Coupled Van Der Pol Oscillators Research Articles

Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV.

Delay Effects on Amplitude Death, Oscillation Death, and Renewed Limit Cycle Behavior in Cyclically Coupled Oscillators

The effects of a distributed `weak generic kernel' delay on cyclically coupled limit cycle and chaotic oscillators are considered. For coupled Van der Pol oscillators (and in fact, other oscillators as well) the delay can produce transitions from amplitude death(AD) or oscillation death (OD) to Hopf bifurcation-induced periodic behavior, with the delayed limit cycle shrinking or growing as the delay is varied towards or away from the bifurcation point respectively. The transition from AD to OD is mediated here via a pitchfork bifurcation, as seen earlier for other couplings as well. Also, the cyclically coupled undelayed van der Pol system here is already in a state of AD/OD, and introducing the delay allows both oscillations and AD/OD as the delay parameter is varied. This is in contrast to other limit cycle systems, where diffusive coupling alone does not result in the onset of AD/OD.\\ For systems where the individual oscillators are chaotic, such as a Sprott oscillator system or a coupled van der Pol-Rayleigh system with parametric forcing, the delay may produce AD/OD (as in the Sprott case), with the AD to OD transition now occurring via a transcritical bifurcation instead. However, this may not be possible, and the delay might just vary the attractor shape. In either of these situations however, increased delay strength tends to cause the system to have simpler behavior, streamlining the shape of the attractor, or shrinking it in cases with oscillations.

Mutual synchronization of self-excited acoustic oscillations in coupled thermoacoustic oscillators

Synchronization refers to the adjustment of rhythms of two nonidentical oscillators due to interaction. In this study, we report for the first time on the ability of computational fluid dynamics (CFD) to simulate the synchronization of self-sustained acoustic oscillations in two coupled thermoacoustic (TA) oscillators that interfere with each other through acoustic radiation. A bifurcation diagram containing both asynchronous and synchronous states is first mapped out by changing the end-to-end distance and the resonator length difference. It is found that the two TA oscillators are analogous to mutually coupled van der Pol oscillators with reactive coupling. Then, the dynamic characteristics of beating and periodic oscillations in asynchronous and synchronous states are analysed separately to facilitate the comprehension of the synchronization process. This research demonstrates that the CFD methodology provides a valuable numerical tool for studying the synchronization of thermally induced acoustic oscillations in TA systems.

On the dynamics of a system of two coupled van der Pol oscillators subjected to a constant excitation force: effects of broken symmetry

On the dynamics of a system of two coupled van der Pol oscillators subjected to a constant excitation force: effects of broken symmetry

Period-doubling route to mixed-mode chaos.

Mixed-mode oscillations (MMOs) are a complex dynamical behavior in which each cycle of oscillation consists of one or more large amplitude spikes followed by one or more small amplitude peaks. MMOs typically undergo period-adding bifurcations under parameter variation. We demonstrate here, in a set of three identical, linearly coupled van der Pol oscillators, a scenario in which MMOs exhibit a period-doubling sequence to chaos that preserves the MMO structure, as well as period-adding bifurcations. We characterize the chaotic nature of the MMOs and attribute their existence to a master-slave-like forcing of the inner oscillator by the outer two with a sufficient phase difference between them. Simulations of a single nonautonomous oscillator forced by two sine functions support this interpretation and suggest that the MMO period-doubling scenario may be more general.

Modelling oscillations in the supply chain: the case of a just-in-sequence supply process from the automotive industry

Pervasive and ubiquitous oscillations, mapping the repetitive variation in time of a specific state, are well known as abundant phenomena in research and practice. Motivated by the success of oscillators in the modelling, analysis and control of dynamical systems, we developed a related approach for the dynamic description of supply chains. This paper aims to introduce a generic oscillator model for supply chains by the original application of oscillator equations. Therefore an established oscillator model for deductive modelling of supply chain echelons is used. With the help of coupled van der Pol oscillators, the dynamical interaction of an inventory system is described and applied to a real-life supply process in automotive industry. According to its reductionist approach only two differential equations are used to analyse a Just-in-Sequence supply process in car industry. Based on the fact that any oscillatory state can be reduced to the phase of the oscillation (phase reduction), a phase space map is generated. This compact visual reference of the supply process can act as the quantitative basis for an adaptive control mechanism during its operation. By delaying or accelerating the inventory oscillations of the supplier stock a detuned coupled supply process can be re-synchronised without changing the amplitude. An additional analysis of Hilbert transform is applied to determine the boundaries of phase-locking between the inventory oscillation phases, where the instantaneous phases are bounded. Furthermore parameters of the synchronisation threshold and the transient phases between synchronous and non-synchronous regimes have been investigated, supported by an Arnold tongue representation. The investigations show that with the help of a generic oscillatory model it is possible to measure and quantify phenomena of inventory dynamics in supply chains. Especially the analysis of synchronisation phenomena with the help of phase space and Arnold tongue representations foster developments of performance measurement in supply chain management.

Repulsive inter-layer coupling induces anti-phase synchronization

We present numerical results for the synchronization phenomena in a bilayer network of repulsively coupled 2D lattices of van der Pol oscillators. We consider the cases when the network layers have either different or the same types of intra-layer coupling topology. When the layers are uncoupled, the lattice of van der Pol oscillators with a repulsive interaction typically demonstrates a labyrinth-like pattern, while the lattice with attractively coupled van der Pol oscillators shows a regular spiral wave structure. We reveal for the first time that repulsive inter-layer coupling leads to anti-phase synchronization of spatiotemporal structures for all considered combinations of intra-layer coupling. As a synchronization measure, we use the correlation coefficient between the symmetrical pairs of network nodes, which is always close to -1 in the case of anti-phase synchronization. We also study how the form of synchronous structures depends on the intra-layer coupling strengths when the repulsive inter-layer coupling is varied.

Quantum Zeno effect in self-sustaining systems: Suppressing phase diffusion via repeated measurements

We study the effect of frequent projective measurements on the dynamics of quantum self-sustaining systems by considering the prototypical example of the quantum van der Pol oscillator. Quantum fluctuations are responsible for phase diffusion which progressively blurs the semiclassical limit-cycle dynamics and synchronization, either to an external driving or between two coupled self-sustained oscillators. We show that by subjecting the system to repeated measurements of heterodyne type at an appropriate repetition frequency one can significantly suppress phase diffusion without spoiling the semiclassical dynamics. This quantum Zeno-like effect may be effective in the case of either one or two coupled van der Pol oscillators, and we discuss its possible implementation in the case of trapped ions.

Multiscale Cross-Recurrence Plot and Recurrence Quantification Analysis Based on Coarse-Grained

In this paper, we proposed multiscale cross-recurrence quantification analysis (MSCRQA) method to analyze the dynamic states of two time series at different time scales. We apply this method to model system (two coupled van der Pol oscillators) and real-world system (SSEC and SZSE). It demonstrates that the MSCRQA can show richer and more recognizable information compared with single time scale. The state of dynamics is different under different time scales. MSCRQA method shows another multiscale perspective to fully mine more hidden internal dynamic information of a time series. This method may provide another method reference for practical application to better explore the laws of the real world.

Anti-phase relay synchronization of wave structures in a heterogeneous multiplex network of 2D lattices

We explore numerically relay synchronization of wave structures in a heterogeneous three-layer network of coupled two-dimensional (2D) lattices of continuous-time systems. Remote layers, which are not directly connected but interact via a relay layer, consist of coupled van der Pol oscillators, while the middle layer is described by a lattice of interacting FitzHugh-Nagumo neurons. We show for the first time that already for weak inter-layer coupling, anti-phase relay synchronization of target wave patterns occurs in the considered network. This is a novel effect which can be observed in multiplex networks of interacting 2D lattices of oscillatory systems. Our numerical studies indicate that strong inter-layer coupling leads to in-phase synchronization of spatio-coherent structures in the network layers. We also analyze the impact of inter-layer coupling ranges in the relay and the remote layers on the relay synchronization effect.

Dynamics of hierarchical weighted networks of van der Pol oscillators.

We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.

Abnormal route to aging transition in a network of coupled oscillators.

In this article, we investigate the dynamical robustness in a network of Van der Pol oscillators. In particular, we consider a network of diffusively coupled Van der Pol oscillators to explore the aging transition phenomena. Our investigation reveals that the route to aging transition in a network of Van der Pol oscillator is different from that of typical sinusoidal oscillators such as Stuart-Landau oscillators. Unlike sinusoidal oscillators, the order parameter does not follow smooth second-order phase transition. Rather, we observe an abnormal phase transition of the order parameter due to the sudden appearance of unbounded trajectories at a critical point. We provide detailed bifurcation analysis of such an abnormal phase transition. We show that the boundary crisis of a limit-cycle oscillator is at the helm of such an unusual discontinuous path of aging transition.

Modulation of contractions in the small intestine indicate desynchronization via supercritical Andronov\u2013Hopf bifurcation

The small intestine is covered by a network of coupled oscillators, the interstitial cells of Cajal (ICC). These oscillators synchronize to generate rhythmic phase waves of contraction. At points of low coupling, oscillations desynchronise, frequency steps occur and every few waves terminates as a dislocation. The amplitude of contractions is modulated at frequency steps. The phase difference between contractions at a frequency step and a proximal reference point increased slowly at first and then, just at the dislocation, increased rapidly. Simultaneous frequency and amplitude modulation (AM/FM) results in a Fourier frequency spectrum with a lower sideband, a so called Lashinsky spectrum, and this was also seen in the small intestine. A model of the small intestine consisting of a chain of coupled Van der Pol oscillators, also demonstrated simultaneous AM/FM at frequency steps along with a Lashinsky spectrum. Simultaneous AM/FM, together with a Lashinsky spectrum, are predicted to occur when periodically-forced or mutually-coupled oscillators desynchronise via a supercritical Andronov–Hopf bifurcation and have been observed before in other physical systems of forced or coupled oscillators in plasma physics and electrical engineering. Thus motility patterns in the intestine can be understood from the viewpoint of very general dynamical principles.

Bistable labyrinth-like structures and chimera states in a 2D lattice of van der Pol oscillators

The present work is devoted to the numerical analysis of the dynamics of a 2D lattice of coupled van der Pol oscillators in the regime of relaxation oscillations. It is shown that the influence of coupling leads to the shift of effective values of the control parameters of individual oscillators. The strong coupling can even cause the transition to bistable dynamics which is never observed in a single van der Pol oscillator. The numerically constructed phase-parametric diagram that takes into account the shifts of parameters shows that the bistability arises through the pitchfork bifurcation when varying the coupling strength. The lattice dynamics is analyzed when the control and coupling parameters are varied within a wide range, and the regime diagrams are constructed in the planes of system parameters. A new type of spatiotemporal pattern, a so-called ”labyrinth-like structure”, is found and described in detail. We also reveal for the first time and study a spiral wave chimera with a new kind of the incoherence core in the form of ”labyrinth-like structure”.

Synchronization effects for dissipative and inertial coupling between multiplex lattices

We analyse the synchronization of spatiotemporal patterns based on spiral and target waves, which are realized in a multiplex network consisting of two nonidentical 2D lattices of the nonlocally coupled van der Pol oscillators. We study the syncronization and its features for two types of the inter-layer unidirectional coupling, namely dissipative and inertial. We show that the synchronization features for these cases are significantly different. When the inter-layer coupling is dissipative the synchronization between the layers is observed only for the strong inter-layer coupling or is even impossible. For the case of inertial interaction the layers are synchronized at sufficiently weak coupling strength. At the same time, there are “desynchronization windows”, in which the layer regimes are no longer synchronized. We explain these phenomena by a noticeably different influence of the inter-layer coupling on the dynamics of individual oscillators in the cases of dissipative and inertial interaction of the layers.

An efficient algorithm for inferring functional connectivity between the drive and the noisy response chaotic oscillators with significant frequency mismatch

In this study, we present a new method for assessing the functional connectivity between the drive and the noisy response chaotic oscillators with significant frequency mismatch. This method is based on the footprint of the drive oscillator on the response oscillator. Specifically, we calculate the magnitude squared coherence between the intrinsic frequency of the drive oscillator and the oscillation frequency of the dynamics of the spectral energy of the response oscillator. The spectral energy of the response oscillator is estimated using a modified S-transform algorithm with a short sliding window. We apply the proposed approach to master–slave Rossler systems and coupled Van der Pol oscillators with a frequency ratio of $$\sim $$ 1:4 and show that when the slave signal is contaminated with white Gaussian noise at different signal-to-noise ratios (SNR), the new algorithm is well suited to assess the presence of coupling with a priori known direction in noisy, unidirectionally coupled chaotic oscillators, especially in the case of weak and moderate coupling.

Synchronization features of target wave structures with an incoherent center

Abstract In the present paper we study numerically the interaction of two 2-dimensional networks each representing a lattice of coupled van der Pol oscillators. The isolated lattices can demonstrate spiral and target wave structures for the cases of no-flux boundary conditions and random initial conditions. We explore the effect of mutual synchronization of target waves and target wave chimeras for dissipative and inertial inter-layer coupling between the lattices and for local and nonlocal intra-layer coupling between the lattice oscillators. We show for the first time the possibility of complete synchronization of target wave chimera states, including synchronization of incoherent cores. Our findings demonstrate that there is a difference with synchronization of spiral wave chimera states in which incoherent cores are not completely synchronized. The obtained results can have an important application and can be particularly used to study the dynamics of cardiac muscle in biophysics.

Role of solitary states in forming spatiotemporal patterns in a 2D lattice of van der Pol oscillators

The behavior of the ensemble of coupled van der Pol oscillators is abundant when coupling parameters are changed. Incoherent and chimera-like regimes are observed at small values of the coupling strength parameter. Synchronization takes place with an increase in the coupling strength and coupling range parameters. Various chimera and solitary states are realised when the coupling strength parameter is sufficiently large. The single van der Pol oscillator is taken in the regime of relaxation oscillations. Therefore, the lattice may realise spiral wave and spiral wave chimera regimes. Besides, target wave and solitary state regimes are observed in the ensemble. Finally, the solitary state and solitary state chimera regimes are firstly shown for the lattice of coupled van der Pol oscillators.

Death Behavior of Mean-Field Coupled van der Pol Oscillators Induced by Time-Delayed Feedback

Explosive death in coupled nonlinear oscillators has been an active area of extensive research in nonlinear dynamics in the recent decades. Depending on proper choice of network topology, coupling scenarios, and feedback strength, explosive death can be revealed. In this work, for the first time, we report the effect of delayed feedback on the death behavior in an ensemble of identical mean-field coupled van der Pol oscillators. In both systems with or without time delay, the normalized amplitude exhibits an abrupt transition between the oscillatory state and the death state. Intriguingly, the presence of time delay in the coupling may induce the normalized amplitude of all oscillators in the network to experience a step-like descent with small jumps in approaching the death state, pulling back the forward and backward transition points. The backward transition point has been explicitly obtained, which is confirmed by the numerical results.

Effect of Fractional-Order on the Dynamic of two Mutually Coupled van der Pol Oscillators: Hubs, Multistability and its Control

This paper presents a novel approach to analyze the dynamic effect of the fractional-order derivative of the two mutually coupled van der Pol oscillators. The stability analysis is presented by two complementary phase diagrams: the isospike diagrams and the two Lyapunov exponent spectra. These diagrams reveal precisely the Hubs, spirals bifurcation and chaos when the derivative order is fixed at q = 0.95. In addition, when the fractional-order is set as a control parameter, various methods for detecting chaos including bifurcation diagrams, spectrum of largest Lyapunov exponent are exploited to establish the connection between the system parameters and various complicated dynamics. A transition was also observed between a desynchronized state and a multistability situation. These diagrams displayed the coexistence of four disconnected attractors (two symmetric). We study the basins of attraction of the system in the multistability regime which thereby reveal the coexistence of attractors in the systems when the fractional-order derivative is taken as a function of initials conditions. Based on the parametric control, we have controlled this striking phenomenon in the system. Finally, the hardware circuit is implemented and the results are found to be in good agreement with the numerical investigations.