Coupled Relaxation Oscillators Research Articles

Dynamical Behavior of Two Coupled Two-Stroke Relaxation Oscillators

Starting from [J.-M. Ginoux et al., Torus breakdown in a uni junction memristor, Int. J. Bifurcation Chaos 28(10) (2018) 1850128; J.-M. Ginoux et al., Torus breakdown in a two-stroke relaxation memristor, Chaos Solitons Fractals 153 (2021) 111594], we define a model which describes the dynamical behavior of a class of two-stroke oscillators (i.e., nonlinear oscillators with two distinct phases per cycle). We highlight some properties of the model still not taken into account, like presence of a global bifurcation and bistability for some regions of the parameters space. We then extend the model to study two coupled two-stroke oscillators, where synchrony and anti-synchrony regimes appear as well as transient chaos and intermittency phenomena. We finally compare the theoretical results on synchrony regimes with the corresponding experimental data obtained from a circuit where two relaxation oscillators, based on the Unijunction Transistor (UJT) electronic component, are coupled together. The comparison shows a good qualitative agreement between the experiments and the theoretical results.

First-order synchronization transition in a large population of strongly coupled relaxation oscillators.

Onset and loss of synchronization in coupled oscillators are of fundamental importance in understanding emergent behavior in natural and man-made systems, which range from neural networks to power grids. We report on experiments with hundreds of strongly coupled photochemical relaxation oscillators that exhibit a discontinuous synchronization transition with hysteresis, as opposed to the paradigmatic continuous transition expected from the widely used weak coupling theory. The resulting first-order transition is robust with respect to changes in network connectivity and natural frequency distribution. This allows us to identify the relaxation character of the oscillators as the essential parameter that determines the nature of the synchronization transition. We further support this hypothesis by revealing the mechanism of the transition, which cannot be accounted for by standard phase reduction techniques.

Reservoir computing using high order synchronization of coupled oscillators

We propose a concept for reservoir computing on oscillators using the high-order synchronization effect. The reservoir output is presented in the form of oscillator synchronization metrics: fractional high-order synchronization value and synchronization efficiency, expressed as a percentage. Using two coupled relaxation oscillators built on VO2 switches, we created an oscillator reservoir that allows simulating the XOR operation. The reservoir can operate as with static input data (power currents, coupling forces), as with dynamic data in the form of spike sequences. Having a small number of oscillators and significant non-linearity, the reservoir expresses a wide range of dynamic states. The proposed computing concept can be implemented on oscillators of diverse nature.

Chimera States on a Ring of Strongly Coupled Relaxation Oscillators

Weakly coupled oscillators can exhibit seemingly incongruous synchronization patterns comprised of coherent and incoherent spatial domains known as chimera states. However, the weak coupling approximation is invalid when the characteristic phase response curve of an oscillator does not scale linearly with the coupling strength and instead changes its shape. In chemical experiments with photo-coupled relaxation oscillators, we find that beyond weak coupling chimera patterns consist of different coexisting cluster states. Numerical modeling reveals that the observed cluster states result from a phase-dependent excitability that is also commonly observed in neural tissue and cardiac pacemaker cells.

Multipliers of an Antiphase Solution in a System of Two Coupled Nonlinear Relaxation Oscillators

We consider a system of two non-linear differential-difference equations that models a behaviour of two coupled neural oscillators with a synaptic connection of a threshold type. To find a solution of the system means to find normalized membrane potentials of neurons. We prove that there exists an antiphase solution. It is a special relaxation periodic regime such that one membrane potential is a half-period shift of the other membrane potential. We prove that the antiphase solution is stable and construct its asymptotic. We show that there exists a solution with a bursting-effect by choosing a specific connection chain delay. That is, for any natural n there is a connection chain delay such that an antiphase solution has exactly n asymptotically high spikes on a period after a refractive segment.

Asymptotic Behavior of the Solutions of a System of Two Weakly Coupled Relaxation Oscillators with Delayed Feedback

We consider a dynamic system that consists of two coupled self-excited oscillators with delayed feedback. Such models occur in applied problems of radio physics and optics. It is assumed that the nonlinear function, which is responsible for the feedback, is finite and contains a great parameter. This reveals the possibility to use a special analytical method to study relaxation oscillations. It is shown that the dynamics of two such oscillators in the case of their asymptotically weak coupling is described by a special finite-dimension mapping and can be rather sophisticated.

Vertex coloring of graphs via phase dynamics of coupled oscillatory networks

While Boolean logic has been the backbone of digital information processing, there exist classes of computationally hard problems wherein this paradigm is fundamentally inefficient. Vertex coloring of graphs, belonging to the class of combinatorial optimization, represents one such problem. It is well studied for its applications in data sciences, life sciences, social sciences and technology, and hence, motivates alternate, more efficient non-Boolean pathways towards its solution. Here we demonstrate a coupled relaxation oscillator based dynamical system that exploits insulator-metal transition in Vanadium Dioxide (VO2) to efficiently solve vertex coloring of graphs. Pairwise coupled VO2 oscillator circuits have been analyzed before for basic computing operations, but using complex networks of VO2 oscillators, or any other oscillators, for more complex tasks have been challenging in theory as well as in experiments. The proposed VO2 oscillator network harnesses the natural analogue between optimization problems and energy minimization processes in highly parallel, interconnected dynamical systems to approximate optimal coloring of graphs. We further indicate a fundamental connection between spectral properties of linear dynamical systems and spectral algorithms for graph coloring. Our work not only elucidates a physics-based computing approach but also presents tantalizing opportunities for building customized analog co-processors for solving hard problems efficiently.

Synchronization of weakly coupled canard oscillators

Synchronization has been studied extensively in the context of weakly coupled oscillators using the so-called phase response curve (PRC) which measures how a change of the phase of an oscillator is affected by a small perturbation. This approach was based upon the work of Malkin, and it has been extended to relaxation oscillators. Namely, synchronization conditions were established under the weak coupling assumption, leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators. Previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points, where canard solutions can arise. In the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles. In particular, we highlight a new special role of the maximal canard in separating two distinct synchronization regimes: the Hopf regime and the relaxation regime. Phase plane analysis of slow–fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard.

A Coupled Memcapacitor Emulator-Based Relaxation Oscillator

This brief proposes a floating emulator circuit for mimicking the dynamic behaviors of flux coupled memcapacitors (MCs) by making use of common off-the-shelf active devices. The equivalent constitutive relation of the coupled MC emulators is theoretically presented in details. The coupled MCs are then utilized to structure new relaxation oscillators (ROs) with the purpose of achieving controllable oscillation frequency and duty cycle. In consideration of different parameter configuration cases, an experimental investigation is carried out using a practical circuit emulating the dual coupled MCs and the ROs. Good agreement between experimental and theoretical results confirms that dual coupled MCs can be utilized to structure ROs with controllability.

Synchronization of pairwise-coupled, identical, relaxation oscillators based on metal-insulator phase transition devices: A model study

Computing with networks of synchronous oscillators has attracted wide-spread attention as novel materials and device topologies have enabled realization of compact, scalable and low-power coupled oscillatory systems. Of particular interest are compact and low-power relaxation oscillators that have been recently demonstrated using MIT (metal-insulator-transition) devices using properties of correlated oxides. Further the computational capability of pairwise coupled relaxation oscillators has also been shown to outperform traditional Boolean digital logic circuits. This paper presents an analysis of the dynamics and synchronization of a system of two such identical coupled relaxation oscillators implemented with MIT devices. We focus on two implementations of the oscillator: (a) a D-D configuration where complementary MIT devices (D) are connected in series to provide oscillations and (b) a D-R configuration where it is composed of a resistor (R) in series with a voltage-triggered state changing MIT device (D). The MIT device acts like a hysteresis resistor with different resistances in the two different states. The synchronization dynamics of such a system has been analyzed with purely charge based coupling using a resistive (RC) and a capacitive (CC) element in parallel. It is shown that in a D-D configuration symmetric, identical and capacitively coupled relaxation oscillator system synchronizes to an anti-phase locking state, whereas when coupled resistively the system locks in phase. Further, we demonstrate that for certain range of values of RC and CC, a bistable system is possible which can have potential applications in associative computing. In D-R configuration, we demonstrate the existence of rich dynamics including non-monotonic flows and complex phase relationship governed by the ratios of the coupling impedance. Finally, the developed theoretical formulations have been shown to explain experimentally measured waveforms of such pairwise coupled relaxation oscillators.

Variety of Alternative Stable Phase-Locking in Networks of Electrically Coupled Relaxation Oscillators

We studied the dynamics of a large-scale model network comprised of oscillating electrically coupled neurons. Cells are modeled as relaxation oscillators with short duty cycle, so they can be considered either as models of pacemaker cells, spiking cells with fast regenerative and slow recovery variables or firing rate models of excitatory cells with synaptic depression or cellular adaptation.It was already shown that electrically coupled relaxation oscillators exhibit not only synchrony but also anti-phase behavior if electrical coupling is weak. We show that a much wider spectrum of spatiotemporal patterns of activity can emerge in a network of electrically coupled cells as a result of switching from synchrony, produced by short external signals of different spatial profiles. The variety of patterns increases with decreasing rate of neuronal firing (or duty cycle) and with decreasing strength of electrical coupling. We study also the effect of network topology - from all-to-all – to pure ring connectivity, where only the closest neighbors are coupled.We show that the ring topology promotes anti-phase behavior as compared to all-to-all coupling. It also gives rise to a hierarchical organization of activity: during each of the main phases of a given pattern cells fire in a particular sequence determined by the local connectivity. We have analyzed the behavior of the network using geometric phase plane methods and we give heuristic explanations of our findings.Our results show that complex spatiotemporal activity patterns can emerge due to the action of stochastic or sensory stimuli in neural networks without chemical synapses, where each cell is equally coupled to others via gap junctions. This suggests that in developing nervous systems where only electrical coupling is present such a mechanism can lead to the establishment of proto-networks generating premature multiphase oscillations whereas the subsequent emergence of chemical synapses would later stabilize generated patterns.

Spatiotemporal order, disorder, and propagating defects in homogeneous system of relaxation oscillators

A variety of complex spatial patterns relevant to chemical and biological systems can be generated through reaction-diffusion mechanisms. In this paper, we show that diffusive coupling through the inactivating component in a system of relaxation oscillators extends such complexity to the temporal domain, generating remarkable spatiotemporal phenomena. We provide analytic explanations of the antiphase synchronization and spatially patterned oscillatory death regimes. We report a chimera state where patches with distinct dynamics coexist and also observe propagating phase defects resembling persistent structures in cellular automata that may be used for computation.

Triggered tremor, phase-locking, and the global clustering of great earthquakes

A system of non-linear coupled relaxation oscillators is used to show how the communication between large earthquakes on a global scale can align their seismic cycles to produce a worldwide clustering of large events. Our model builds on recent observations that the seismic waves from a large earthquake can trigger an episode of non-volcanic tremor at the base of a distant fault. We assume that tremor is indicative of creep on the ductile extension of the fault zone that loads its overlying seismogenic layer thus advancing the fault toward failure. If this advance is larger toward the end of the seismic cycle, we show that two or more interacting faults will align their cycles, even if their recurrence intervals are not identical.

Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay

In this paper, we study the following system of two coupled relaxation oscillators of the van der Pol type with delayε$\ddot{x}_1-(1-x_1^2)\dot{x}_1+x_1=h_1(x_2(t-\tau)-x_1(t-\tau)),$ε$\ddot{x}_2-(1-x_2^2)\dot{x}_2+x_2=h_2(x_1(t-\tau)-x_2(t-\tau)),$where $h_1$ and $h_2$ are nonlinear functions.It is shown that this system can exhibit Hopf bifurcation as the time delay $\tau$ passes certain critical values. The distribution of the eigenvalues of the linearized system is studied thoroughly in terms of the parameter $\ep$ and the linear parts of functions $h_1$ and $h_2$. The normal form theory for general retarded functional equations developed by Faria and Magalhães is applied to perform center manifold reduction and hence to obtain the explicit normal form Hopf bifurcation which can be used to determine the stability of the bifurcating periodic solutions and and the direction of Hopf bifurcation. Examples are given to confirm the theoretical results.

Dynamics of a model of two delay-coupled relaxation oscillators

This paper investigates the dynamics of a new model of two coupled relaxation oscillators. The model replaces the usual DDE (differential-delay equation) formulation with a discrete-time approach with jumps. Existence, bifurcation and stability of in-phase periodic motions is studied. Simple periodic motions, which involve exactly two jumps per period, are found to have large plateaus in parameter space. These plateaus are separated by regions of complicated dynamics, reminiscent of the Devil’s Staircase. Stability of motions in the in-phase manifold are contrasted with stability of motions in the full phase space.

Dynamics of a ring of three coupled relaxation oscillators

The dynamics of a ring of three identical relaxation oscillators is shown to exhibit a variety of periodic motions, including clockwise and counter-clockwise wave-like modes, and a synchronous mode in which all three oscillators are in phase. The model involves individual oscillators which exhibit sudden jumps, modeling the relaxation oscillations of van der Pol oscillators. Methods include (i) numerical integration, (ii) a semi-analytical method involving solving transcendental equations numerically, and (iii) perturbation methods. A variety of bifurcations of the periodic motions are identified. This work is motivated by application to the design of a decision-making machine which can sort initial conditions according to their steady state.

Local network parameters can affect inter-network phase lags in central pattern generators

Weakly coupled phase oscillators and strongly coupled relaxation oscillators have different mechanisms for creating stable phase lags. Many oscillations in central pattern generators combine features of each type of coupling: local networks composed of strongly coupled relaxation oscillators are weakly coupled to similar local networks. This paper analyzes the phase lags produced by this combination of mechanisms and shows how the parameters of a local network, such as the decay time of inhibition, can affect the phase lags between the local networks. The analysis is motivated by the crayfish central pattern generator used for swimming, and uses techniques from geometrical singular perturbation theory.

Forced oscillation modes in a birhythmic system of two coupled relaxation oscillators near Andronoy - Hopf bifurcation

A system of two identical relaxation oscillators of the FitzZHugh-Nagumo type with the parameters chosen in the vicinity of a bifurcation of limit cycle emergence was examined for the dynamic modes arising in the presence of a weak harmonic signal applied to both elements. Slow variable exchange between them gives rise to three stable limit cycles called in-phase, anti-phase, and extremely asymmetrical, in which only one of the oscillators generates spikes. In this study, we show that slow variable exchange also causes this system to respond to weak harmonic forcing in a manner quite different from what is known in the classical dynamics of forced oscillations. In addition to the expected synchronization tongues generated by interaction of the signal with the in-phase attractor, we observed at least three other consequences of the coexistence of different solutions: (i) there appeared broad bands of synchronization of the signal with the anti-phase solution at high frequencies multiple to the frequency of the anti-phase oscillations, whereas the base synchronization frequency band became much narrower; (ii) signal period ranges were found in which the limit cycles were such that several spikes were produced over the complete period and each oscillator was characterized with the same set of discrete

Stochastic multiresonance in the coupled relaxation oscillators

We study the noise-dependent dynamics in a chain of four very stiff excitable oscillators of the FitzHugh-Nagumo type locally coupled by inhibitor diffusion. We could demonstrate frequency- and noise-selective signal acceptance which is based on several noise-supported stochastic attractors that arise owing to slow variable diffusion between identical excitable elements. The attractors have different average periods distinct from that of an isolated oscillator and various phase relations between the elements. We explain the correspondence between the noise-supported stochastic attractors and the observed resonance peaks in the curves for the linear response versus signal frequency.

Limit Cycles Arising in a Chain of Inhibitorily Coupled Identical Relaxation Oscillators Near the Self-Oscillation Threshold

We consider the dynamic regimes arising in a linear chain of four identical stiff FitzHugh- Nagumo oscillators existing in the vicinity of the bifurcation of limit cycle emergence. It is shown that in a broad range of coupling forces, slow-variable exchange between the oscillators gives rise to multiple limit cycles with different periods and different phase relations. In addition to the expected antiphase solutions, three families of stable limit cycles that differ in the number of bursts of the fast variable in the neighboring elements and in the number of bursts per period are detected. The boundaries of attractor stability are calculated and the parameter regions of their coexistence are found.