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The phase reduction approach has manifested its efficacy in investigating synchronization behaviors in limit-cycle oscillators. However, spatial distributions of the phase value on the limit cycle may lead to illusions of synchronizations for oscillators close to bifurcations. In this paper, we compared the phase sensitivity function in the spatial domain and time domain for oscillators close to saddle-node homoclinic (SNH) bifurcation, also known as saddle-node bifurcation on an invariant circle. It was found that the phase sensitivity function in the spatial domain can show the phase accumulation feature on the limit cycle, which can be ignored in the phase sensitivity function in the time domain. As an example, the synchronization distributions of uncoupled SNH oscillators driven by common and independent noises were investigated, where the space-dependent coupling function was considered on common noise. These results shed some light on the phase dynamics of oscillators close to bifurcations.

The radiation dynamics of a closed chain of lasers with optoelectronic delayed coupling is analyzed. We consider bidirectional couplings of two designs: (i) diffusion type with feedback and (ii) bidirectional type without feedback. Assuming that the number of lasers is sufficiently large, we propose integro-differential models for spatially distributed variables with periodic boundary conditions. The critical value of the coupling coefficient is determined at which the stationary state of laser generation spontaneously becomes unstable due to the presence of a delay in the coupling lines. Bifurcations of asymptotically infinite dimension are described. For both types of coupling, we obtain the same two-dimensional complex partial differential equation of the Ginzburg–Landau type (with the difference only in the coefficients) for the slowly varying amplitude of the fundamental harmonic. Using the simplest homogeneous solution of such a quasi-normal form, the oscillations of laser radiation in chains, which can be anti-phase or in-phase depending on the time delay and the design of the couplings, are analytically described. • The models of a large laser chain with bidirectional couplings are proposed. • Bifurcations of infinite dimension are studied in dependence on coupling delay. • Boundary-value problems are obtained as quasi-normal forms. • The delay-dependent direction (super- or subcritical) of bifurcation is determined. • Delay intervals leading to anti- or in-phase oscillations in laser chain are obtained. [ABSTRACT FROM AUTHOR]

The phase reduction approach has manifested its efficacy in investigating synchronization behaviors in limit-cycle oscillators. However, spatial distributions of the phase value on the limit cycle may lead to illusions of synchronizations for oscillators close to bifurcations. In this paper, we compared the phase sensitivity function in the spatial domain and time domain for oscillators close to saddle-node homoclinic (SNH) bifurcation, also known as saddle-node bifurcation on an invariant circle. It was found that the phase sensitivity function in the spatial domain can show the phase accumulation feature on the limit cycle, which can be ignored in the phase sensitivity function in the time domain. As an example, the synchronization distributions of uncoupled SNH oscillators driven by common and independent noises were investigated, where the space-dependent coupling function was considered on common noise. These results shed some light on the phase dynamics of oscillators close to bifurcations. [ABSTRACT FROM AUTHOR]

A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems. [ABSTRACT FROM AUTHOR]

Coherent terahertz (THz) radiations based on intrinsic Josephson junctions (IJJs) from a Bi2Sr2CaCu2O (BSCCO) mesa under a bias voltage were observed in 2007. Later on, a novel phase kink state was proposed which can explain the key experimental results. However, how to drive the system into the phase kink state efficiently still remains an open problem. In this work, we propose several ways to synchronize the system to the phase kink state. We start from random and typical regular initial phase configurations. It is found that starting from some initial phase configurations, the system can be driven efficiently into the phase kink state. The time spent before entering the phase kink states for these initial phase configurations is shorter when the initial phase configuration is further away from the uniform phase configuration. [ABSTRACT FROM AUTHOR]

One of the most sensitive methods for determining the presence and character of coupling between two oscillators is based on the modeling of their phase dynamics. This method is applicable if the observed oscillations are not synchronous and the length of available time series is no less than 50 characteristic periods. Now it is established that the method can be applied to still shorter time series with a length of 20 to 50 periods, provided that the estimated phase coherence coefficient (measuring the degree of synchronization) does not exceed a certain threshold dependent on the time series length and the frequency detuning. This result expands the possibilities of detecting the coupling between oscillatory systems under conditions of nonstationary signals and deficient data. [ABSTRACT FROM AUTHOR]

Complex continuous wavelet transforms are used to study the dynamics of instantaneous phase difference δϕ between the fluctuations of arterial blood pressure (ABP) and cerebral blood flow velocity (CBFV) in a middle cerebral artery. For healthy individuals, this phase difference changes slowly over time and has an almost uniform distribution for the very low-frequency (0.02-0.07 Hz) part of the spectrum. We quantify phase dynamics with the help of the synchronization index γ = ⟨sinΔϕ⟩² + ⟨cosδϕ⟩² that may vary between 0 (uniform distribution of phase differences, so the time series are statistically independent of one another) and 1 (phase locking of ABP and CBFV, so the former drives the latter). For healthy individuals, the group-averaged index γ has two distinct peaks, one at 0.11 Hz [γ = 0.59 ± 0.09] and another at 0.33 Hz (γ = 0.55 ± 0.17). In the very low-frequency range (0.02-0.07 Hz), phase difference variability is an inherent property of an intact autoregulation system. Consequently, the average value of the synchronization parameter in this pan of the spectrum is equal to 0.13 ± 0.03. The phase difference variability sheds new light on the nature of cerebral hemodynamics, which so far has been predominantly characterized with the help of the high-pass filter model. In this intrinsically stationary approach, based on the transfer function formalism, the efficient autoregulation is associated with the positive phase shift between oscillations of CBFV and ABP. However, the method is applicable only in the part of the spectrum (0.1-0.3 Hz) where the coherence of these signals is high. We point out that synchrony analysis through the use of wavelet transforms is more general and allows us to study nonstationary aspects of cerebral hemodynamics in the very low-frequency range where the physiological significance of autoregulation is most strongly pronounced. [ABSTRACT FROM AUTHOR]

Complex-valued Hopfield networks which possess the energy function are analyzed. The dynamics of the network with certain forms of an activation function is decomposable into the dynamics of the amplitude and phase of each neuron. Then the phase dynamics is described as a coupled system of phase oscillators with a pair-wise sinusoidal interaction. Therefore its phase synchronization mechanism is useful for the area-wide offset control of the traffic signals. The computer simulations show the effectiveness under the various traffic conditions. [ABSTRACT FROM AUTHOR]

A new method is proposed for the investigation of transitions from synchronous dynamics of coupled oscillators to asynchronous regimes attained upon synchronization breakage by a sequence of short phase resetting pulses. The possibility to use the proposed method for determining the coefficient of coupling between interacting systems from an analysis of the signals generated by these systems is considered. [ABSTRACT FROM AUTHOR]

In this paper we extend the results of Frankel and Kiemel [SIAM J. Appl. Math, 53 (1993), pp. 1436-1446] to a network of slowly coupled oscillators. First, we use Malkin's theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network. Then, we illustrate the result using slowly coupled oscillators (1) near Andronov-Hopf bifurcations, (2) near saddle-node on invariant circle bifurcations, and (3) near relaxation oscillations. We compare and contrast synchronization properties of slowly and weakly coupled oscillators. [ABSTRACT FROM AUTHOR]

We investigate theoretically and experimentally the detrimental effect of fiber dispersion on the synchronization of an optoelectronic phase chaos cryptosystem. We evaluate the root-mean square synchronization error and the cancellation spectra between the emitter and the receiver in order to characterize the quality of the optical fiber communication link. These two indicators explicitly show in temporal and spectral domain how fiber dispersion does negatively affect the phase chaos cancellation at the receiver stage. We demonstrate that the dispersion management techniques used in conventional optical fiber networks, such as dispersion-compensating modules/fibers or dispersion shifted fibers, are also efficient to strongly reduce the detrimental effects of fiber propagation in phase chaos communications. This compatibility therefore opens the way to a successful integration of more than 10-Gb/s phase chaos communications systems in existing networks, even when the fiber link spans over more than 100 km. [ABSTRACT FROM PUBLISHER]

We study fast and slow clustering dynamics of Cucker–Smale ensemble with internal phase dynamics via the Cucker–Smale–Kuramoto (in short, CSK) model. The CSK model describes the emergent dynamics of flocking particles with phase dynamics. It consists of the Cucker–Smale flocking model and the Kuramoto model, and their interplay is registered in the communication weight function between particles. We present a sufficient framework for mono-cluster flocking and complete synchronization in terms of system parameters and initial data. In particular, the mono-cluster flocking will emerge exponentially fast (fast dynamics). On the other hand, when initial spatial-velocity configuration is close to the bi-cluster flocking, we also provide a sufficient framework leading to convergence of bi-cluster flocking algebraically slow depending on the decay rate of the communication weight (slow flocking dynamics). We also provide several numerical examples and compare them with analytical results. [ABSTRACT FROM AUTHOR]

This study numerically investigates the influence of the taper on the flow-induced vibrations of an elastically mounted circular cylinder under vortex-induced vibrations (VIV). The dynamic response of three different taper ratios [ T R defined as T R = l / (d 2 − d 1) = 12 (highly tapered cylinder), 20 (medium tapered cylinder), and 40 (low tapered cylinder)], where d1 and d2 are the largest and smallest diameter of the tapered cylinder, respectively, with length l is studied at a fixed Reynolds number [ Re D , defined based on the averaged cylinder diameter, D = (d 1 + d 2) / 2 ] of 150. The amplitude and frequency response of the tapered cylinder is characterized by a low mass ratio (defined as the ratio of the total oscillating mass to the displaced fluid mass) of m* = 2 over the wide range of reduced velocity (3 ≤ U r ≤ 16) covering the full amplitude-response spectrum (based on the oscillation amplitude) of the VIV. The results show the existence of difference in the spanwise shedding of vortices owing to the poor spanwise pressure correlation. The flow field analysis in the wake of the oscillating cylinder reveals the dominance of the three-dimensional structures in the wake (near the top end with the larger diameter) behind the cylinder with the increase in the taper ratio (even at such low Re D where the uniform cylinder exhibits the two-dimensional wake). Also, the tapered cylinder exhibits a wide range of frequency synchronization (i.e., wide lock-in area) compared to the uniform cylinder. Tapering the cylinder results in the shift of the peak of the max oscillation amplitude or, in turn, the shift in the transitioning of the response branches. Further, force decomposition, energy transfer, and phase dynamics are also discussed for the taper cylinders. [ABSTRACT FROM AUTHOR]

Swarmalators are entities with the simultaneous presence of swarming and synchronization that reveal emergent collective behavior due to the fascinating bidirectional interplay between phase and spatial dynamics. Although different coupling topologies have already been considered, here we introduce time-varying competitive phase interaction among swarmalators where the underlying connectivity for attractive and repulsive coupling varies depending on the vision (sensing) radius. Apart from investigating some fundamental properties like conservation of center of position and collision avoidance, we also scrutinize the cases of extreme limits of vision radius. The concurrence of attractiveâ€"repulsive competitive phase coupling allows the exploration of diverse asymptotic states, like static Ď€, and mixed phase wave states, and we explore the feasible routes of those states through a detailed numerical analysis. In sole presence of attractive local coupling, we reveal the occurrence of static cluster synchronization where the number of clusters depends crucially on the initial distribution of positions and phases of each swarmalator. In addition, we analytically calculate the sufficient condition for the emergence of the static synchronization state. We further report the appearance of the static ring phase wave state and evaluate its radius theoretically. Finally, we validate our findings using Stuartâ€"Landau oscillators to describe the phase dynamics of swarmalators subject to attractive local coupling. [ABSTRACT FROM AUTHOR]

Abstract Modeling and interpreting (partial) synchronous neural activity can be a challenge. We illustrate this by deriving the phase dynamics of two seminal neural mass models: the Wilson-Cowan firing rate model and the voltage-based Freeman model. We established that the phase dynamics of these models differed qualitatively due to an attractive coupling in the first and a repulsive coupling in the latter. Using empirical structural connectivity matrices, we determined that the two dynamics cover the functional connectivity observed in resting state activity. We further searched for two pivotal dynamical features that have been reported in many experimental studies: (1) a partial phase synchrony with a possibility of a transition towards either a desynchronized or a (fully) synchronized state; (2) long-term autocorrelations indicative of a scale-free temporal dynamics of phase synchronization. Only the Freeman phase model exhibited scale-free behavior. Its repulsive coupling, however, let the individual phases disperse and did not allow for a transition into a synchronized state. The Wilson-Cowan phase model, by contrast, could switch into a (partially) synchronized state, but it did not generate long-term correlations although being located close to the onset of synchronization, i.e. in its critical regime. That is, the phase-reduced models can display one of the two dynamical features, but not both. Highlights • Networks of neural masses are reduced to phase oscillator models. • Freeman- and Wilson-Cowan-based phase models can resemble empirically observed functional connectivity. • Neither a network of Freeman nor of Wilson-Cowan phase models can. • Capture both partial synchronization and scale-free temporal correlations. [ABSTRACT FROM AUTHOR]

Summary: Control‐channel interactions in a linear diffusive network model are studied, with the aim of determining the role of the network's topology in such interactions. Specifically, the influence of a built controller on the infinite and finite zero structure of a second control channel is characterized. The analysis shows how the network's graph topology, the relative positions of the two channels, and the architecture of the built controller influence the zero structure of the second control channel. Further, the influence of low or high gain controller designs on remote channels is examined. The analyses demonstrate that some control architectures and positions can introduce undesirable nonminimum‐phase dynamics at remote locations, while others are guaranteed to maintain or promote minimum‐phase dynamics. [ABSTRACT FROM AUTHOR]

We experimentally investigate the forced synchronization of a self-excited chaotic thermoacoustic oscillator with two natural frequencies, f1 and f2. On increasing the forcing amplitude, xf, at a fixed forcing frequency, ff, we find two different types of synchronization: (i) ff/f1 = 1: 1 or 2: 1 chaos-destroying synchronization (CDS), and (ii) phase synchronization of chaos (PSC). En route to 1: 1 CDS, the system transitions from an unforced chaotic state (CH1,2) to a forced chaotic state (CH1,2,f), then to a two-frequency quasiperiodic state where chaos is destroyed (T²2,f), and finally to a phase-locked period-1 state (P1f). The route to 2: 1 CDS is similar, but the quasiperiodic state hosts a doubled torus (2T²2,f) that transforms into a phase-locked period-2 orbit (P2f) when CDS occurs. En route to PSC, the system transitions to a forced chaotic state (CH1,2,f) followed by a phase-locked chaotic state, where f1, f2 and ff still coexist but their phase difference remains bounded. We find that the maximum reduction in thermoacoustic amplitude occurs near the onset of CDS, and that the critical xf required for the onset of CDS does not vary significantly with ff. We then use two unidirectionally coupled Anishchenko-Astakhov oscillators to phenomenologically model the experimental synchronization dynamics, including (i) the route to 1: 1 CDS, (ii) various phase dynamics, such as phase drifting, slipping and locking, and (iii) the thermoacoustic amplitude variations in the ff/f1-xf plane. This study extends the applicability of open-loop control further to a chaotic thermoacoustic system, demonstrating (i) the feasibility of using an existing actuation strategy to weaken aperiodic thermoacoustic oscillations, and (ii) the possibility of developing new active suppression strategies based on both established and emerging methods of chaos control. [ABSTRACT FROM AUTHOR]

We study the collective behaviors of the fractional Kuramoto model (FKM) proposed in [16] with general interaction network topologies and present a new hybrid fractional Kuramoto (HFKM) model for synchronization. Here, our proposed HFKM model consists of continuous evolution processes and a sequence of discrete processes. The continuous dynamics between discrete times is governed by the fractional Kuramoto (FKM) model, whereas a sequence of discrete processes governs the reinitialization of data by deleting past memory of phase dynamics except for the current state and changing interaction network topologies. For the proposed models, we provide sufficient conditions for the emergence of practical synchronization for nonidentical oscillators and phase synchronization for identical oscillators, respectively. This improves the earlier work [16] on the slow relaxation of the FKM model. [ABSTRACT FROM AUTHOR]

We develop a numerical approach to reconstruct the phase dynamics of driven or coupled self-sustained oscillators. Employing a simple algorithm for computation of the phase of a perturbed system, we construct numerically the equation for the evolution of the phase. Our simulations demonstrate that the description of the dynamics solely by phase variables can be valid for rather strong coupling strengths and large deviations from the limit cycle. Coupling functions depend crucially on the coupling and are generally non-decomposable in phase response and forcing terms. We also discuss the limitations of the approach. [ABSTRACT FROM AUTHOR]

We develop an approach for a fast experimental inference of synchronization properties of an oscillator. While the standard technique for determination of synchronization domains implies that the oscillator under study is forced with many different frequencies and amplitudes, our approach requires only several observations of a driven system. Reconstructing the phase dynamics from data, we successfully determine synchronization domains of noisy and chaotic oscillators. Our technique is especially important for experiments with living systems where an external action can be harmful and shall be minimized. [ABSTRACT FROM AUTHOR]

We propose a general method for optimizing periodic input waveforms for global entrainment of weakly forced limit-cycle oscillators based on phase reduction and nonlinear programming. We derive averaged phase dynamics from the mathematical model of a limit-cycle oscillator driven by a weak periodic input and optimize the Fourier coefficients of the input waveform to maximize prescribed objective functions. In contrast to the optimization methods that rely on the calculus of variations, the proposed method can be applied to a wider class of optimization problems including global entrainment objectives. As an illustration, we consider two optimization problems, one for achieving fast global convergence of the oscillator to the entrained state and the other for realizing prescribed global phase distributions in a population of identical uncoupled noisy oscillators. We show that the proposed method can successfully yield optimal input waveforms to realize the desired states in both cases. [ABSTRACT FROM AUTHOR]

The role of the phase dynamics in turbulence is investigated. As a demonstration of the importance of the phase dynamics, a simplified system is used, namely the one-dimensional Burgers equation, which is evolved numerically. The system is forced via a known external force, with two components that are added into the evolution equations of the amplitudes and the phase of the Fourier modes, separately. In this way, we are able to control the impact of the force on the dynamics of the phases. In the absence of the direct forcing in the phase equation, it is observed that the phases are not stochastic as assumed in the Random Phase Approximation (RPA) models, and in contrast, the non-linear couplings result in intermittent locking of the phases to ± π/2. The impact of the force, applied purely on the phases, is to increase the occurrence of the phase locking events in which the phases of the modes in a wide k range are now locked to ± π/2, leading to a change in the dynamics of both phases and amplitudes, with a significant localization of the real space flow structures. [ABSTRACT FROM AUTHOR]

We studied the properties of low-frequency (LF) heart rate variability (HRV) and photoplethysmographic waveform variability (PPGV) and their interaction under conditions where the hemodynamic connection between them is obviously absent, as well as the LF regulation of PPGV in the absence of heart function. The parameters of HRV and finger PPGV were evaluated in 10 patients during cardiac surgery under cardiopulmonary bypass (on-pump cardiac surgery) with or without cardioplegia. The following spectral indices of PPGV and HRV were ertimated: the total spectral power (TP), the high-frequency (HF) and the LF ranges of TP in percents (HF% and LF%), and the LF/HF ratio. We assessed also the index S of synchronization between the LF oscillations in finger photoplethysmogram (PPG) and heart rate (HR) signals. The analysis of directional couplings was carried out using the methods of phase dynamics modeling. It is shown that the mechanisms leading to the occurrence of oscillations in the LF range of PPGV are independent of the mechanisms causing oscillations in the LF range of HRV. At the same time, the both above-mentioned LF oscillations retain their activity under conditions of artificial blood circulation and cardioplegia (the latter case applies only to LF oscillations in PPG). In artificial blood circulation, there was a coupling from the LF oscillations in PPG to those in HR, whereas the coupling in the opposite direction was absent. The coupling from the LF oscillations in PPG to those in HR has probably a neurogenic nature, whereas the opposite coupling has a hemodynamic nature (due to cardiac output). [ABSTRACT FROM AUTHOR]

We consider the dynamics of a lattice of phase oscillators with a nearest-neighbor coupling. The clustering hierarchy is described for the case of linear distribution of natural frequencies. We demonstrate that for small couplings prior to the appearance of the first cluster the dynamics is quasi-Hamiltonian: the phase volume is conserved in average, and the spectrum of the Lyapunov exponents is symmetric. We explain this unexpected for a dissipative system phenomenon using the concept of reversibility. We show that for a certain coupling a smooth transition from the quasi-Hamiltonian to the dissipative dynamics occurs, which is a novel type of chaos–chaos transition. [Copyright &y& Elsevier]

Synchronization is common in both nature and human physiology, often used to illustrate nonlinear dynamics. Interestingly, one can initiate their comprehension of this phenomenon from pure linear systems. In this Tutorial, we begin with a theoretical exploration of coupled oscillators' dynamic behavior, enabling us to discern and contrast the unique attributes of dissipative coupling as opposed to commonly observed coherent coupling. We then examine synchronization in two dissipative coupled linear systems: one with two pendulums mutually linked via the Lenz effect and the other with two RLC oscillators coupled via a resistor. This Tutorial is designed to serve as a concise starting point for researchers interested in exploring synchronization phenomena using a simplified model driven solely by dissipative coupling. [ABSTRACT FROM AUTHOR]

Macroscopic oscillations in the brain are involved in various cognitive and physiological processes, yet their precise function is not completely understood. Communication through coherence (CTC) theory proposes that these rhythmic electrical patterns might serve to regulate the information flow between neural populations. Thus, to communicate effectively, neural populations must synchronize their oscillatory activity, ensuring that input volleys from the presynaptic population reach the postsynaptic one at its maximum phase of excitability. We consider an Excitatory–Inhibitory (E–I) network whose macroscopic activity is described by an exact mean-field model. The E–I network receives periodic inputs from either one or two external sources, for which effective communication will not be achieved in the absence of control. We explore strategies based on optimal control theory for phase–amplitude dynamics to design a periodic control that sets the target population in the optimal phase to synchronize its activity with a specific presynaptic input signal and establish communication. The control mechanism resembles the role of a higher cortical area in the context of selective attention. To design the control, we use the phase–amplitude reduction of a limit cycle and leverage recent developments in this field in order to find the most effective control strategy regarding a defined cost function. Furthermore, we present results that guarantee the local controllability of the system close to the limit cycle. • Optimal-control strategies provide controls that suitably adjust the dynamics for enhanced communication. • Controlled synchronization is decisive for effective communication in mean-field models of E–I networks. • Using phase–amplitude description, optimal periodic controls minimally alter the oscillation, while enhance communication. [ABSTRACT FROM AUTHOR]

We consider the effect of an external periodic force on chimera states in the phase oscillator model proposed in [Phys. Rev. Lett, v. 101, 00319007 (2008)]. Using the Ott – Antonsen reduction, the dynamical equations for the global order parameter characterizing the degree of synchronization are constructed. The frequency locking by an external periodic force region is constructed. The possibility of stable chimeras synchronization and unstable chimeras stabilization is established. The instability development of the chimera states leads to the appearance of breather chimeras or complete synchronization. [ABSTRACT FROM AUTHOR]

Swarmalators are oscillators that swarm through space as they synchronize in time. Introduced a few years ago to model many systems that mix synchrony with self-assembly, they remain poorly understood theoretically. Here, we obtain the first analytic results on swarmalators moving in two spatial dimensions by introducing a simplified model where the swarmalators have no hard-shell interaction terms and move on a periodic plane. These simplifications allow expressions for order parameters, stabilities and bifurcations to be derived exactly. [ABSTRACT FROM AUTHOR]

The synchronization of two groups of electrochemical oscillators is investigated during the electrodissolution of nickel in sulfuric acid. The oscillations are coupled through combined capacitance and resistance, so that in a single pair of oscillators (nearly) in-phase synchronization is obtained. The internal coupling within each group is relatively strong, but there is a phase difference between the fast and slow oscillators. The external coupling between the two groups is weak. The experiments show that the two groups can exhibit (nearly) anti-phase collective synchronization. Such synchronization occurs only when the external coupling is weak, and the interactions are delayed by the capacitance. When the external coupling is restricted to those between the fast and the slow elements, the anti-phase synchronization is more prominent. The results are interpreted with phase models. The theory predicts that, for antiphase collective synchronization, there must be a minimum internal phase difference for a given shift in the phase coupling function. This condition is less stringent with external fast-to-slow coupling. The results provide a framework for applications of collective phase synchronization in modular networks where weak coupling between the groups can induce synchronization without rearrangements of the phase dynamics within the groups. [ABSTRACT FROM AUTHOR]

Investigating the dynamics of a network of oscillatory systems is a timely and urgent topic. Phase synchronization has proven paradigmatic to study emergent collective behavior within a network. Defining the phase dynamics, however, is not a trivial task. The literature provides an arsenal of solutions, but results are scattered and their formulation is far from standardized. Here, we present, in a unified language, a catalogue of popular techniques for deriving the phase dynamics of coupled oscillators. Traditionally, approaches to phase reduction address the (weakly) perturbed dynamics of an oscillator. They fall into three classes. (i) Many phase reduction techniques start off with a Hopf normal form description, thereby providing mathematical rigor. There, the caveat is to first derive the proper normal form. We explicate several ways to do that, both analytically and (semi-)numerically. (ii) Other analytic techniques capitalize on time scale separation and/or averaging over cyclic variables. While appealing for their more intuitive implementation, they often lack accuracy. (iii) Direct numerical approaches help to identify oscillatory behavior but may limit an overarching view how the reduced phase dynamics depends on model parameters. After illustrating and reviewing the necessary mathematical details for single oscillators, we turn to networks of coupled oscillators as the central issue of this report. We show in detail how the concepts of phase reduction for single oscillators can be extended and applied to oscillator networks. Again, we distinguish between numerical and analytic phase reduction techniques. As the latter dwell on a network normal form, we also discuss associated reduction methods. To illustrate benefits and pitfalls of the different phase reduction techniques, we apply them point-by-point to two classic examples: networks of Brusselators and a more elaborate model of coupled Wilson–Cowan oscillators. The reduction of complex oscillatory systems is crucial for numerical analyses but more so for analytical estimates and model prediction. The most common reduction is towards phase oscillator networks that have proven successful in describing not only the transition between incoherence and global synchronization, but also in predicting the existence of less trivial network states. Many of these predictions have been confirmed in experiments. As we show, however, the phase dynamics depends to large extent on the employed phase reduction technique. In view of current and future trends, we also provide an overview of various methods for augmented phase reduction as well as for phase–amplitude reduction. Weindicate how these techniques can be extended to oscillator networks and, hence, may allow for an improved derivation of the phase dynamics of coupled oscillators. [ABSTRACT FROM AUTHOR]

The discrepancy between structural and functional connectivity in neural systems forms the challenge in understanding general brain functioning. To pinpoint a mapping between structure and function, we investigated the effects of (in)homogeneity in coupling structure and delays on synchronization behavior in networks of oscillatory neural masses by deriving the phase dynamics of these generic networks. For homogeneous delays, the structural coupling matrix is largely preserved in the coupling between phases, resulting in clustered stationary phase distributions. Accordingly, we found only a small number of synchronized groups in the network. Distributed delays, by contrast, introduce inhomogeneity in the phase coupling so that clustered stationary phase distributions no longer exist. The effect of distributed delays mimicked that of structural inhomogeneity. Hence, we argue that phase (de-)synchronization patterns caused by inhomogeneous coupling cannot be distinguished from those caused by distributed delays, at least not by the naked eye. The here-derived analytical expression for the effective coupling between phases as a function of structural coupling constitutes a direct relationship between structural and functional connectivity. Structural connectivity constrains synchronizability that may be modified by the delay distribution. This explains why structural and functional connectivity bear much resemblance albeit not a one-to-one correspondence. We illustrate this in the context of resting-state activity, using the anatomical connectivity structure reported by Hagmann and others. [ABSTRACT FROM AUTHOR]

Synchronization properties of model systems containing large numbers of phase oscillators have many potential biophysical and other applications. Biophysical examples include networks of pacemaker cells in the heart and suprachiasmatic nucleus of the brain. In physical systems, the phase dynamics of high-Tc superconducting materials continue to attract attention, since systems of intrinsic Josephson junctions form natural arrays of coupled phase oscillators. As such, they have the potential to act as systems in which various exotic synchronization effects may be observed. In this paper, we make a more detailed exploration of the parameter space containing regions of spontaneous chaos synchronization in the CCJJ + DC model of intrinsic Josephson junctions. Extensive regions of phase synchronization-corresponding to the Shapiro steps with zero charge density in the S-layers-are found through numerical simulation. By computing the Lyapunov exponent spectra, we see that the spontaneous chaos synchronization occurs within certain subregions that overlap with uncharged steps in the IV-characteristics. We tried to stabilize the spontaneous chaos synchronization over a wider range of parameters, by applying a global, noninvasive, and proportional control scheme. The control is affected by applying a time-dependent phase shift to the external electromagnetic radiation, proportional to the total voltage. The effect of the control is found to be three-fold. First, it tends to broaden the current interval over which lower harmonic Shapiro steps occur. Second, it does not change the width in the current range over which the chaos synchronization occurs. Third, it makes the chaos synchronization more robust to thermal noise. The chaos synchronization we report here may be useful in any applications requiring more powerful, high-frequency, and chaotic signals, such as in secure communication. [ABSTRACT FROM AUTHOR]

Background: This study utilized recent advancements in electroencephalography (EEG) technology that enable the measurement of prefrontal event-related potentials (ERPs) to facilitate the early detection of mild cognitive impairment (MCI). We investigated two-channel prefrontal ERP signals obtained from a large cohort of elderly participants and compare among cognitively normal (CN), subjective cognitive decline (SCD), amnestic MCI (aMCI), and nonamnestic MCI (naMCI) groups. Methods: Signal processing and ERP component analyses, specifically adapted for two-channel prefrontal ERP signals evoked by the auditory oddball task, were performed on a total of 1,754 elderly participants. Connectivity analyses were conducted to assess brain synchronization, especially in the beta band involving the phase locking value (PLV) and coherence (COH). Time-frequency, time-trial, grand average, and further statistical analyses of the standard and target epochs were also conducted to explore differences among the cognition groups. Results: The MCI group’s response to target stimuli was characterized by greater response time variability (p < 0.001) and greater variability in the P300 latency (p < 0.05), leading to less consistent responses than those of the healthy control (HC) group (CN+SCD subgroups). In the connectivity analyses of PLV and COH waveforms, significant differences were observed, indicating a loss of synchronization in the beta band in response to standard stimuli in the MCI group. In addition, the absence of event-related desynchronization (ERD) indicated that information processing related to readiness and task performance in the beta band was not efficient in the MCI group. Furthermore, the observed decline in the P200 amplitude as the standard trials progressed suggests the impaired attention and inhibitory processes in the MCI group compared to the HC group. The aMCI subgroup showed high variability in COH values, while the naMCI subgroup showed impairments in their overall behavioral performance. Conclusion: These findings highlight the variability and connectivity measures can be used as markers of early cognitive decline; such measures can be assessed with simple and fast two-channel prefrontal ERP signals evoked by both standard and target stimuli. Our study provides deeper insight of cognitive impairment and the potential use of the prefrontal ERP connectivity measures to assess early cognitive decline. [ABSTRACT FROM AUTHOR]

The brain is a complex non-equilibrium system capable of expressing many different dynamics as well as the transitions between them. We hypothesized that the level of non-equilibrium can serve as a signature of a given brain state, which was quantified using the arrow of time (the level of irreversibility). Using this thermodynamic framework, the irreversibility of emergent cortical activity was quantified from local field potential recordings in male Lister-hooded rats at different anesthesia levels and during the sleep-wake cycle. This measure was carried out on five distinct brain states: slow-wave sleep, awake, deep anesthesia–slow waves, light anesthesia–slow waves, and microarousals. Low levels of irreversibility were associated with synchronous activity found both in deep anesthesia and slow-wave sleep states, suggesting that slow waves were the state closest to the thermodynamic equilibrium (maximum symmetry), thus requiring minimum energy. Higher levels of irreversibility were found when brain dynamics became more asynchronous, for example, in wakefulness. These changes were also reflected in the hierarchy of cortical dynamics across different cortical areas. The neural dynamics associated with different brain states were characterized by different degrees of irreversibility and hierarchy, also acting as markers of brain state transitions. This could open new routes to monitoring, controlling, and even changing brain states in health and disease. [ABSTRACT FROM AUTHOR]

Phase synchronization of parametric excited Rössler system has been investigated in this paper. By introducing the conception of phase for a chaotic motion, it has been demonstrated that the mean frequency of chaotic attractor and the frequency of the parametric excitation may be locked in different ratios for certain parameter conditions, implying phase synchronization can be observed. The evolution from nonsynchronized state to phase synchronization has been discussed in detail, which reveals different phase dynamics may exist during the process. With the variation of parameters, the imaging point on the Poincaré plane may finally settle down onto the attractor, which yields phase synchronization. [ABSTRACT FROM AUTHOR]

We present results on experimental and theoretical studies of chaos in a solid-state ring laser with periodic pump modulation. We show that the synchronized chaos in the counter-propagating waves is observed for the values of pump modulation frequency f[sub p] satisfying the inequality f[sub 1]

In this work, we consider a ring of coupled electronic (Wien-bridge) oscillators from a perspective combining modeling, simulation, and experimental observation. Following up on earlier work characterizing the pairwise interaction of Wien-bridge oscillators by Kuramoto–Sakaguchi phase dynamics, we develop a lattice model for a chain thereof, featuring an exponentially decaying spatial kernel. We find that for certain values of the Sakaguchi parameter a, states of traveling phasedomain fronts involving the coexistence of two clearly separated regions of distinct dynamical behavior, can establish themselves in the ring lattice. Experiments and simulations show that stationary coexistence domains of synchronization only manifest themselves with the introduction of a local impurity; here an incoherent cluster of oscillators can arise reminiscent of the chimera states in a range of systems with homogeneous oscillators and suitable nonlocal interactions between them. [ABSTRACT FROM AUTHOR]

A common observation is that large groups of oscillatory biological units often have the ability to synchronize. A paradigmatic model of such behavior is provided by the Kuramoto model, which achieves synchronization through coupling of the phase dynamics of individual oscillators, while each oscillator maintains a different constant inherent natural frequency. Here we consider the biologically likely possibility that the oscillatory units may be capable of enhancing their synchronization ability by adaptive frequency dynamics. We propose a simple augmentation of the Kuramoto model which does this. We also show that, by the use of a previously developed technique [Ott and Antonsen, Chaos 18, 037113 (2008)], it is possible to reduce the resulting dynamics to a lower dimensional system for the macroscopic evolution of the oscillator ensemble. By employing this reduction, we investigate the dynamics of our system, finding a characteristic hysteretic behavior and enhancement of the quality of the achieved synchronization. [ABSTRACT FROM AUTHOR]

The synchronization phenomenon in nature has been utilized in sensing and timekeeping fields due to its numerous advantages, including amplitude and frequency stabilization, noise reduction, and sensitivity improvement. However, the limited synchronization bandwidth hinders its broader application, and few techniques have been explored to enhance this aspect. In this paper, we conducted theoretical and experimental studies on the unidirectional synchronization characteristics of a resonator with phase lock loop oscillation. A novel enhancement method for the synchronization bandwidth using a parametrically excited MEMS oscillator is proposed, which achieves a remarkably large synchronization bandwidth of 8.85 kHz, covering more than 94% of the hysteresis interval. Importantly, the proposed method exhibits significant potential for high-order synchronization and frequency stabilization compared to the conventional directly excited oscillator. These findings present an effective approach for expanding the synchronization bandwidth, which has promising applications in nonlinear sensing, fully mechanical frequency dividers, and high-precision time references. [ABSTRACT FROM AUTHOR]