Ring Of Oscillators Research Articles

Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with Mexican hat connectivity.

Spatiotemporal waves of synchronized activity are known to arise in oscillatory neural networks with lateral inhibitory coupling. How such patterns respond to dynamic changes in coupling strength is largely unexplored. The present study uses analysis and simulation to investigate the evolution of wave patterns when the strength of lateral inhibition is varied dynamically. Neural synchronization was modeled by a spatial ring of Kuramoto oscillators with Mexican hat lateral coupling. Broad bands of coexisting stable wave solutions were observed at all levels of inhibition. The stability of these waves was formally analyzed in both the infinite ring and the finite ring. The broad range of multi-stability predicted hysteresis in transitions between neighboring wave solutions when inhibition is slowly varied. Numerical simulation confirmed the predicted transitions when inhibition was ramped down from a high initial value. However, non-wave solutions emerged from the uniform solution when inhibition was ramped upward from zero. These solutions correspond to spatially periodic deviations of phase that we call ripple states. Numerical continuation showed that stable ripple states emerge from synchrony via a supercritical pitchfork bifurcation. The normal form of this bifurcation was derived analytically, and its predictions compared against the numerical results. Ripple states were also found to bifurcate from wave solutions, but these were locally unstable. Simulation also confirmed the existence of hysteresis and ripple states in two spatial dimensions. Our findings show that spatial synchronization patterns can remain structurally stable despite substantial changes in network connectivity.

Optimal configuration for vibration frequencies in a ring of harmonic oscillators: The nonidentical mass effect

The parameter diversity effect in coupled nonidentical elements has attracted persistent interest in nonlinear dynamics. Of fundamental importance is the so-called optimal configuration problem for how the spatial position of elements with different parameters precisely determines the dynamics of the whole system. In this work, we study the optimal configuration problem for the vibration spectra in the classical mass–spring model with a ring configuration, paying particular attention to how the configuration of different masses affects the second smallest vibration frequency (ω2) and the largest one (ωN ). For the extreme values of ω2 and ωN ,n amely, ( ω2)min ,( ω2)max ,( ωN )min ,a nd ( ωN )max, we find some explicit organization rules for the optimal configurations and some approximation rules when the explicit organization rules are not available. The different distributions of ω2 and ωN are compared. These findings are interesting and valuable for uncovering the underlying mechanism of the parameter diversity effect in more general cases.

Multiple synchronous states in a ring of coupled phase oscillators

A significant phenomenon in nature is that of collective synchronization, in which a large population of coupled oscillators spontaneously synchronizes at a common frequency. Nonlinearly coupled systems with local interactions are of special importance, in particular, the Kuramoto model in its nearest-neighbor version. In this paper the dynamics of a ring of Kuramoto phase oscillators with unidirectional couplings is investigated. We simulate numerically the bifurcation tree of average frequency observed and the multiple stable states in the synchronization region with the increase of the coupling strength for N4, which cannot be found for N3. Oscillators synchronize at a common frequency =0 when K is larger than a critical value of N=3. Multiple branches with 0 will appear besides the zero branch, and the number of branches increases with increasing oscillators for the system N3. We further present a theoretical analysis on the feature and stability of the multiple synchronous states and obtain the asymptotically stable solutions. When the system of N=2 reaches synchronization, the dynamic equation has two solutions: one is stable and the other is unstable. And there is also one stable solution for N=3 when the system is in global synchronization. For the larger system (N3), we study the identical oscillators and can find all the multiple branches on the bifurcation tree. Our results show that the phase difference between neighboring oscillators has different fixed values corresponding to the numbers of different branches. The behaviors in the synchronization region computed by numerical simulation are consistent with theoretical calculation very well. The systems in which original states belong to different stable states will evolve to the same incoherent state with an adiabatic decreasing of coupling strength. Behaviors of synchronization of all oscillators are exactly the same in non-synchronous region whenever the system evolves from an arbitrary branch according to the bifurcation trees. This result suggests that the only incoherent state can be attributed to the movement ergodicity in the phase space of coupled oscillators in an asynchronous region. When the system achieves synchronization, the phenomenon of the coexistence of multiple stable states will emerge because of the broken ergodicity. All these analyses indicate that the multiple stable states of synchronization in nonlinear coupling systems are indeed generically observable, which can have potential engineering applications.

Experimental synchronization of chaos in a large ring of mutually coupled single-transistor oscillators: phase, amplitude, and clustering effects.

In this paper, experimental evidence of multiple synchronization phenomena in a large (n = 30) ring of chaotic oscillators is presented. Each node consists of an elementary circuit, generating spikes of irregular amplitude and comprising one bipolar junction transistor, one capacitor, two inductors, and one biasing resistor. The nodes are mutually coupled to their neighbours via additional variable resistors. As coupling resistance is decreased, phase synchronization followed by complete synchronization is observed, and onset of synchronization is associated with partial synchronization, i.e., emergence of communities (clusters). While component tolerances affect community structure, the general synchronization properties are maintained across three prototypes and in numerical simulations. The clusters are destroyed by adding long distance connections with distant notes, but are otherwise relatively stable with respect to structural connectivity changes. The study provides evidence that several fundamental synchronization phenomena can be reliably observed in a network of elementary single-transistor oscillators, demonstrating their generative potential and opening way to potential applications of this undemanding setup in experimental modelling of the relationship between network structure, synchronization, and dynamical properties.

Rotational Waves in Rings of Electrochemical Oscillators

Chemical and biological systems exhibit network properties dependent upon the behavior of individual units and the interactions between units.[1] The presence or absence of interactions and the strength of the interactions determine the topology of the network and in turn the network properties. From a mathematical perspective, a variety of non-rotating and rotating waves can exist for a ring of locally coupled units[2] and the non-rotating solutions are the dominant outcome when coupling is the only mean of interactions regardless of the initial conditions.[3] Therefore, engineering is required to achieve solutions aside from the dominant non-rotating state. The formation of waves is examined in oscillatory nickel dissolution where units of individual oscillators interact through external resistance which controls the strength of interaction. An experimental technique using a locally coupled twenty nickel electrode ring is presented in which through a combination of temporary alterations in topology and the application of global feedback provides all possible rotational solutions. There are four rotational waves which can propagate about the ring and are distinguished by the velocity and the number of waves. The propagation rate is inversely proportional to the number of waves in the rotation (m) for all rotations. Solutions with no rotation have an m value of zero while rotational solutions are obtained for m=1,2,3,4. Larger m values (m ≥ 5) are shown to not be stable in the network. The results thus 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.

Asynchronous Oscillations of Double-Mode and N-Phase in a Ring of Simultaneous Oscillators

In this study, a ring of coupled simultaneous oscillators with two LC resonators is investigated. By computer simulations, asynchronous oscillations of doublemode and N-phase are confirmed to be stably generated. Careful observation reveals various interesting synchronization phenomena which have not reported before.

Single-clustering synchronization in a ring of Kuramoto oscillators

Multiple-clustering synchronization is a common scenario of global phase synchronization. However, a novel single-clustering type of phase synchronization in a ring of Kuramoto oscillators has been recently reported in studying the influence of the permutation of the natural frequencies of oscillators on the synchronization efficiency (Wu et al 2012 Europhys. Lett. 97 40005). It was found that it occurs for a particular spatial frequency distribution and gives rise to a very small critical coupling strength Kc even if the oscillator number is large. Here we focus on this particular type of synchronization and study its generality. We provide some solid evidence for the convergence of Kc to a small constant in the thermodynamic limit, based on the finite size analysis. Further we demonstrate that it is robust in the sense of either switching the natural frequencies of any two oscillators or randomly perturbing the frequencies of all coupled oscillators. All these findings prove that the single-clustering synchronization is indeed generically observable, with merit for potential engineering applications.

Collective Behavior of a Ring of Van Der Pol Oscillators under Gaussian and Random Coupling Schemes

The collective behavior of a ring of coupled identical van der Pol oscillators is numerically studied in this work. Constant, gaussian and random distributions of the coupling parameter along the ring are considered. Three values of the oscillators constant are assumed in order to cover from quasilinear to nonlinear dynamic performance. Single and multiple coupled frequencies are obtained using power spectra of the long term time series. Phase portraits are obtained from numerical simulations, and the coupled behavior is analyzed, compared and discussed.

Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays

We experimentally study the complex dynamics of a unidirectionally coupled ring of four identical optoelectronic oscillators. The coupling between these systems is time-delayed in the experiment and can be varied over a wide range of delays. We observe that as the coupling delay is varied, the system may show different synchronization states, including complete isochronal synchrony, cluster synchrony, and two splay-phase states. We analyze the stability of these solutions through a master stability function approach, which we show can be effectively applied to all the different states observed in the experiment. Our analysis supports the experimentally observed multistability in the system.

ALGORITHMS FOR OBTAINING A SADDLE TORUS BETWEEN TWO ATTRACTORS

We develop two algorithms for obtaining an index 1 saddle torus between two attractors of which basin boundary is smooth by using the bisection method. In the first algorithm, which we name "overall template method", we make a template approximating a whole attractor and use it for pattern matching. The attractor can be a periodic orbit, a torus and even a chaos. In the second algorithm, which we name "partial template method", we make a template approximating only a part of an attractor. In this algorithm, the attractor is restricted to a periodic orbit and a two-torus, but calculation time is much smaller than that of the overall template method. We demonstrate these two algorithms for two solutions, observed in a ring of six-coupled bistable oscillators. One is a saddle torus in the basin boundary of the two switching solutions, and the other is of the two quasi-periodic propagating wave solutions. We calculate the Lyapunov spectrum of the obtained saddle torus, and confirm one positive Lyapunov exponent.

The Dynamical Impact of a Shortcut in Unidirectionally Coupled Rings of Oscillators

We study the destabilization mechanism in a unidirectional ring of identical oscillators, perturbed by the introduction of a long-range connection. It is known that for a homogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivial equilibrium undergoes a sequence of Hopf bifurcations eventually leading to the coexistence of multiple stable periodic states resembling the Eckhaus scenario. We show that this destabilization scenario persists under small non-local perturbations. In this case, the Eckhaus line is modulated according to certain resonance conditions. In the case when the shortcut is strong, we show that the coexisting periodic solutions split up into two groups. The first group consists of orbits which are unstable for all parameter values, while the other one shows the classical Eckhaus behavior.

Dynamic Behavior of a Large Ring of Coupled Self-Excited Oscillators

Abstract Interconnected, self-excited oscillators are often found in nature and in engineered devices. In this work, a ring of van der Pol oscillators, each of which is connected to its immediate neighbors, is considered. The focus is on the emergent behavior of a large number of oscillators. Conditions are determined under which time-independent solutions are obtained, and the linear stability of these solutions is investigated. The effect of the singularity of the coupling matrix on the ring dynamics is explored. When this becomes singular, an infinite number of steady states is present, and the phenomenon of oscillation death arises. It is also possible to have, depending on initial conditions, all oscillators with in-phase synchrony, metachronal traveling waves with different wavelengths going around the ring, or standing waves. Interconnected oscillators can propagate information at a group velocity, and the information signal is present as an amplitude modulation.

BIFURCATION ANALYSIS OF THE PROPAGATING WAVE AND THE SWITCHING SOLUTIONS IN A RING OF SIX-COUPLED BISTABLE OSCILLATORS — BIFURCATION STARTING FROM TYPE 2 STANDING WAVE SOLUTION

In this paper, we analyze the bifurcation of Type 2 periodic solution in a ring of six-coupled bistable oscillators. We show that pitchfork and heteroclinic bifurcations, which induce chaos, cause a change from periodic (standing wave) solution to quasi-periodic (propagating wave) solution or the inverse when the coupling strength is varied. We also explain the existence of the switching solution, and presume that the birth and death of this switching solution are due to a pitchfork bifurcation of a quasi-periodic solution.

CHAOS OF THE PROPAGATING PULSE WAVE IN A RING OF SIX-COUPLED BISTABLE OSCILLATORS

In this paper, some properties of the chaotic propagating pulse wave in a ring of six-coupled bistable oscillators are investigated. When coupling factor α becomes large beyond a certain critical value, the standing pulse wave converts to a propagating pulse wave. Further, as α is increased, the propagating pulse wave behaves chaotically. We find some interesting properties of the chaotic propagating pulse wave such as random change of propagating direction and stepwise change of pulse position with respect to time. In particular, we notice that the shape of probability density of one-direction propagation time and distance is similar to that of Logistic map.

Discrete multivortex solitons

We introduce discrete multivortex solitons in a ring of nonlinear oscillators coupled to a central site. Regular clusters of discrete vortices appear as a result of mode collisions, and we show that their stability is determined by global symmetries rather than the stability of constituent vortices. Stable multivortex solitons support complex vortex dynamics including charge flipping and spiraling.

Multiple periodic solutions in a delay-coupled system of neural oscillators

In this paper, effects of the synaptic delay of signal transmissions on the pattern formation of nonlinear waves in a bidirectional ring of neural oscillators is studied. Firstly, the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Meanwhile, using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of Lie groups, we discuss the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Finally, Hopf bifurcation directions and corresponding stabilities of bifurcating periodic orbits are derived by using the normal form approach and the center manifold theory. These theoretical results are significant to complement experimental and numerical observations made in living neuronal systems and artificial neural networks, in order to better understand the mechanisms underlying the system’s dynamics.

Periodic patterns in a ring of delay-coupled oscillators

We describe the appearance and stability of spatiotemporal periodic patterns (rotating waves) in unidirectional rings of coupled oscillators with delayed couplings. We show how delays in the coupling lead to the splitting of each rotating wave into several new ones. The appearance of rotating waves is mediated by the Hopf bifurcations of the symmetric equilibrium. We also conclude that the coupling delays can be effectively replaced by increasing the number of oscillators in the chain. The phenomena are shown for the Stuart-Landau oscillators as well as for the coupled FitzHugh-Nagumo systems modeling an ensemble of spiking neurons interacting via excitatory chemical synapses.

Bistable chimera attractors on a triangular network of oscillator populations

We study a triangular network of three populations of coupled phase oscillators with identical frequencies. The populations interact nonlocally, in the sense that all oscillators are coupled to one another, but more weakly to those in neighboring populations than to those in their own population. This triangular network is the simplest discretization of a continuous ring of oscillators. Yet it displays an unexpectedly different behavior: in contrast to the lone stable chimera observed in continuous rings of oscillators, we find that this system exhibits two coexisting stable chimeras. Both chimeras are, as usual, born through a saddle-node bifurcation. As the coupling becomes increasingly local in nature they lose stability through a Hopf bifurcation, giving rise to breathing chimeras, which in turn get destroyed through a homoclinic bifurcation. Remarkably, one of the chimeras reemerges by a reversal of this scenario as we further increase the locality of the coupling, until it is annihilated through another saddle-node bifurcation.

COMPLEX PATTERN IN A RING OF VAN DER POL OSCILLATORS COUPLED BY TIME-VARYING RESISTORS

Synchronization phenomena in coupled oscillatory systems are very important models to describe various higher-dimensional nonlinear phenomena in the field of natural science. In this paper, phase synchronization in a ring of van der Pol oscillators coupled by time-varying resistors is studied. The coexistence of in-phase and anti/N-phase states and various interesting phase synchronization patterns are observed when the parameters are changed. Further, the influence of duty cycle of time-varying resistors for the observed phase synchronization patterns is investigated.

Stability of the synchronization manifold in nearest neighbor nonidentical van der Pol-like oscillators

We investigate the stability of the synchronization manifold in a ring and an open-ended chain of nearest neighbors coupled self-sustained systems, each self-sustained system consisting of multi-limit cycles van der Pol oscillators. Such model represents, for instance, coherent oscillations in biological systems through the case of an enzymatic-substrate reaction with ferroelectric behavior in brain waves model. The ring and open-ended chain of identical and non-identical oscillators are considered separately. By using the Master Stability Function approach (for the identical case) and the complex Kuramoto order parameter (for the non-identical case), we derive the stability boundaries of the synchronized manifold. We have found that synchronization occurs in a system of many coupled modified van der Pol oscillators and it is stable even in presence of a spread of parameters.