International Journal of Bifurcation and Chaos

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.

By considering cascaded heterogeneous chaotic systems with drive-response configurations, information signal is encoded in the first chaotic system of the drive-module. A small amplitude chaotic signal from this first system is used as new information signal to the second different chaotic system within the drive-module. The chaotic signal from the second system of the drive-module is now transmitted to the response-module. An appropriate feedback loop is constructed in the response-module to achieve synchronization among the variables of the drive- and response-modules. We apply this approach to transmit analog and digital information signals in which the quality of the recovered signal is higher and the encoding is potentially secure.

We studied synchronization of electrosensitive cells of the paddlefish by means of electrophysiological experiments. We found that primary afferents of the paddlefish are represented by noisy nonlinear oscillators. Different types of phase locked regimes are observed. The influence of internal noise is discussed.

The 300 year record of the yearly sunspot numbers and numerically generated trajectory of the solar inertial motion (SIM) were subjects of a synchronization analysis. Phase synchronization of the sunspot cycle and a fast component of the SIM have been found and confirmed with statistical significance in three epochs (1727–1757, 1802–1832 and 1863–1922) of the entire 1700–1997 record. This result can be considered as a quantitative support for the hypothesis that there is a weak interaction of gravity and solar activity.

It is shown that phase synchronization of chaotic dynamical systems can be achieved using a binary coupling signal that takes only two possible values. This phenomenon is demonstrated using analog electronic implementations of the Rössler system and Chua's oscillator.

Phase synchronization is shown to occur between opposite cells of a ring consisting of chaotic Lorenz oscillators coupled unidirectionally through driving. As the coupling strength is diminished, full phase synchronization cannot be achieved due to random generation of phase jumps. The Brownian dynamics underlying this process is studied in terms of a stochastic diffusion model of a particle in a one-dimensional medium.

The effect of coherence resonance can change the firing process in noise-driven excitable systems towards rather regular dynamics. This effect provides a mechanism of the generation of stochastic oscillations whose characteristics are controlled by noise intensity. Following this, a noisy excitable system can be considered as a corehence resonance oscillator. For such functional units, we investigate the mutual and forced synchronization in terms of locking of the peak frequencies in the power spectrum and also in terms of phase locking. The connection of synchronization phenomenon of noise-induced oscillations and coherence resonance effect is discussed. The examples, studied numerically and experimentally, include Morris–Lecar neuron model and a monovibrator electronic circuit, respectively.

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.

Neuronal activity during information processing and muscle activity are generally characterized by oscillations. Mostly, widespread areas are involved and electrophysiological signals are measured on different sites of the cortex or of the muscle.

In order to investigate functional relationships between different components of multidimensional electrophysiological signals, coherence and phase analyses turned out to be useful tools. These parameters allow the investigation of synchronization phenomena with regard to oscillations of defined frequencies or frequency bands.

Coherence and phase are closely connected spectral parameters. Coherence may be understood as a measure of phase stability. Whereas coherence describes the amount of common information with regard to oscillations within certain frequency bands, the corresponding phase, from which time delays of these oscillations can be computed, hints at the direction of information transfer through oscillation.

Coherence and phase analysis of surface EMG during continuous activity of deep and superficial muscles show distinct differences due to volume conduction properties of myoelectrical signals. Superficial activity therefore is characterized by significant coherence and stable phase relationships, which, additionally, can be used to determine motor unit action potential (MUAP) propagation velocity along the fibre direction without application of invasive methods. Deep muscle activity lacks significant coherence.

Mental processes can be very brief and cooperation between different areas may be highly dynamic. For this reason in addition to usual Fourier estimation of coherence and phase, a two-dimensional approach of adaptive filtering was developed to estimate coherence and phase continuously in time. Statistical and dynamic properties of instantaneous phase are discussed.

In order to demonstrate the value of this method for studying higher cognitive processes the method was applied to EEG recorded during word processing. During visual presentation of abstract nouns an information transfer through the propagation of oscillations from visual areas to frontal association areas in the α₁-frequency band could be verified within the first 400 ms. In contrast, in case of auditory presentation positive phases from the temporal electrode locations T3 and T4 towards the occipital areas appear within the time interval of 300 ms–600 ms. The α₁-band predominately seems to reflect sensory processing and attention processes.

Chaotic lag synchronization is a subtle dynamical phenomenon in which the states of two mutually coupled, nonidentical chaotic oscillators follow each other with a fixed time delay. We investigate to what extent lag synchronization can be observed in laboratory experiments. Our measurements indicate that due to the influence of noise, lag synchronization appears to occur intermittently in time. Numerical confirmation and a heuristic explanation to the observed intermittent behavior are given.

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.

Synchronization in chaotic systems has become an active area of research since the pioneering work of Pecora and Carroll. Most existing works, however, rely on a passive approach: A coupling between chaotic systems is necessary for their mutual synchronization. We describe here a feedback approach for synchronizing chaotic systems that is applicable in high dimensions. We show how two chaotic systems can be synchronized by applying small feedback perturbations to one of them. We detail our strategy to design the control based on the pole-placement method, and give numerical examples.

We consider a relatively new application of mutually interacting, synchronized oscillators. The idea is to intentionally introduce variations among the elements to induce phase shifts between the oscillators. Though often unwanted, in certain instances these phase shifts are highly desirable: We discuss how to manipulate array parameters in order to steer and scan the beam emanating from radiating arrays.

Considering two identical, coupled Rössler systems, the paper first examines the bifurcations through which low-periodic orbits embedded into the synchronized chaotic state lose their transverse stability and produce the characteristic picture of riddled basins of attraction. The paper hereafter addresses the issue of the robustness of the synchronized chaotic state to a mismatch of the parameter values between the two subsystems. It is shown that the synchronized state is shifted away from the symmetric manifold, and the magnitude of this shift is expressed in terms of the coupling strength and the mismatch parameter. Finally, the paper illustrates how similar phenomena can be observed in a system of two coupled chaotic oscillators describing the spiking behavior of biological cells.

A periodic action onto a chaotic system can result in phase synchronization: Although the oscillations remain chaotic, their pace is prescribed by the rhythm of the force. We show how this adjustment of characteristic timescales to the period of the forcing is violated when, due to the presence of a saddle point in the autonomous attractor, the time intervals between returns onto a Poincaré plane are unbounded. In such a situation, the segments of a chaotic trajectory in which its phase follows the phase of the force, can alternate with short segments of phase-locking in ratios different from 1:1. This phenomenon is demonstrated for several nonlinear systems.