Synchronization in the human cardiorespiratory system

We investigate synchronization between cardiovascular and respiratory systems in healthy humans under free-running conditions. For this aim we analyze nonstationary irregular bivariate data, namely, electrocardiograms and measurements of respiratory flow. We briefly discuss a statistical approach to synchronization in noisy and chaotic systems and illustrate it with numerical examples; effects of phase and frequency locking are considered. Next, we present and discuss methods suitable for the detection of hidden synchronous epochs from such data. The analysis of the experimental records reveals synchronous regimes of different orders n:m and transitions between them; the physiological significance of this finding is discussed.     

一个不连续映象中的混沌稳定或混沌抑制

借助于一个张弛振子模型和与之相应的不连续映象说明了在这类系统中瞬态集的映蔽效应会产生三种区域：1）稳定混沌区，在此区域中不存在周期窗口，混沌轨道是结构稳定的；2）完全锁相区，在此区域中混沌被抑制，只存在周期运动；3）准周期区域，在此区域中混沌被抑制，只存在准周期或临界稳定的周期运动．这种思想被用来解释在一个实际张弛振子电路中观察到的稳定混沌区和完全锁相区 关键词：     

Phase synchronization in unidirectionally coupled chaotic ratchets

We study chaotic phase synchronization of unidirectionally coupled deterministic chaotic ratchets. The coupled ratchets were simulated in their chaotic states and perfect phase locking was observed as the coupling was gradually increased. We identified the region of phase synchronization for the ratchets and show that the transition to chaotic phase synchronization is via an interior crisis transition to strange attractor in the phase space.     

振幅耦合动态网络中相邻结点间的相同步

研究用适当的量化指标来刻画动态网络的相同步,为此定义了新的量化指标:相邻结点的网络平均锁相值和网络平均相频差.动态网络结点选择的是多旋转中心的Lorenz混沌振子,对Lorenz系统进行柱面坐标变换,用振幅耦合方法构造动态网络.分别对星形网络和小世界网络进行了仿真计算,结果表明随着耦合强度的增大,网络中相邻结点的两个系统之间存在相同步现象,而且相同步行为与定义的量化指标之间存在较准确的对应关系.     

Frequency locking by external force from a dynamical system with strange nonchaotic attractor

Usually, phase synchronization is studied in chaotic systems driven by either periodic force or chaotic force. In the present work, we consider frequency locking in chaotic Rössler oscillator by a special driving force from a dynamical system with a strange nonchaotic attractor. In this case, a transition from generalized marginal synchronization to frequency locking is observed. We investigate the bifurcation of the dynamical system and explain why generalized marginal synchronization can occur in this model.     

Locking response in coupled phase systems

Some distinctive features of the locking response to a chaotic action are pointed out using the example of coupled phase systems of the delay type: a) the possibility of locking for any values of the parameters of the acting subsystem in the case of precise matching of the response-subsystem parameters; b) relatively low sensitivity to mismatching of the subsystem parameters; and c) mild loss of locking in a nonuniform chain of phase systems.Nizhny Novgorod State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 8, pp. 811–814, August, 1993.     

Suppression of Chaos and Phase Locking in Two Coupled Nonidentical Neurons under Periodic Input

Dynamical behaviour of two coupled neurons with at least one of them being chaotic is presented. Bifurcation diagrams and Lyapunov exponents are calculated to diagnose the dynamical behaviour of the coupled neurons with the increasing coupling strength. It is found that when the coupling strength increases, a chaotic neuron can be controlled by the coupling between neurons. At the same time, phase locking is studied by the maxima of the differences of instantaneous phases and average frequencies between two coupled neurons, and the inherent connection of phase locking and the suppression of chaos is formulated. It is observed that the onset of phase locking is closely related to the suppression of chaos. Finally, a way for suppression of chaos in two coupled nonidentical neurons under periodic input is suggested.     

n:m phase synchronization with mutual coupling phase signals

We generalize the n:m phase synchronization between two chaotic oscillators by mutual coupling phase signals. To characterize this phenomenon, we use two coupled oscillators to demonstrate their phase synchronization with amplitudes practically noncorrelated. We take the 1:1 phase synchronization as an example to show the properties of mean frequencies, mean phase difference, and Lyapunov exponents at various values of coupling strength. The phase difference increases with 2pi phase slips below the transition. The scaling rules of the slip near and away from the transition are studied. Furthermore, we demonstrate the transition to a variety of n:m phase synchronizations and analyze the corresponding coupling dynamics. (c) 2002 American Institute of Physics.     

Study on the method of controlling chaos in an Er-doped fiber dual-ring laser via external optical injection and shifting optical feedback light

Controlling chaos in an erbium-doped fiber dual-ring laser is studied by injecting an external light and shifting the phase of an optical feedback light. First, the parameters of the externally injected optical signal are adjusted to bring the chaotic laser into a single-period state and a multiperiod state, respectively. In addition, the single-mode locking in a single ring of the laser as well as the dual-mode locking in dual ring of the laser are found. Secondly, the method of optical feedback is used to stabilize chaos by controlling the parameters of the optical devices in an external optical path. We show that the optical negative feedback can stabilize the chaotic laser into a single-period state, while the optical positive feedback can stabilize the chaotic laser into another single-period state and a period-doubling state. Lastly, various dynamical states are produced in the laser by shifting the phase of the optical feedback light.     

Analysis of phase synchronization of chaotic oscillations in terms of symbolic CTQ-analysis

The application of symbolic CTQ-analysis for studying synchronization of chaotic oscillations is considered. This approach differs substantially from its analogs since it makes it possible to diagnose and measure quantitatively the characteristics of intermittency regimes in synchronization of chaotic systems and, hence, to analyzer the temporal structure of synchronization. The application of the symbolic analysis apparatus based on the T alphabet to systems with phase locking and synchronization of time scales is demonstrated for the first time. As an example, a complex system of two mutually coupled nonidentical Rössler oscillators in the helical chaos regime with attractors having an ill-conditioned phase is considered. The results show that the method considered here makes it possible to reliably diagnose synchronism sooner than a phase locking and/or time-scale synchronization threshold is detected.     

Spatial disorder and pattern formation in lattices of coupled bistable elements

The spatio-temporal dynamics of discrete lattices of coupled bistable elements is considered. It is shown that both regular and chaotic spatial field distributions can be realized depending on parameter values and initial conditions. For illustration, we provide results for two lattice systems: the FitzHugh-Nagumo model and a network of coupled bistable oscillators. For the latter we also prove the existence of phase clusters, with phase locking of elements in each cluster.     

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic R?ssler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.     

Chaotic phase synchronization in small-world networks of bursting neurons

We investigate the chaotic phase synchronization in a system of coupled bursting neurons in small-world networks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that phase synchronization is largely facilitated by a large fraction of shortcuts, but saturates when it exceeds a critical value. We also study the external chaotic phase synchronization of bursting oscillators in the small-world network by a periodic driving signal applied to a single neuron. It is demonstrated that there exists an optimal small-world topology, resulting in the largest peak value of frequency locking interval in the parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this interval increases with the driving amplitude, but decrease rapidly with the network size. We infer that the externally applied driving parameters outside the frequency locking region can effectively suppress pathologically synchronized rhythms of bursting neurons in the brain.     

Phase locking and chaotic synchronization in an array of three lasers

A linear array of three lasers that are coupled mutually in space is investigated. It is shown that the phase of the laser fields is locked with intermediate coupling while the laser intensities are totally chaotic and chaotically synchronized. When the intensities of lasers reenter the regime of chaotic synchronization at smaller coupling constant, the laser fields show low degree of phase locking. The phase differences in the fields between three lasers show rich patterns when the coupling is changed. Received 3 August 2001 and Received in final form 27 September 2001     

Locking-based frequency measurement and synchronization of chaotic oscillators with complex dynamics

We propose a method for the determination of a characteristic oscillation frequency for a broad class of chaotic oscillators generating complex signals. It is based on the locking of standard periodic self-sustained oscillators by an irregular signal. The method is applied to experimental data from chaotic electrochemical oscillators, where other approaches of frequency determination (e.g., based on Hilbert transform) fail. Using the method we characterize the effects of phase synchronization for systems with ill-defined phase by external forcing and due to mutual coupling.     

Hyperchaos of arbitrary order generated by a single feedback circuit, and the emergence of chaotic walks

It is shown that hyperchaos of order m (i.e., with m positive Lyapunov exponents) can be generated by a single feedback circuit in n = 2m + 1 variables. This feedback circuit is constructed such that, dividing phase space into hypercubes, it changes sign wherever the trajectory passes from one hypercube into an adjacent one. Letting the negative diagonal elements in the Jacobian tend to zero, the dynamics becomes conservative. Instead of chaotic attractors, unbounded chaotic walks are then generated. Here we report chaotic walks emerging from a continuous system rather than the well known chaotic walks present in "Lorentz gas" and "couple map lattices."     

A geometric theory of chaotic phase synchronization

A rigorous mathematical treatment of chaotic phase synchronization is still lacking, although it has been observed in many numerical and experimental studies. In this article we address the extension of results on phase synchronization in periodic oscillators to systems with phase coherent chaotic attractors with small phase diffusion. As models of such systems we consider special flows over diffeomorphisms in which the neutral direction is periodically perturbed. A generalization of the Averaging Theorem for periodic systems is used to extend Kuramoto's geometric theory of phase locking in periodically forced limit cycle oscillators to this class of systems. This approach results in reduced equations describing the dynamics of the phase difference between drive and response systems over long time intervals. The reduced equations are used to illustrate how the structure of a chaotic attractor is important in its response to a periodic perturbation, and to conclude that chaotic phase coherent systems may not always be treated as noisy periodic oscillators in this context. Although this approach is strictly justified for periodic perturbations affecting only the phase variable of a chaotic oscillator, we argue that these ideas are applicable much more generally.     

耦合混沌系统的相同步：从高维混沌到低维混沌

混沌系统的相同步现象是近几年混沌同步研究的热点,它反映了混沌运动中的有序行为.用分岔树来研究耦合系统相同步的进程,并用Lyapunov指数谱来探讨系统动力学在相同步时从高维混沌向低维混沌过渡的进程.发现了从多个有理同步的时间交替到完全相同步的道路.还 发现了相同步中的混沌抑制及通过倍周期分岔向混沌同步的恢复.此外,研究表明,非对称 耦合可以大大加强耦合系统的相同步,这对实际应用有重要的意义. 关键词： 相同步 分岔树 李指数     

Automatic control of phase synchronization in coupled complex oscillators

We present an automatic control method for phase locking of regular and chaotic nonidentical oscillations, when all subsystems interact via feedback. This method is based on the well known principle of feedback control which takes place in nature and is successfully used in engineering. In contrast to unidirectional and bidirectional coupling, the approach presented here supposes the existence of a special controller, which allows to change the parameters of the controlled systems. First we discuss general principles of automatic phase synchronization (PS) for arbitrary coupled systems with a controller whose input is given by a special quadratic form of coordinates of the individual systems and its output is a result of the application of a linear differential operator. We demonstrate the effectiveness of our approach for controlled PS on several examples: (i) two coupled regular oscillators, (ii) coupled regular and chaotic oscillators, (iii) two coupled chaotic Rössler oscillators, (iv) two coupled foodweb models, (v) coupled chaotic Rössler and Lorenz oscillators, (vi) ensembles of locally coupled regular oscillators, (vii) ensembles of locally coupled chaotic oscillators, and (viii) ensembles of globally coupled chaotic oscillators.     

Synchronization of polarization mode of two unidirectionally coupled vertical-cavity surface-emitting laser and its application in communication

We theoretically investigate the synchronization performance of polarization mode of two unidirectionally coupled vertical-cavity surface-emitting laser (VCSEL) and propose an encryption scheme, where the phase of the polarization states of the master laser are modulated. It is demonstrated that high synchronization can be achieved between corresponding polarization components of the master laser and the slave laser. Moreover, two distinct synchronization regimes exist, complete synchronization and injection locking. The transmitted digital information is successfully recovered at each polarization branch of the slaver laser. The results open an opportunity for multichannel chaotic communication by utilizing different polarization components in single-mode or multi-transverse-mode VCSELs.