In-phase Synchronization Research Articles

Weakly Nonlinear Theory for Oscillatory Dynamics in a One-Dimensional PDE-ODE Model of Membrane Dynamics Coupled by a Bulk Diffusion Field

We study the dynamics of systems consisting of two spatially segregated ODE compartments coupled through a one-dimensional bulk diffusion field. For this coupled PDE-ODE system, we first employ a multi-scale asymptotic expansion to derive amplitude equations near codimension-one Hopf bifurcation points for both in-phase and anti-phase synchronization modes. The resulting normal form equations pertain to any vector nonlinearity restricted to the ODE compartments. In our first example, we apply our weakly nonlinear theory to a coupled PDE-ODE system with Sel'kov membrane kinetics, and show that the symmetric steady state undergoes supercritical Hopf bifurcations as the coupling strength and the diffusivity vary. We then consider the PDE diffusive coupling of two Lorenz oscillators. It is shown that this coupling mechanism can have a stabilizing effect, characterized by a significant increase in the Rayleigh number required for a Hopf bifurcation. Within the chaotic regime, we can distinguish between synchronous chaos, where both the left and right oscillators are in-phase, and chaotic states characterized by the absence of synchrony. Finally, we compute the largest Lyapunov exponent associated with a linearization around the synchronous manifold that only considers odd perturbations. This allows us to predict the transition to synchronous chaos as the coupling strength and the diffusivity increase.

On the Fluid Dynamical Effects of Synchronization in Side-by-Side Swimmers.

In-phase and anti-phase synchronization of neighboring swimmers is examined experimentally using two self-propelled independent flexible foils swimming side-by-side in a water tank. The foils are actuated by pitching oscillations at one extremity—the head of the swimmers—and the flow engendered by their undulations is analyzed using two-dimensional particle image velocimetry in their frontal symmetry plane. Following recent observations on the behavior of real fish, we focus on the comparison between in-phase and anti-phase actuation by fixing all other geometric and kinematic parameters. We show that swimming with a neighbor is beneficial for both synchronizations tested, as compared to swimming alone, with an advantage for the anti-phase synchronization. We show that the advantage of anti-phase synchronization in terms of swimming performance for the two-foil “school” results from the emergence of a periodic coherent jet between the two swimmers.

Phase-Inversion Waves Propagating in an In-Phase Synchronization on Oscillators Coupled as a Cross

A large number of synchronization phenomena are observed in this world, and investigations and analyses of these phenomena are important in various fields. In this study, phase-inversion waves, which continuously propagate the phase-state between adjacent oscillators on a cross system in which van der Pol oscillators are coupled by inductors, are observed using our simulators and actual circuits. We analyze the phase-inversion waves by using the electric power, phase differences between adjacent oscillators, theoretical frequencies, and instantaneous frequencies, and clarify the penetration mechanism of phase-inversion waves at a cross point by using the phase differences as well as the theoretical and simulated instantaneous frequencies.

Anti-phase collective synchronization with intrinsic in-phase coupling of two groups of electrochemical oscillators.

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 anti-phase 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. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Bifurcations of Time-Delay-Induced Multiple Transitions Between In-Phase and Anti-Phase Synchronizations in Neurons with Excitatory or Inhibitory Synapses

Time-delay-induced synchronous behaviors and synchronization transitions have been widely investigated for coupled neurons, and they play important roles for physiological functions. In the present study, time-delay-induced synchronized subthreshold oscillations were simulated, and the bifurcations underlying the synchronized behaviors were identified for a pair of coupled FitzHugh–Nagumo neurons. Multiple transitions between in-phase and anti-phase synchronizations induced by the time delay were simulated for the inhibitory and excitatory couplings. Subcritical or supercritical Hopf bifurcations and the stability of the Hopf-bifurcating periodic subthreshold oscillations were acquired using center manifold reduction and normal form theory. The in-phase or anti-phase synchronizations of the stable periodic subthreshold oscillations, which appear for multiple values of the time delay, were interpreted with the related eigenspace. The distributions of the different dynamical behaviors, including the synchronizations and bifurcations in the two-parameter plane of the time delay and coupling strength, were acquired for both types of synapses, and the different roles of the inhibitory and excitatory couplings on the synchronization transitions were compared.

Erratum: In-phase synchronization in complex oscillator networks by adaptive delayed feedback control [Phys. Rev. E 98 , 042302 (2018)

Erratum: In-phase synchronization in complex oscillator networks by adaptive delayed feedback control [Phys. Rev. E <b>98</b> , 042302 (2018)

Synchronization of dual diffusion flame in co-flow

In this study, the interaction of two adjacent diffusion flames (dual flame) were experimentally investigated using the direct imaging method and particle image velocimetry (PIV) measurements for varied distances between the two flames under the influence of the co-flow. The direct imaging method indicates the presence of various synchronization modes, such as merging, in-phase synchronization, amplitude death, anti-phase synchronization, and desynchronization, as the distance between the flames was increased. The dual flame synchronization was characterized using an image analysis of the time-series variation of flame heights summarized by the mean flame height, root-mean-square of the flame height, Strouhal number, and the cross-correlation coefficient of flame heights against the burner distance. Furthermore, the unsteady velocity fields of the synchronized flames were measured using PIV combined with the proper orthogonal decomposition (POD) analysis to extract the flow structure of the dual flame. The experimental results indicated that the in-phase mode is characterized by the formation of symmetrical vortices, whereas the anti-phase mode features the formation of asymmetrical vortices inside and outside of the flame. The POD analysis demonstrated higher fluctuating energy in the anti-phase mode than in the in-phase mode, which suggested the formation of a highly organized flame structure in the anti-phase mode.

Synchronization in ensembles of delay-coupled nonidentical neuronlike oscillators

We study both numerically and experimentally the synchronization in an ensemble of nonidentical neuronlike oscillators described by the FitzHugh–Nagumo equations. The cases of constant values of time-delayed couplings between the oscillators and adaptively controlled values of time-delayed couplings are considered. For the experimental study of the ensemble of neuronlike oscillators, we construct a radio engineering setup, in which the ability to specify both constant values and adaptively tuned values of couplings between the oscillators is implemented. Moreover, it is possible to specify an arbitrary architecture and type of dynamical couplings between oscillators in the setup. By the example of a system of two bidirectionally coupled nonidentical oscillators and a ring consisting of ten unidirectionally coupled nonidentical FitzHugh–Nagumo systems, it is shown that the using of an adaptively controlled time-delayed coupling allows one to achieve the in-phase synchronization of all oscillators in the ensemble even in the case of a large parameter mismatch. The results obtained in the physical experiment are in good agreement with the results of the numerical simulation.

Plasticity facilitates pattern selection of networks of chemical oscillations.

Rotating wave synchronization patterns are explored with a ring of 20 electrochemical oscillators during nickel electrodissolution in sulfuric acid. With desynchronized initial states, coupling alone yields predominance of nonrotating solutions, i.e., in-phase synchronization. An experimental technique is presented in which, through a combination of temporary alterations in topology, the application of global feedback provides rotational solutions. With phase repulsive global feedback, the in-phase synchronization is destabilized and a rotating wave is obtained. This feedback induced rotating wave can be employed to establish an initial condition for the rotating wave with coupling only. Higher order rotating solutions with 2, 3, and 4 waves corotating around the ring are observed, where the initial conditions are generated by temporary network rewiring to a structure with 2, 3, and 4 loops, respectively, and by global feedback. The experimental observations are supported by numerical simulations with a phase model. The results indicate that while network plasticity is thought to be significant in the operation of neural systems, it can also play a role in pattern selection of chemical systems.

Synchronization and amplitude death in a pair of Van der Pol oscillators under conjugate coupling

This work presents an analysis of synchronization and amplitude death (oscillation decay) phenomena in a pair of coupled Van der Pol oscillators under two types of conjugate couplings. The coupling is through diffusive flow of the conjugate or dissimilar variables and referred to as type 1 and type 2 coupling respectively. The analytical conditions for in-phase and out-of-phase synchronizations and amplitude death are determined. It is found that the amplitude death occurs only with type 2 coupling over a region of coupling strength. In this region, with decaying amplitude the oscillators approach a stationary state of negligible amplitude in anti-phase fashion. Beyond this range, on lower side the stable in-phase synchronization occurs if the frequencies of both the oscillators are equal, and on higher side uncontrolled amplitude growth occur. The type 1 coupling produces stable in-phase synchronization for positive coupling strength values and stable anti-phase synchronization for negative values under the condition that the coupling strength must be greater than or equal to the frequency detuning. In coupled systems the synchronization conditions are achieved maintaining the linear phase characteristics similar to the uncoupled oscillators. This study may be of interest in modeling the behavior of coupled biological and electrical oscillators.

In-phase synchronization between two auto-oscillating bubbles

Acoustically driven bubbles are nonlinear oscillators showing a wide range of behaviors such as period-doubling bifurcations, deterministic chaos, and synchronization to an external signal. Here we demonstrate that bubbles driven with a constant heat source can couple with each other and yield in-phase synchronization, even in the absence of an external sound field. Besides perfect synchronization, we resolve the parameter space where independent oscillations and bubble merging occur. The main finding that only weakly driven oscillating bubbles synchronize provides a path for experimental scale-up to study spatiotemporal synchronization. In the wider scope the findings are relevant for heat transfer applications from structured heaters where complex multibubble oscillations are expected.

Synchronization of spin-torque oscillators via spin pumping

A theoretical treatise is presented on coupled dynamics between spin torque oscillators consisting of perpendicularly magnetized free layers and in-plane magnetized pinned layers. As a coupling mechanism, an injection of pure spin current from one oscillator to the other is considered because the free layer is capable of emitting the pure spin current by spin pumping effect. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates that an in-phase synchronization is excited in the oscillators for a wide range of applied current. The results indicate that injection of the spin current by spin pumping is the efficient approach for an enhancement of emission power in spintronics devices using an array of spin torque oscillators.

Distinct online and offline effects of alpha and beta transcranial alternating current stimulation (tACS) on continuous bimanual performance and task-set switching

In the present study we examined the effect of bihemispheric in-phase synchronization of motor cortical rhythms on complex bimanual coordination. Twenty young healthy volunteers received 10 Hz or 20 Hz tACS in a double-blind crossover design while performing a bimanual task-set switching paradigm. We used a bilateral high-density montage centred over the hand knob representation within the primary motor cortices to apply tACS time-locked to the switching events. Online tACS in either frequency led to faster but more erroneous switching transitions compared to trials without active stimulation. When comparing stimulation frequencies, 10 Hz stimulation resulted in higher error rates and slower switching transitions than 20 Hz stimulation. Furthermore, the stimulation frequencies showed distinct carry-over effects in trials following stimulation trains. Non-stimulated switching transitions were generally faster but continuous performance became more erroneous over time in the 20 Hz condition. We suggest that the behavioural effects of bifocal in-phase tACS are explained by online synchronization of long-range interhemispheric sensorimotor oscillations, which impacts on interhemispheric information flow and the top-down control required for flexible control of complex bimanual actions. Different stimulation frequencies may lead to distinct offline effects, which potentially accumulate over time and therefore need to be taken into account when evaluating subsequent performance.

Weak multiplexing in neural networks: Switching between chimera and solitary states.

We investigate spatio-temporal patterns occurring in a two-layer multiplex network of oscillatory FitzHugh-Nagumo neurons, where each layer is represented by a nonlocally coupled ring. We show that weak multiplexing, i.e., when the coupling between the layers is smaller than that within the layers, can have a significant impact on the dynamics of the neural network. We develop control strategies based on weak multiplexing and demonstrate how the desired state in one layer can be achieved without manipulating its parameters, but only by adjusting the other layer. We find that for coupling range mismatch, weak multiplexing leads to the appearance of chimera states with different shapes of the mean velocity profile for parameter ranges where they do not exist in isolation. Moreover, we show that introducing a coupling strength mismatch between the layers can suppress chimera states with one incoherent domain (one-headed chimeras) and induce various other regimes such as in-phase synchronization or two-headed chimeras. Interestingly, small intra-layer coupling strength mismatch allows to achieve solitary states throughout the whole network.

Coupled Dynamics of Magnetizations in Spin-Hall Oscillators via Spin-Current Injection

An array of spin torque oscillators (STOs) for practical applications such as pattern recognition was recently proposed, where several STOs are connected by a common nonmagnet. In this structure, in addition to the electric and/or magnetic interactions proposed in previous works, the STOs are spontaneously coupled to each other through the nonmagnetic connector, due to the injection of spin current. Solving the Landau-Lifshitz-Gilbert equation numerically for such system consisting of three STOs driven by the spin Hall effect, it is found that both in-phase and antiphase synchronization of the STOs can be achieved by adjusting the current density and appropriate distance between the oscillators.

Bursting and Synchronization of Coupled Neurons under Electromagnetic Radiation

Bursting is an important firing activity of neurons, which is caused by a slow process that modulates fast spiking activity. Based on the original second‐order Morris‐Lecar neuron model, an improved third‐order Morris‐Lecar neuron model can produce bursting activity is proposed, in which the effect of electromagnetic radiation is considered as a slow process and the original equation of Morris‐Lecar neuron model as a fast process. Extensive numerical simulation results show that the improved neuron model can produce different types of bursting, and bursting activity shows a deep dependence on system parameters and electromagnetic radiation parameters. In addition, synchronization transitions of identical as well as no‐identical coupled third‐order Morris‐Lecar neurons are studied, the results show that identical coupled neurons experience a complex synchronization process and reach complete synchronization finally with the increase of coupling intensity. For no‐identical coupled neurons, only anti‐phase synchronization and in‐phase synchronization can be reached. The studies of bursting activity of single neuron and synchronization transition of coupled neurons have important guiding significance for further understanding the information processing of neurons and collective behaviors in neuronal network under electromagnetic radiation environment.

Angular Velocity and Phase Shift Control of Mechatronic Vibrational Setup

The paper is devoted to experimental study of possibilities for consensus control of the unbalanced rotors for the machine used in the vibration technology. This problem is treated as an ensuring the prescribed phase shift between the vibrating rotor actuators and has a significant value for the various kinds of the vibration technology, such as vibrating transportation, drilling, and so on. It can not be considered without taking into account the natural tendency of in-phase or anti-phase synchronization of the mechanically coupled oscillating units (cf. Huygens clock synchronization). This phenomenon prevents the controller efforts in maintaining the desired value of the phase shift. In the paper, the control laws for frequency stabilization simultaneously with cooperative control for assuring the desired phase shift between the rotors angular positions are derived and experimentally studied by means of the mechatronic vibration testbed. It is obtained that for the low and medium frequencies the self-synchronization of unbalanced rotors does not prevent achieving the desired phase shift between the rotors. For a high frequency band, the Huygens self-synchronization of rotors manifests itself, narrowing the range of the achievable phase shift.

Role of buoyancy-driven vortices in inducing different modes of coupled behaviour in candle-flame oscillators

We investigate the coupled behaviour of two oscillatory flames produced by separate bundles of candles, referred to as candle-flame oscillators, as the distance between them is varied. Each bundle consists of four candles whose individual flames are fused so that the resultant flame produces self-sustained limit cycle oscillations. The recent study by Manoj et al. [Scientific Reports 8, 11626 (2018)] has reported the occurrence of four different modes of coupled behaviour, which include in-phase synchronization, amplitude death, anti-phase synchronization, and desynchronization by observing the flame dynamics of such coupled candle-flame oscillators. Here, we investigate the physical mechanism behind the occurrence of these different dynamical modes. Towards this purpose, we perform simultaneous measurements of the flow field around the candle flames using high-speed shadowgraph and of the reaction zone of each flame using high-speed CH* chemiluminescence imaging. We notice that these modes are distinguished by the distinct features of the flame dynamics and the corresponding buoyancy-induced flows surrounding the flames. We observe that the difference in the interaction of vortices, formed due to the instability of buoyancy-induced flows around each flame at various distances, plays a significant role in inducing different modes of coupled dynamics between the oscillators. Furthermore, we find that the change in the length scales of vortices shed around the flames is a contributing factor in increasing the frequency of the oscillators during the transition from in-phase to anti-phase mode of synchronization.

Ordinal synchronization: Using ordinal patterns to capture interdependencies between time series

We introduce Ordinal Synchronization (OS) as a new measure to quantify synchronization between dynamical systems. OS is calculated from the extraction of the ordinal patterns related to two time series, their transformation into D-dimensional ordinal vectors and the adequate quantification of their alignment. OS provides a fast and robust-to noise tool to assess synchronization without any implicit assumption about the distribution of data sets nor their dynamical properties, capturing in-phase and anti-phase synchronization. Furthermore, varying the length of the ordinal vectors required to compute OS it is possible to detect synchronization at different time scales. We test the performance of OS with data sets coming from unidirectionally coupled electronic Lorenz oscillators and brain imaging datasets obtained from magnetoencephalographic recordings, comparing the performance of OS with other classical metrics that quantify synchronization between dynamical systems.

Synchronization of asymmetrically coupled systems

In this paper, the dynamics of two mechanical oscillators interacting through an asymmetric coupling is considered. In the uncontrolled system, the asymmetry in the coupling prevents the oscillators from reaching synchronization. Therefore, a state feedback controller is designed and applied to the coupling system in order to remove the asymmetry. The resulting closed-loop dynamics is analyzed by using the Poincare method, and analytic conditions for the existence of stable synchronous solutions are derived. It is demonstrated that two synchronous solutions coexist in the closed-loop system, namely in-phase and anti-phase synchronization. Additionally, analytic expressions for the amplitude, frequency, and phase of these solutions—which are not determined by a reference signal, but rather by the intrinsic properties of the coupled systems—are provided and a characteristic equation for determining the stability of the synchronous solution is given. Finally, the obtained theoretical results are illustrated by numerical simulations, including a numerical study on the robustness of the synchronous solution.