Showing papers in "International Journal of Bifurcation and Chaos in 2000"
TL;DR: The phenomenon of neural bursting is described, and geometric bifurcation theory is used to extend the existing classification of bursters, including many new types, and it is shown that different bursters can interact, synchronize and process information differently.
Abstract: Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neuro-computational properties of the cells. For example, when the rest state is near a saddle-node bifurcation, the cell can fire all-or-none spikes with an arbitrary low frequency, it has a well-defined threshold manifold, and it acts as an integrator; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov–Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not all-or-none, it does not have a well-defined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing. We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize and process information differently.
1,765 citations
TL;DR: Some basic dynamical properties and various bifurcations of Chen's equation are investigated, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.
Abstract: Anticontrol of chaos by making a nonchaotic system chaotic has led to the discovery of some new chaotic systems, particularly the continuous-time three-dimensional autonomous Chen's equation with only two quadratic terms. This paper further investigates some basic dynamical properties and various bifurcations of Chen's equation, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.
381 citations
TL;DR: BIFurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input.
Abstract: Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying t...
350 citations
TL;DR: This paper introduces a new method, single-trial phase locking statistics (S-PLS), to estimate phase locking in single trials of brain signals between two electrodes, and compares these results with those provided by PLS and shows that they are qualitatively very similar, although S-PLs provides better discrimination of synchronic episodes.
Abstract: This paper introduces a new method, single-trial phase locking statistics (S-PLS) to estimate phase locking in single trials of brain signals between two electrodes. The possibility of studying single trials removes an important limitation in the study of long-range synchrony in brain signals. S-PLS is closely related to our previous method, phase locking statistics (PLS) that estimates phase locking over a set of trials. The S-PLS method is described in detail and applied to human surface recordings during the task of face-recognition. We compare these results with those provided by PLS and show that they are qualitatively very similar, although S-PLS provides better discrimination of synchronic episodes.
301 citations
TL;DR: This contribution presents a brief introduction to the theory of synchronization of selfsustained oscillators, and the basic notions of phase and frequency locking are reconsidered within a common framework.
Abstract: In this contribution we present a brief introduction to the theory of synchronization of selfsustained oscillators. Classical results for synchronization of periodic motions and eects of noise on this process are reviewed and compared with recently found phase synchronization phenomena in chaotic oscillators. The basic notions of phase and frequency locking are reconsidered within a common framework. The application of phase synchronization to data analysis is discussed.
221 citations
TL;DR: This paper shows that many chaotic systems as paradigms in the research of chaos can be transformed into a class of nonlinear systems in the so-called nonautonomous "strict-feedback" form, and it is shown that the output of the nonaut autonomous system can asymptotically track theoutput of any known, bounded and smooth nonlinear reference model.
Abstract: This paper is concerned with the control of a class of chaotic systems using adaptive backstepping, which is a systematic design approach for constructing both feedback control laws and associated Lyapunov functions. Firstly, we show that many chaotic systems as paradigms in the research of chaos can be transformed into a class of nonlinear systems in the so-called nonautonomous "strict-feedback" form. Secondly, an adaptive backstepping control scheme is extended to the nonautonomous "strict-feedback" system, and it is shown that the output of the nonautonomous system can asymptotically track the output of any known, bounded and smooth nonlinear reference model. Finally, the Duffing oscillator with key constant parameters unknown, is used as an example to illustrate the feasibility of the proposed control scheme. Simulation studies are conducted to show the effectiveness of the proposed method.
163 citations
TL;DR: It is mathematically prove that the controlled system is indeed chaotic in the sense of Li and Yorke.
Abstract: In this paper, the problem of making a stable nonlinear autonomous system chaotic or enhancing the existing chaos of an originally chaotic system by using a small-amplitude feedback controller is studied. The designed controller is a linear feedback controller composed with a nonlinear modulo or sawtooth function, which can lead to uniformly bounded state vectors of the controlled system with positive Lyapunov exponents, thereby yielding chaotic dynamics. We mathematically prove that the controlled system is indeed chaotic in the sense of Li and Yorke. A few potential applications of the new chaotification algorithm are briefly discussed.
155 citations
TL;DR: It is shown that phase synchronization has important applications in the study of ecological communities where the spatial coupling of populations can lead to large scale complex synchronization eects.
Abstract: An ecological population model is presented for the purposes of exploring complex synchronization phenomena in biological systems. The model describes a three level predator{prey{ resource system which oscillates with Uniform Phase evolution, yet has Chaotic Abundance levels or Amplitudes (UPCA). We investigate the phase synchronization of two nonidentical diusively coupled phase coherent models (i.e. with UPCA dynamics) and extend the analysis to study the models’ \funnel" regimes and response to noise forcing. Similar synchronization effects are reported for a two-dimensional lattice of chaotic population models coupled via nearest neighbors. With weak coupling, a collective phase synchronization emerges yet the peak population abundance levels are chaotic and largely uncorrelated. The synchronization patterns and traveling wave structures found in the spatial model correspond to those observed in natural systems | in particular, Ecology’s well-known Canadian hare{lynx cycle. We show that phase synchronization has important applications in the study of ecological communities where the spatial coupling of populations can lead to large scale complex synchronization eects.
121 citations
TL;DR: In this article, a simple design of an adaptive observer for estimating the unknown parameters of the transmitter is proposed based on the design of quadratic Lyapunov function for the error system.
Abstract: The problem of synchronizing two nonlinear systems (transmitter and receiver) is considered. A simple design of an adaptive observer for estimating the unknown parameters of the transmitter is proposed based on the design of quadratic Lyapunov function for the error system. The results are illustrated by an example of signal transmission based on a pair of synchronizing Chua circuits.
121 citations
TL;DR: A very general formulation of the identical synchronization problem is developed, it is shown that asymptotic results can be derived for very general cases, and it is demonstrated that simple oscillator configurations can probe the Master Stability Function.
Abstract: 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.
118 citations
TL;DR: This paper compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D using the mountain–pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA).
Abstract: In this paper, we compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D. We present the mountain–pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA). Semilinear elliptic equations are well known to be rich in their multiplicity of solutions. Many such physically significant solutions are also known to lack stability and, thus, are elusive to capture numerically. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domains on the multiplicity. Special emphasis is placed on SIA and MPA, by which multiple unstable solutions are computed. The domains include the disk, symmetric or nonsymmetric annuli, dumbbells, and dumbbells with cavities. The nonlinear partial differential equations include the Lane–Emden equation, Chandrasekhar's equation, Henon's equation, a singularly perturbed equation, and equations with sublinear growth. Relevant numerical data of solutions are listed as possible benchmarks for other researchers. Commentaries from the existing literature concerning solution behavior will be made, wherever appropriate. Some further theoretical properties of the solutions obtained from visualization will also be presented.
TL;DR: This work presents an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron that was numerically computed using continuation of solutions of boundary value problems.
Abstract: Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.
TL;DR: Comparing the analysis of the relation between the tremor of both hands in normal subjects and subjects with a rare abnormal organization of certain neural pathways proves the involvement of central structures in enhanced physiological tremor.
Abstract: We discuss cross-spectral analysis and report applications for the investigation of human tremors. For the physiological tremor in healthy subjects, the analysis enables to determine the resonant contribution to the oscillation and allows to test for a contribution of reflexes to this tremor. Comparing the analysis of the relation between the tremor of both hands in normal subjects and subjects with a rare abnormal organization of certain neural pathways proves the involvement of central structures in enhanced physiological tremor. The relation between the left and the right side of the body in pathological tremor shows a specific difference between orthostatic and all other forms of tremor. An investigation of EEG and tremor in patients suffering from Parkinson's disease reveals the tremor-correlated cortical activity. Finally, the general issue of interpreting the results of methods designed for the analysis of bivariate processes when applied to multivariate processes is considered. We discuss and apply partial cross-spectral analysis in the frame of graphical models as an extention of bivariate cross-spectral analysis for the multivariate case.
TL;DR: The problem of transmitting digital information using chaotic signals over a channel with Gaussian white noise perturbation is introduced rigorously and how previously published methods, in particular those based on chaos synchronization, fit into this framework is shown.
Abstract: The problem of transmitting digital information using chaotic signals over a channel with Gaussian white noise perturbation is introduced rigorously. It is shown that discrete time base-band chaotic communication systems with discrete time Gaussian white noise in the channel are sufficiently general in this context. The optimal receiver is given explicitly in terms of conditional probabilities. For the example of chaos shift keying using iterations of the tent map, the optimal classifier is constructed explicitly. Finally, it is shown how previously published methods, in particular those based on chaos synchronization, fit into this framework.
TL;DR: This work studies the synchronization problem in discrete-time via an extended Kalman filter (EKF) and shows that the filter is indeed suitable for synchronization of (noisy) chaotic transmitter dynamics.
Abstract: We study the synchronization problem in discrete-time via an extended Kalman filter (EKF). That is, synchronization is obtained of transmitter and receiver dynamics in case the receiver is given via an EKF that is driven by a noisy drive signal from a noisy transmitter dynamics. The convergence of the filter dynamics towards the transmitter dynamics is rigorously shown using recent results in extended Kalman filtering. Two extensive simulation examples show that the filter is indeed suitable for synchronization of (noisy) chaotic transmitter dynamics. An application to private communication is also given.
TL;DR: A sliding mode hyperplane design for a class of chaotic systems with uncertainties is considered and it is guaranteed that under the proposed control law, uncertain chaotic systems can asymptotically track target orbits.
Abstract: A sliding mode hyperplane design for a class of chaotic systems with uncertainties is considered in this paper. The concept of extended systems is used such that continuous control input is obtained using a sliding mode design scheme. It is guaranteed that under the proposed control law, uncertain chaotic systems can asymptotically track target orbits. The converging speed of error states can be arbitrarily set by assigning the corresponding dynamics to the sliding surfaces. Illustrative examples of a controlled uncertain Duffing–Holmes system are presented.
TL;DR: It is concluded that one of these measures — the conditional probability index — allows reliable detection of synchronous epochs of different order n:m and, thus, makes possible an automatic processing of large data sets.
Abstract: We investigate the phase synchronization of heartbeat and respiration in a group of healthy infants. Having presented and compared two quantitative measures of synchronization, we conclude that one of these measures — the conditional probability index — allows reliable detection of synchronous epochs of different order n:m and, thus, makes possible an automatic processing of large data sets. In our analysis of experimental time series, we have found numerous epochs of phase synchronization. It turned out that the average degree of synchronization varies with the age of the newborns.
TL;DR: Stochastic resonance is brought into the contemporary framework of applied mathematics which has begun to unify the study of fluctuations, statistical mechanics and information theory through the transfer of renormalization methods from statistical mechanics to information theory.
Abstract: A growing body of work suggests an important unifying perspective in the study of stochastic resonance: Examining the "prehistory probability density" of the ensemble of pathways by which a system approaches the trigger of the nonlinear oscillator. This view, in the context of the Shannon–McMillan Theorem of information theory, leads to a draconian simplification of the complex of "signal," "noise" and oscillator in terms of a single object, an ergodic information source dual to a certain class of resonances. For "spatial" stochastic resonator arrays, in the most general sense, the Shannon–McMillan Theorem further provides an analogy between the Shannon uncertainty of the dual information source and the free energy density of a physical system which allows imposition of real-space renormalization to give essential results in a straightforward manner. We find in particular threshold behavior in the onset of an epileptiform spatiotemporal coherence, and a likely usefulness of tuned spatial arrays for the detection of very subtle pattern. These simplifications may provide, in addition, a natural context for discussing hierarchical structure in neural systems. In sum, we attempt to bring stochastic resonance into the contemporary framework of applied mathematics which has begun to unify the study of fluctuations, statistical mechanics and information theory. Our critical generalization lies in the transfer of renormalization methods from statistical mechanics to information theory, via a parametization of source uncertainty.
TL;DR: The dynamics of a fourth-order autonomous nonlinear electric circuit with two active elements, one linear negative conductance and one nonlinear resistor with a symmetrical piecewise-linear v–i characteristic is studied.
Abstract: In this paper we study the dynamics of a fourth-order autonomous nonlinear electric circuit with two active elements, one linear negative conductance and one nonlinear resistor with a symmetrical piecewise-linear v–i characteristic. Using the capacitances C1 and C2 as the control parameters, we observe the phenomenon of antimonotonicity and the formation of "bubbles" in the development of bifurcations, resulting typically in reverse period-doubling sequences. We also find a crisis-induced intermittency, when the spiral attractor suddenly widens to a double-scroll attractor. We have plotted several bifurcation diagrams of reverse period-doubling sequences and computed the scaling parameter δ versus the control parameter C2 for the different regimes, where bubbles evolve. Thus, besides the usual Feigenbaum constant δ → δF = 4.6692…, we also observe, in some cases, a convergence of δ to , as expected from theoretical considerations. Finally, by plotting a return map associated with one of the state variables, we demonstrate the strongly one-dimensional character of the dynamics and discuss the dependence of this map on the parameters of the system.
TL;DR: It is shown that a system exhibiting chaos can be driven to a desired periodic motion by designing a combination of feedforward controller and a time-varying controller.
Abstract: A general framework for local control of nonlinearity in nonautonomous systems using feedback strategies is considered in this work. In particular, it is shown that a system exhibiting chaos can be driven to a desired periodic motion by designing a combination of feedforward controller and a time-varying controller. The design of the time-varying controller is achieved through an application of Lyapunov–Floquet transformation which guarantees the local stability of the desired periodic orbit. If it is desired that the chaotic motion be driven to a fixed point, then the time-varying controller can be replaced by a constant gain controller which can be designed using classical techniques, viz. pole placement, etc. A sinusoidally driven Duffing's oscillator and the well-known Rossler system are chosen as illustrative examples to demonstrate the application.
TL;DR: This work carries out a two-parameter bifurcation analysis of a model of the Colpitts oscillator and shows that the birth of the harmonic cycle is associated with a Hopf bIfurcation and the effects of idealization in the model are discussed.
Abstract: In this work we consider the Colpitts oscillator as a paradigm for sinusoidal oscillation and we investigate its nonlinear dynamics In particular, we carry out a two-parameter bifurcation analysis of a model of the oscillator This analysis is conducted by combining numerical continuation techniques and normal form theory First, we show that the birth of the harmonic cycle is associated with a Hopf bifurcation and we discuss the effects of idealization in the model Various families of limit cycles are identified and their bifurcations are analyzed in detail In particular, we demonstrate that the bifurcation diagram in the parameter space is organized by an infinite family of homoclinic bifurcations Finally, local and global coexistence phenomena are described
TL;DR: This paper is concerned with the study of nonlinear phenomena in a closed loop voltage-controlled DC–DC Buck–Boost converter when suitable parameters are varied and it is shown that the winding number plotted as a function of the bifurcation parameter is a devil's staircase.
Abstract: This paper is concerned with the study of nonlinear phenomena in a closed loop voltage-controlled DC–DC Buck–Boost converter when suitable parameters are varied. The dynamics is analyzed using both the continuous-time model and the numerically computed stroboscopic map. The analysis of the one-dimensional bifurcation diagram shows that Neimarck–Sacker bifurcation occurs at certain values of the parameters. Phase-locking periodic windows, the period-adding sequence, and transition from quasiperiodicity to period-doubling via torus breakdown are also obtained. The two-dimensional bifurcation diagram is carefully computed. This shows that phase-locking orbits produce so-called Arnold tongues in the parameter space. It is shown that the winding number plotted as a function of the bifurcation parameter is a devil's staircase. As typically occurs in general circle maps, the fine structures of the Arnold tongues and the devil's staircase show self-similarity.
TL;DR: This work descretize nonlinear elliptic PDEs by the MQ method, which results in modest-size systems of nonlinear algebraic equations which can be efficiently continued by standard continuation software such as AUTO and CONTENT.
Abstract: The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest-size systems of nonlinear algebraic equations which can be efficiently continued by standard continuation software such as AUTO and CONTENT. Examples are given of detection of bifurcations in 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.
TL;DR: Evidence is on hand showing that the principal form of synchrony is by establishment of a low degree of covariance among very large numbers of otherwise autonomous neurons, which allows for rapid state transitions of neural populations between successive chaotic basins of attraction along itinerant trajectories.
Abstract: The electrical activity of neurons in brains fluctuates erratically both in terms of pulse trains of single neurons and the dendritic currents of populations of neurons. Obviously the neurons interact with one another in the production of intelligent behavior, so it is reasonable to expect to find evidence for varying degrees of synchronization of their pulse trains and dendritic currents in relation to behavior. However, synaptic communication between neurons depends on propagation of action potentials between neurons, often with appreciable distances between them, and the transmission delays are not compatible with synchronization in any simple way. Evidence is on hand showing that the principal form of synchrony is by establishment of a low degree of covariance among very large numbers of otherwise autonomous neurons, which allows for rapid state transitions of neural populations between successive chaotic basins of attraction along itinerant trajectories. The small fraction of covariant activity is extracted by spatial integration upon axonal transmission over divergent–convergent pathways, through which a remarkable improvement in signal-to-noise ratio is achieved. The raw traces of local activity show little evidence for synchrony, other than zero-lag correlation, which appears to be largely a statistical artifact. Brains rely less on tight phase-locking of small numbers of periodically firing neurons and more on low degrees of cooperativity achieved by order parameters influencing very large numbers of neurons. Brains appear to be indifferent to and undisturbed by widely varying time and phase relations between individual neurons and even large semi-autonomous areas of cortex comprising their mesoscopic neural masses.
TL;DR: It is demonstrated that the heart rate of a healthy human can be synchronized by means of weak external noninvasive forcing in the form of a sequence of sound and light pulses, being either periodic or aperiodic.
Abstract: We demonstrate that the heart rate of a healthy human can be synchronized by means of weak external noninvasive forcing in the form of a sequence of sound and light pulses, being either periodic or aperiodic, the latter forcing given by interbeat intervals of the heart of another subject. The phenomenon of phase locking of n:m type is observed for both situations in about 90% of our experiments. The plot for the ratio of forcing frequency to the average frequency of response versus detuning possesses a plateau and is in agreement with classical synchronization theory.
TL;DR: From the study, chaos of pulsation in capillary vessels reflected mental conditions through the autonomic nervous system and found that the chaotic fluctuations decreased in the symptomatic phase of psychiatric patients.
Abstract: The presence of deterministic chaos has been clarified experimentally in the time series of pulsation of capillary vessels [Tsuda et al., 1992]. In this paper, first we reconfirmed the nonlinearity of the time series of pulsation of capillary vessels with surrogate data testing. Second, we investigated the characteristics of chaos of capillary data on normal subjects in different states of the autonomic nervous system and on psychiatric patients. As a result, the chaotic fluctuations decreased for the state of sympathetic dominance of normal subjects. On the contrary, the chaotic fluctuations increased for the state of parasympathetic predominance of normal subjects. We also found that the chaotic fluctuations decreased in the symptomatic phase of psychiatric patients. Interpretation of this fact is as follows. Psychiatric patients in symptomatic phase are in a condition of poor flexibility of emotion and motivated behavior in the limbic system. This condition affects the regulation of the autonomic nervous system in the hypothalamus. So it is easy to fall into the imbalance of the autonomic nervous system even with slight stress. Furthermore, in the attractor of capillary chaos we found two parts that showed the role of origin of stimulation of the sympathetic nervous system and that of stimulation of the parasympathetic nervous system, respectively. The geometrical change of topology was caused by the interplay of two parts. From our study we concluded that chaos of pulsation in capillary vessels reflected mental conditions through the autonomic nervous system.
TL;DR: Results are presented on the application and evaluation of chaos control for slowing and regularizing local electrical activation of the right atrium of humans during induced atrial fibrillation.
Abstract: Chaos control has been applied to control atrial fibrillation in humans. Results are presented on the application and evaluation of chaos control for slowing and regularizing local electrical activation of the right atrium of humans during induced atrial fibrillation.
TL;DR: Experimental phase synchronization of chaos is demonstrated for a plasma discharge tube subject to a high dc voltage (800–900 V), and paced with a low amplitude wave generator.
Abstract: Experimental phase synchronization of chaos is demonstrated for a plasma discharge tube subject to a high dc voltage (800–900 V), and paced with a low amplitude (less than 1 V) wave generator.
TL;DR: It is shown that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient, using the idea of Melnikov equivalence.
Abstract: This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.
TL;DR: This work shows via numerical simulations that the proposed feedback control strategy of chemotherapy in an early treatment setting can handle strong uncertainties in the HIV dynamics induced by imperfect modeling and sampled/delayed cell measurements.
Abstract: Using a model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce a feedback control strategy of chemotherapy in an early treatment setting, where the control represents the percentage of effect chemotherapy has on the viral production. We seek to regulate the viral count by manipulating the percentage of effect chemotherapy has on the viral production. We show via numerical simulations that the proposed feedback control strategy can handle strong uncertainties in the HIV dynamics induced by imperfect modeling and sampled/delayed cell measurements.