory pattern is retrieved at the time, but cannot enable the simultaneous retrieval of more than one pattern. Stable phase-locking of periodic oscillations or limit cycle attractors leads to incorrect feature bindings if the simultaneously retrieved patterns share some of their features. We investigate retrieval dynamics of multiple active patterns in a network of chaotic model neurons. Several memory patterns are kept simultaneously active and separated from each other by a dynamic itinerant synchronization between neurons. Neurons representing shared features alternate their synchronization between patterns, thus multiplexing their binding relationships. Our model includes a mechanism for self-organized readout or decoding of memory pattern coherence in terms of short-term potentiation and short-term depression of synaptic weights.
TL;DR: Topological techniques are used to show that there exists a minimal required set of escape segments within the escape-time plot; develop an algorithm which computes this minimal set and show that the minimal set eventually displays a recursive structure governed by an "Epistrophe Start Rule".
Abstract: We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each end point of every escape segment. In the present paper, we use topological techniques to: (1) show that there exists a minimal required set of escape segments within the escape-time plot; (2) develop an algorithm which computes this minimal set; (3) show that the minimal set eventually displays a recursive structure governed by an “Epistrophe Start Rule:” a new epistrophe is spawned Δ=D+1 iterates after the segment to which it converges, where D is the minimum delay time of the complex.
TL;DR: By adjusting the initial conditions, this work proposes a method to control the unboundedness, which leads the quantum coherent atomic tunneling to predictable periodical oscillation.
Abstract: The chaotic coherent atomic tunneling between two periodically driven and weakly coupled Bose–Einstein condensates has been investigated. The perturbed correction to the homoclinic orbit is constructed and its boundedness conditions are established that contain the Melnikov criterion for the onset of chaos. We analytically reveal that the chaotic coherent atomic tunneling is deterministic but not predictable. Our numerical calculation shows good agreement with the analytical result and exhibits nonphysically numerical instability. By adjusting the initial conditions, we propose a method to control the unboundedness, which leads the quantum coherent atomic tunneling to predictable periodical oscillation.
TL;DR: A spatial extension of the logistic map-termed spatial logisticmap-is found to display the same basic universality classes as the commonly studied diffusively coupled logistic lattice despite being vastly simpler.
Abstract: A spatial extension of the logistic map—termed spatial logistic map—is found to display the same basic universality classes as the commonly studied diffusively coupled logistic lattice despite being vastly simpler. By analyzing the escape rates and the Lyapunov spectra it is shown that the main attractors of the spatial logistic map are stable and hence that it is a good candidate for serving as a prototype for the class of coupled map lattices which it is a part of. The spatial logistic map is then employed to provide an analytical derivation for the recently discovered linear scaling of the wavelength under increasing coupling ranges.
TL;DR: The ring shows partial synchronization and all the five patterns predicted analyzing the symmetries of the ring are obtained experimentally and to compare with the experiment, the ring has been integrated numerically and the results are in good qualitative agreement with the experimental ones.
Abstract: The dynamical behavior of a ring of six diffusively coupled Rossler circuits, with different coupling schemes, is experimentally and numerically investigated using the coupling strength as a control parameter. The ring shows partial synchronization and all the five patterns predicted analyzing the symmetries of the ring are obtained experimentally. To compare with the experiment, the ring has been integrated numerically and the results are in good qualitative agreement with the experimental ones. The results are analyzed through the graphs generated plotting the y variable of the ith circuit versus the variable y of the jth circuit. As an auxiliary tool to identify numerically the behavior of the oscillators, the three largest Lyapunov exponents of the ring are obtained.
TL;DR: It is demonstrated that the direction of coupling of two interacting self-sustained electronic oscillators can be determined from the realizations of their signals.
Abstract: We demonstrate that the direction of coupling of two interacting self-sustained electronic oscillators can be determined from the realizations of their signals. In our experiments, two electronic generators, operating in a periodic or a chaotic state, were subject to symmetrical or unidirectional coupling. In data processing, first the phases have been extracted from the observed signals and then the directionality of coupling was quantitatively estimated from the analysis of mutual dependence of the phase dynamics.