Complete Phase Synchronization Research Articles

Synchronization of Kuramoto–Sakaguchi model with the distributed time-delayed interactions

The synchronous behavior of Kuramoto–Sakaguchi models with distributed time-delayed interactions is investigated. We prove complete and partial phase synchronization for both the distributed semi time-delay model and the full time-delay model. Additionally, we present several numerical experiments supporting our main results and illustrating interesting patterns.

Multi-stable states and synchronicity of a cellular neural network with memristive activation function

The cellular neural network (CNN) is an implementable solution for fully connected neural networks. Using nanoscale memristor to realize its nonlinear activation function can simplify the circuit implementation of CNN effectively. This paper presents a paradigm of the basic CNN cell by introducing a voltage-controlled memristor as the activating module of its output circuit. A three-cell memristor-based CNN (mCNN) is constructed to demonstrate the parameter- and initial condition-influenced dynamical behaviors induced by the activating memristor. Furtherly, two identical three-cell mCNNs are chosen as the subnets to construct a memristor-coupled mCNN, based on which the multi-stable states and the synchronous behaviors are investigated. Numerical results show that the multi-stable states of the memristor-coupled mCNN are flexibly switched by adjusting the coupling strength and initial conditions. Under the control of the memristor coupler, the two subnets can achieve complete synchronization, lag synchronization and phase synchronization. Finally, the FPGA-based hardware experiments are executed to verify the numerical results.

Synchronization in STDP-driven memristive neural networks with time-varying topology

Synchronization is a widespread phenomenon in the brain. Despite numerous studies, the specific parameter configurations of the synaptic network structure and learning rules needed to achieve robust and enduring synchronization in neurons driven by spike-timing-dependent plasticity (STDP) and temporal networks subject to homeostatic structural plasticity (HSP) rules remain unclear. Here, we bridge this gap by determining the configurations required to achieve high and stable degrees of complete synchronization (CS) and phase synchronization (PS) in time-varying small-world and random neural networks driven by STDP and HSP. In particular, we found that decreasing P (which enhances the strengthening effect of STDP on the average synaptic weight) and increasing F (which speeds up the swapping rate of synapses between neurons) always lead to higher and more stable degrees of CS and PS in small-world and random networks, provided that the network parameters such as the synaptic time delay tau _c, the average degree langle k rangle, and the rewiring probability beta have some appropriate values. When tau _c, langle k rangle, and beta are not fixed at these appropriate values, the degree and stability of CS and PS may increase or decrease when F increases, depending on the network topology. It is also found that the time delay tau _c can induce intermittent CS and PS whose occurrence is independent F. Our results could have applications in designing neuromorphic circuits for optimal information processing and transmission via synchronization phenomena.

Synchronization in memristive HR neurons with hidden coexisting firing and lower energy under electrical and magnetic coupling

Employing memristors to mimic biological synapses and to describe electromagnetic induction effects attracts much attention in current researches. In this paper, a memristive Hindmarsh–Rose neuron model is proposed by using an electromagnetic inductive current involving quadratic memconductance to replace the adaptive current in the classical 3-D HR model. The memristive neuron shows hidden firing and coexisting state. The energy function of memristive neuron is deduced by the generalized Hamiltonian formalism. In particular, two memristive neurons with irregular firing reach phase synchronization under the electrical coupling, while achieve complete synchronization under the magnetic coupling. The coupled neurons operate with economy energy in the region of whether complete synchronization or phase synchronization. Corresponding results are further demonstrated by designing digital circuit. These theoretical investigations can better choose appropriate model to explore dynamics of larger-scale neuronal network.

Coexisting Attractors and Multistate Noise-Induced Intermittency in a Cycle Ring of Rulkov Neurons

We study dynamics of a unidirectional ring of three Rulkov neurons coupled by chemical synapses. We consider both deterministic and stochastic models. In the deterministic case, the neural dynamics transforms from a stable equilibrium into complex oscillatory regimes (periodic or chaotic) when the coupling strength is increased. The coexistence of complete synchronization, phase synchronization, and partial synchronization is observed. In the partial synchronization state either two neurons are synchronized and the third is in antiphase, or more complex combinations of synchronous and asynchronous interaction occur. In the stochastic model, we observe noise-induced destruction of complete synchronization leading to multistate intermittency between synchronous and asynchronous modes. We show that even small noise can transform the system from the regime of regular complete synchronization into the regime of asynchronous chaotic oscillations.

Evolutionary Kuramoto dynamics

Biological systems have a variety of time-keeping mechanisms ranging from molecular clocks within cells to a complex interconnected unit across an entire organism. The suprachiasmatic nucleus, comprising interconnected oscillatory neurons, serves as a master-clock in mammals. The ubiquity of such systems indicates an evolutionary benefit that outweighs the cost of establishing and maintaining them, but little is known about the process of evolutionary development. To begin to address this shortfall, we introduce and analyse a new evolutionary game theoretic framework modelling the behaviour and evolution of systems of coupled oscillators. Each oscillator is characterized by a pair of dynamic behavioural dimensions, a phase and a communication strategy, along which evolution occurs. We measure success of mutations by comparing the benefit of synchronization balanced against the cost of connections between the oscillators. Despite the simple set-up, this model exhibits non-trivial behaviours mimicking several different classical games-the Prisoner's Dilemma, snowdrift games, coordination games-as the landscape of the oscillators changes over time. Across many situations, we find a surprisingly simple characterization of synchronization through connectivity and communication: if the benefit of synchronization is greater than twice the cost, the system will evolve towards complete communication and phase synchronization.

Synchronization of Kuramoto oscillators with the distributed time-delays and inertia effect

We prove the complete and partial phase synchronization for the Kuramoto oscillators with distributed time-delays and inertia effect. Our results assert that the Kuramoto models incorporated with a small variation of distributed time-delays and inertia effect still exhibit synchronization. This shows the robustness of the synchronization phenomena of the original Kuramoto model in the perturbation of time-delay and inertia effects. We also present several numerical experiments supporting our main results and exhibiting interesting patterns.

Electrical coupling of individual electrocatalytic oscillators.

The catalytic electro-oxidation of some small organic molecules is known to display kinetic instabilities, which reflect on potential and/or current oscillations. Under oscillatory conditions, those systems can be considered electrocatalytic oscillators and, therefore, can be described by their amplitude, frequency, and waveform. Just like mechanical oscillators, the electrocatalytic ones can be coupled and their dynamics can be changed by setting different coupling parameters. In the present work, we study the unidirectional coupling of electrocatalytic oscillators, namely, those comprehending the catalytic electro-oxidation of methanol and formic acid on polycrystalline platinum in acidic media under potentiostatic control. Herein, we explore two different scenarios (the coupling of compositionally identical and non-identical oscillators) and investigate the effects of the master's identity and of the coupling constant on the slave's dynamics. For the master (methanol)-slave (methanol) coupling, the oscillators exhibited phase lag synchronization and complete phase synchronization. On the other hand, for the master (formic acid)-slave (methanol) coupling, the oscillators exhibited complete phase synchronization with phase-locking with a 2:3 ratio, complete phase synchronization with phase-locking with a 1:2 ratio, phase lag synchronization, and complete phase synchronization. The obtained results suggest that both the master's identity and the coupling constant (sign and magnitude) are parameters that play an important role on the coupled systems, in such a way that even for completely different systems, synchronization could emerge by setting a suitable coupling constant. To the best of our knowledge, this is the first report concerning the electrical coupling of hidden N-shaped-negative differential resistance type systems.

Dynamics and Synchronization of Complex-Valued Ring Networks

Networks of coupled oscillators have been used to model various real-world self-organizing systems. However, the dynamics, especially chaos and bifurcation, of complex-valued networks are rarely investigated. In this paper, a ring network of interacting complex-valued van der Pol oscillators is studied to model the formation of ring dynamics. Although there are only stable limit cycles in a complex-valued van der Pol oscillator, chaos, hyperchaos, and coexisting chaotic attractors are observed from the ring network, which are analyzed by using the Lyapunov exponent spectrum, bifurcation diagram and 0–1 test. In addition, complexity analysis on nonlinear coefficients and coupling strengths illustrates that the range of parameters within the chaotic (hyperchaotic) region has positive correlation with the number of oscillators. It is shown that the chaotic bifurcation path is highly robust against the size variation of the ring network, which always evolves to chaos directly from period-1 and quasi-periodic states, respectively. Moreover, it is demonstrated that complete synchronization and phase synchronization of oscillations are stable in a large-scale ring network, while chaotic phase synchronization is unstable in a small-scale network.

Stable chimeras of non-locally coupled Kuramoto–Sakaguchi oscillators in a finite array

We consider chimera states of coupled identical phase oscillators where some oscillators are phase synchronized (have the same phase) while others are desynchronized. Chimera states of non-locally coupled Kuramoto–Sakaguchi oscillators in arrays of finite size are known to be chaotic transients; after a transient time, all the oscillators are phase synchronized, with the transient time increasing exponentially with the number of oscillators. In this work, we consider a small array of six non-locally coupled Kuramoto–Sakaguchi oscillators and modify the range of the phase lag parameter to destabilize their complete phase synchronization. Under these circumstances, we observe a chimera state spontaneously formed by the partition of oscillators into two independently synchronizable clusters of both stable and unstable synchronous states. In the chimera state, the trajectory of the phase differences of the desynchronized oscillators relative to the synchronous cluster is a stable periodic orbit, and as a result, the chimera state is a stable but not long-lived transient. We observe the chimera state with random initial conditions in a restricted range of the phase lag parameter and clarify why the state is observable in the restricted range using Floquet theory for periodic orbit stability.

Dynamics of multilayer networks with amplification.

We study the dynamics of a multilayer network of chaotic oscillators subject to amplification. Previous studies have proven that multilayer networks present phenomena such as synchronization, cluster, and chimera states. Here, we consider a network with two layers of Rössler chaotic oscillators as well as applications to multilayer networks of the chaotic jerk and Liénard oscillators. Intra-layer coupling is considered to be all to all in the case of Rössler oscillators, a ring for jerk oscillators and global mean field coupling in the case of Liénard, inter-layer coupling is unidirectional in all these three cases. The second layer has an amplification coefficient. An in-depth study on the case of a network of Rössler oscillators using a master stability function and order parameter leads to several phenomena such as complete synchronization, generalized, cluster, and phase synchronization with amplification. For the case of Rössler oscillators, we note that there are also certain values of coupling parameters and amplification where the synchronization does not exist or the synchronization can exist but without amplification. Using other systems with different topologies, we obtain some interesting results such as chimera state with amplification, cluster state with amplification, and complete synchronization with amplification.

Synchronization of a Kuramoto-like model for power grids with frustration

We discuss the complete synchronization for a Kuramoto-like model for power grids with frustration. For identical oscillators without frustration, it will converge to complete phase and frequency synchronization exponentially fast if the initial phases are distributed in a half circle. For nonidentical oscillators with frustration, we present a framework leading to complete frequency synchronization where the initial phase configurations are located inside the half of a circle. Our estimates are based on the monotonicity arguments of extremal phase and frequency.

Synchronization of Kuramoto oscillators with time-delayed interactions and phase lag effect

We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase and frequency synchronization are in view. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complete and partial phase synchronization for both models with the uniform time-delay and phase lag. Our results show that the Kuramoto models incorporated with small variation of time-delays and/or phase lag effect still exhibit the synchronization. These support that the original Kuramoto model (i.e., no time-delays/phase lags) is qualitatively robust in the perturbation of time-delay and phase lag effects. We also present several numerical experiments supporting our main results.

Capacitor coupling induces synchronization between neural circuits

The signal propagation and information encoding in neurons are much dependent on the activation of synapses function. Electric synapse often plays short-range modulation so that neurons can give fast responses to external stimulus while chemical synapse keeps active under long-range modulation on neural activities. Each neuron can be considered as an intelligent signal processor and appropriate setting in parameters in the electric devices can build reliable neural circuits to detect effective signal exchange and propagation. In this paper, a two-variable FitzHugh–Nagumo electrical oscillator driven by a sinusoidal voltage source is used to investigate the synchronization when the coupling capacitor is activated, which both of the plates of the coupling capacitor can trigger time-varying electric field in the capacitor, and energy flow is propagated to modulate the outputs of neural circuits. The circuit equations are obtained, and scale transformation is applied to get a dimensionless dynamical system under field coupling, and the error function for output voltage and phases is calculated to judge whether complete synchronization and phase synchronization can be realized by selecting different capacitances for the coupling capacitor. It is found that synchronization realization between neurons can be controlled by adjusting the capacitance of coupling capacitor. For two identical neurons driven by the same stimulus, complete synchronization is reached, while phase synchronization is stabilized when neurons are driven by different stimuli as the capacitor coupling is switched on. Furthermore, the realization of complete synchronization is verified on the analog circuits. It provides another new scheme for synchronization realization between neurons and a new mechanism for signal encoding in neurons via field coupling is explained.

Synchronization of pitch and plunge motions during intermittency route to aeroelastic flutter.

Interaction of fluid forces with flexible structures is often prone to dynamical instabilities, such as aeroelastic flutter. The onset of this instability is marked by sustained large amplitude oscillations and is detrimental to the structure's integrity. Therefore, investigating the possible physical mechanisms behind the onset of flutter instability has attracted considerable attention within the aeroelastic community. Recent studies have shown that in the presence of oncoming fluctuating flows, the onset of flutter instability is presaged by an intermediate regime of oscillations called intermittency. Further, based on the intensity of flow fluctuations and the relative time scales present in the flow, qualitatively different types of intermittency at different flow regimes have been reported hitherto. However, the coupled interaction between the pitch (torsion) and plunge (bending) modes during the transition to aeroelastic flutter has not been explored. With this, we demonstrate with a mathematical model that the onset of flutter instability under randomly fluctuating flows occurs via a mutual phase synchronization between the pitch and the plunge modes. We show that at very low values of mean flow speeds, the response is by and large noisy and, consequently, a phase asynchrony between the modes is present. Interestingly, during the regime of intermittency, we observe the coexistence of patches of synchronized periodic bursts interspersed amidst a state of desynchrony between the pitch and the plunge modes. On the other hand, at the onset of flutter, we observe a complete phase synchronization between the pitch and plunge modes. This study concludes by utilizing phase locking value as a quantitative measure to demarcate different states of synchronization in the aeroelastic response.

Emergence of Phase-Locking in the Kuramoto Model for Identical Oscillators with Frustration

We present an exponential synchronization for the finite-dimensional Kuramoto model for identical oscillators under the effect of frustration. In the presence of frustration, the total phase balance law and gradient-flow structure are destroyed. Hence the gradient flow approach cannot be applied for the emergence of phase-locking, and the Lyapunov functional approach based on $\ell^2$-energy and phase diameter only show some results in a restricted set of initial configurations smaller than a half circle. In this paper, we show that an initial phase configuration whose order parameter is bounded below by some positive value, which is only depending on the size of frustration, leads to the complete phase synchronization or the bipolar state exponentially fast. This extends earlier results on the asymptotic phase-locking for the identical oscillators in which initial phase configurations are confined in a half circle.

Phase synchronization of baroclinic waves in a differentially heated rotating annulus experiment subject to periodic forcing with a variable duty cycle.

A series of laboratory experiments in a thermally driven, rotating fluid annulus are presented that investigate the onset and characteristics of phase synchronization and frequency entrainment between the intrinsic, chaotic, oscillatory amplitude modulation of travelling baroclinic waves and a periodic modulation of the (axisymmetric) thermal boundary conditions, subject to time-dependent coupling. The time-dependence is in the form of a prescribed duty cycle in which the periodic forcing of the boundary conditions is applied for only a fraction δ of each oscillation. For the rest of the oscillation, the boundary conditions are held fixed. Two profiles of forcing were investigated that capture different parts of the sinusoidal variation and δ was varied over the range 0.1≤δ≤1. Reducing δ was found to act in a similar way to a reduction in a constant coupling coefficient in reducing the width of the interval in forcing frequency or period over which complete synchronization was observed (the "Arnol'd tongue") with respect to the detuning, although for the strongest pulse-like forcing profile some degree of synchronization was discernible even at δ=0.1. Complete phase synchronization was obtained within the Arnol'd tongue itself, although the strength of the amplitude modulation of the baroclinic wave was not significantly affected. These experiments demonstrate a possible mechanism for intraseasonal and/or interannual "teleconnections" within the climate system of the Earth and other planets that does not rely on Rossby wave propagation across the planet along great circles.

A model of synchronization over quantum networks

We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.

Quantum synchronization in an optomechanical system based on Lyapunov control.

We extend the concepts of quantum complete synchronization and phase synchronization, which were proposed in A. Mari et al., Phys. Rev. Lett. 111, 103605 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.103605, to more widespread quantum generalized synchronization. Generalized synchronization can be considered a necessary condition or a more flexible derivative of complete synchronization, and its criterion and synchronization measure are proposed and analyzed in this paper. As examples, we consider two typical generalized synchronizations in a designed optomechanical system. Unlike the effort to construct a special coupling synchronization system, we purposefully design extra control fields based on Lyapunov control theory. We find that the Lyapunov function can adapt to more flexible control objectives, which is more suitable for generalized synchronization control, and the control fields can be achieved simply with a time-variant voltage. Finally, the existence of quantum entanglement in different generalized synchronizations is also discussed.

Criterion of quantum synchronization and controllable quantum synchronization based on an optomechanical system

We propose a quantitative criterion to determine whether the coupled quantum systems can achieve complete synchronization or phase synchronization in the process of analyzing quantum synchronization. Adopting the criterion, we discuss the quantum synchronization effects between optomechanical systems and find that the error between the systems and the fluctuation of error is sensitive to coupling intensity by calculating the largest Lyapunov exponent of the model and quantum fluctuation, respectively. By taking the appropriate coupling intensity, we can control quantum synchronization even under different logical relationships between switches. Finally, we simulate the dynamical evolution of the system to verify the quantum synchronization criterion and to show the ability of synchronization control.