Coupled Van Der Pol Oscillators Research Articles

Impact of coupling on the road to synchronization of two coupled Van der Pol oscillators

Synchronized dynamics of two coupled van der Pol oscillators x and y balances between the attracting forces and the prevention of synchrony represented by the coupling μ and the half difference of their natural frequencies Δω. The current paper investigates two regimes of such balance: (i) μ slightly exceeds Δω, and (ii) μ is large. In case of regime (i), the oscillators are shown to quickly attain the neighborhood of the limit cycle of a complex geometry; their pre-limit behavior can be characterized by the vertical mismatch x−y. If μ→∞, case (ii), the shape of the solutions of the underlying differential equations promptly follows that of the limit cycle, which is the solution of a single van der Pol equation, but the tendency to the limit cycle is slow, being proportional to 1/μ. Further, the mismatch between the oscillators disappears as μ increases. During the transition between the regimes occurred with the growth of μ, the vertical mismatch and the time required to solutions to reach the neighborhood of the limit cycle exhibit the Λ- and V-shapes. These shapes attain the maximum and, respectively, minimum at moderate values of μ that were earlier suggested by studies of synchronization between solar activity components.

Repertoire of dynamical states in dissimilarly coupled Van der Pol oscillators

We have considered dissimilarly coupled Van der Pol oscillators with an offset parameter which determines the degree of heterogeneity of the dissimilar coupling strength. Increasing degree of heterogeneity for decreasing values of the offset parameter results in a rich repertoire of bifurcation transitions and dynamical states including epochs of period doubling bifurcation. Two distinct multi-stable states are also observed along with several symmetry breaking dynamical states. We have deduced analytical stability conditions for Hopf and pitch-fork bifurcations through a linear stability analysis of symmetry preserving states, namely, trivial steady state and oscillation death state. The analytical conditions are found to match exactly with the simulation results in the two-parameter phase diagram. In addition to torus bifurcation, crisis and crisis induced intermittency routes to chaos are also observed for an appropriate heterogeneity of the dissimilar coupling strength. The period doubling bifurcation is characterized using the largest Lyapunov exponents of the dissimilarly coupled Van der Pol oscillators.

Multiple Hopf Bifurcations of Four Coupled van der Pol Oscillators with Delay

In this paper, a system of four coupled van der Pol oscillators with delay is studied. Firstly, the conditions for the existence of multiple periodic solutions of the system are given. Secondly, the multiple periodic solutions of spatiotemporal patterns of the system are obtained by using symmetric Hopf bifurcation theory. The normal form of the system on the central manifold and the bifurcation direction of the bifurcating periodic solutions are derived. Finally, numerical simulations are attached to demonstrate our theoretical results.

Certification of almost global phase synchronization of all-to-all coupled phase oscillators

Coupled oscillators may exhibit almost global phase synchronization, namely their phases tend to asymptotically overlap for almost all initial conditions. We consider certification of this property using Rantzer’s dual Lyapunov approach with sum of squares (SOS) programming. To this aim, we use a stereographic transformation from a hypertorus to an Euclidean space. For the case of all-to-all coupling, this transformation converts the problem of certifying stability into the problem of certifying divergence of almost all solutions to infinity. We show that the latter can be solved using a polynomial Lyapunov density, which can be constructed via SOS programming. This leads to the certification of almost global phase synchronization of all-to-all coupled phase oscillators. We apply our method to an example of coupled phase oscillators and to an example of coupled van der Pol oscillators, and show that it can support the existing tools of local stability analysis by ensuring almost global phase synchronization.

Proportional and Derivative Coupling: a Way to Achieve Synchronization for Coupled Oscillators

In order to achieve synchronization, the concept of proportional coupling and derivative coupling, as analogue to PD control theory, is proposed. The author analyzes from the coupled simple harmonic oscillators to Kuromoto model, and to the coupled Van der Pol oscillators. It is conjectured that only proportional coupling for two or multiple linear and nonlinear coupled oscillators is not sufficient most of the time to achieve synchronization, whereas derivative coupling is the dominant factor for two or multiple linear and nonlinear coupled oscillators to get synchronization. In addition, it is noticed that if there is no coupling, there is no synchronization if starting from different initial conditions. In order to achieve synchronization, coupling is required. However, coupling does not necessary mean it will achieve synchronization, it has to be coupled in a particular way, and otherwise it will not get synchronization. As a result of this study, two major questions for the linear and nonlinear coupled oscillators to get synchronization are proposed. Generally, only proportional coupling (i.e. spring coupling) is earlier to analyze as it is a simple physics problem. Once we couple them using the derivative coupling approach, it will get major complications.

Dynamics of Three Coupled Generators of Quasi-periodic Oscillations

The dynamics of three coupled generators capable of demonstrating autonomous quasi-periodic oscillations is studied. The complex structure of the Lyapunov charts of the system is discussed, which reveals invariant tori of different dimensions, quasi-periodic bifurcations of tori, and Arnold’s resonant web based on tori of different dimensions. Cases of different types of tuning of individual generators (periodic oscillations, quasi-periodic oscillations) are considered. A detailed numerical bifurcation analysis of the equilibrium state and limit cycles, which form a complex picture of dynamic regimes, is carried out.Common features and differences compared to the case of three coupled van der Pol oscillators are discussed.

Application of Heavy and Underestimated Dynamic Models in Adaptive Receding Horizon Control Without Constraints

In the heuristic “Adaptive Receding Horizon Controller” (ARHC) the available dynamic model of the controlled system usually is placed in the role of a constraint under which various cost functions can be minimized over a horizon. A possible secure design can be making calculations for a “heavy dynamic model” that may produce high dynamical burden that is efficiently penalized by the cost functions and instead of the original nominal trajectory results a “deformed” one that can be realized by the controlled system of “less heavy dynamics”. In the lack of accurate system model a fixed point iterationbased adaptive approach is suggested for the precise realization of this deformed trajectory. To reduce the computational burden of the control the usual approach in which the dynamic model is considered as constraint and Lagrange-multipliers are introduced as co-state variables is evaded. The heavy dynamic model is directly built in the cost and the computationally greedy Reduced Gradient Algorithm is replaced by a transition between the simple and fast Newton-Raphson and the slower Gradient Descent algorithms (GDA). In the paper simulation examples are presented for two dynamically coupled van der Pol oscillators as a strongly nonlinear system. The comparative use of simple nondifferentiable and differentiable cost functions is considered, too.

К вопросу о периодических решениях системы дифференциальных уравнений, описывающих колебания двух слабосвязанных осцилляторов Ван дер Поля

Изучается система двух слабосвязанных полностью идентичных осцилляторов Ван дер Поля в случае диффузионной связи.В работе изучен в полном объеме вопрос о существовании и устойчивости периодических решений рассматриваемой системы. Показано, что у нее могут быть периодические решения трех типов, которые порождают циклы Андронова-Хопфа, противофазный, и третий тип циклов синхронизации: асимметричные циклы. Анализ задачи использовал метод нормальных форм Пуанкаре-Дюлака, а также метод интегральных многообразий. We study a system of two weakly coupled completely identical van der Pol oscillators in the case of diffusion coupling. the question of the existence and stability of periodic solutions of the system under consideration. It is shown that it can have periodic solutions of three types, which generate Andronov-Hopf cycles, antiphase, and the third type of synchronization cycles: asymmetric cycles. The analysis of the problem used the Poincare-Dulac method of normal forms, as well as the method of integral manifolds.

Nonintegrability of Dynamical Systems Near Degenerate Equilibria

We prove that general three- or four-dimensional systems %of differential equations are real-analytically nonintegrable near degenerate equilibria in the Bogoyavlenskij sense under additional weak conditions when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of the corresponding Poincare-Dulac normal forms and further to that of simple planar systems, and use a novel approach for proving the analytic nonintegrability of planar systems. Our result also implies that general three- and four-dimensional systems exhibiting fold-Hopf and double-Hopf codimension-two bifurcations, respectively, are real-analytically nonintegrable under the weak conditions. To demonstrate these results, we give two examples for the Rossler system and coupled van der Pol oscillators.

Quantum synchronization in quadratically coupled quantum van der Pol oscillators

We implement nonlinear anharmonic interaction in the coupled van der Pol oscillators to investigate the quantum synchronization behavior of the systems. We study the quantum synchronization in two oscillator models, coupled quantum van der Pol oscillators and anharmonic self-oscillators. We demonstrate that the considered systems exhibit a high-order synchronization through coupling in both classical and quantum domains. We show that, due to the anharmonicity of the nonlinear interaction between the oscillators, the system exhibits phonon blockade in the phase-locking regime, which is a pure nonclassical effect and has not been observed in the classical domain. We also demonstrate that, for coupled anharmonic oscillators, the system shows a multiple resonance phase-locking behavior due to nonlinear interaction. We point out that the synchronization blockade arises due to strong anticorrelation between the oscillators, which leads to phonon antibunching in the same parametric regime. In the anharmonic oscillator case we illustrate the simultaneous occurrence of bunching and antibunching effects as a consequence of simultaneous negative and positive correlation between the anharmonic oscillators. We examine the aforementioned characteristic features in the frequency entrainment of the oscillators using a power spectrum where one can observe normal mode splitting and the Mollow triplet in the strong coupling regime. Finally, we propose a possible experimental realization for the considered system in the trapped ion and optomechanical setting.

Coupled van der Pol and Duffing oscillators: emergence of antimonotonicity and coexisting multiple self-excited and hidden oscillations

Coupled van der Pol and Duffing oscillators: emergence of antimonotonicity and coexisting multiple self-excited and hidden oscillations

Multistability, transient chaos and hyperchaos, synchronization, and chimera states in wireless magnetically coupled VDPCL oscillators

Systems in nature are for the most part rich in connection, interaction, and communication in various ways that are generally complex. Synchronization is one of the main phenomenons associated with couple systems. Synchronization may be understood as a process where several systems adjust time scales of their oscillations or a given property of their motion over time due to their interactions. In compact designs, unconnected circuits may interact through the magnetic field. This interaction can be avoided or exploited in some cases. This work investigated the dynamics and the synchronization study of a set of magnetically coupled van der Pol oscillators coupled to a linear circuit (the VDPCL oscillator) in wireless interaction. The coupling coefficient between the coupled but unconnected oscillators is used as the only control parameter allowing investigating the dynamics and the synchronization of the oscillators. We first derive a mathematical model of the ring of the magnetically coupled oscillators. After this, some basic properties of the model are derived. The dynamical study of the model is carried out in the case of two and three sub-circuits including the phase synchronization, by using the well-known classical tools, that is, the bifurcation diagrams, the Lyapunov exponents, time series, and phase portraits. It is found that the system displays a rich catalog of behaviors such as regular, chaotic, and hyperchaotic (for a weak coupling) behaviors in terms of the control parameter. Moreover, it is found that the systems exhibit the coexistence of many attractors. The system is shown to exhibit global and partial synchronization (chimera state) in the case of three sub-circuits. Transient chaos/hyperchaos are also highlighted in the two sub-circuits cases. PSPICE based simulations and real laboratory measurements are included and are consistent with the theoretical/numerical results.

Stochastic synchronization in nonlinear network systems driven by intrinsic and coupling noise

In this paper, we consider a noisy network of nonlinear systems in the sense that each system is driven by two sources of state-dependent noise: (1) an intrinsic noise that can be generated by the environment or any internal fluctuations and (2) a noisy coupling which is generated by interactions with other systems. Our goal is to understand the effect of noise and coupling on synchronization behaviors of such networks. First, we assume that all the systems are driven by a common noise and show how a common noise can be detrimental or beneficial for network synchronization behavior. Then, we assume that the systems are driven by independent noise and study network approximate synchronization behavior. We numerically illustrate our results using the example of coupled Van der Pol oscillators.

Asynchronous and synchronous quenching of a globally unstable jet via axisymmetry breaking

We explore experimentally whether axisymmetry breaking can be exploited for the open-loop control of a prototypical hydrodynamic oscillator, namely a low-density inertial jet exhibiting global self-excited axisymmetric oscillations. We find that when forced transversely or axially at a low amplitude, the jet always transitions first from a period-1 limit cycle to $\mathbb {T}^2$ quasiperiodicity via a Neimark–Sacker bifurcation. However, we find that the subsequent transition, from $\mathbb {T}^2$ quasiperiodicity to $1:1$ lock-in, depends on the spatial symmetry of the applied perturbations: axial forcing induces a saddle-node bifurcation at small detuning but an inverse Neimark–Sacker bifurcation at large detuning, whereas transverse forcing always induces an inverse Neimark–Sacker bifurcation irrespective of the detuning. Crucially, we find that only transverse forcing can enable both asynchronous and synchronous quenching of the natural mode to occur without resonant or non-resonant amplification of the forced mode, resulting in substantially lower values of the overall response amplitude across all detuning values. From this, we conclude that breaking the jet axisymmetry via transverse forcing is a more effective control strategy than preserving the jet axisymmetry via axial forcing. Finally, we show that the observed synchronization phenomena can be modelled qualitatively with just two forced coupled Van der Pol oscillators. The success of such a simple low-dimensional model in capturing the complex synchronization dynamics of a multi-modal hydrodynamic system opens up new opportunities for axisymmetry breaking to be exploited for the open-loop control of other globally unstable flows.

Synchronization or cluster synchronization in coupled Van der Pol oscillators networks with different topological types

In this paper, we discuss the mechanism of synchronization or cluster synchronization in the coupled van der Pol oscillator networks with different topology types by using the theory of rotating periodic solutions. The synchronous solutions here are transformed into rotating periodic solutions of some dynamical systems. By analyzing the bifurcation of rotating periodic solutions, the critical conditions of synchronous solutions are given in three different networks. We use the rotating periodic matrix in the rotating periodic theory to judge various types of synchronization phenomena, such as complete synchronization, anti-phase synchronization, periodic synchronization, or cluster synchronization. All rotating periodic matrices which satisfy the exchange invariance of multiple oscillators form special groups in these networks. By using the conjugate classes of these groups, we obtain various possible synchronization solutions in three networks. In particular, we find symmetry has different effects on synchronization in different networks. The network with more types of symmetry has more elements in the corresponding group, which may have more types of synchronous solutions. However, different types of symmetry may get the same type of synchronous solutions or different types of synchronous solutions, depending on whether their corresponding rotating periodic matrices are similar.

Explosive Transition in Coupled Oscillators Through Mixed Attractive-Repulsive Interactions

The emergence of many fascinating dynamic behaviors is affected by more than one interaction among the elements or cells in a network. In fact, the concurrence and competition of different types of effects among subsystems show a strong connection to the dynamic transition process between oscillation patterns. Here, a network of generic oscillators with mixed attractive-repulsive couplings is introduced to demonstrate the transition from oscillatory states to stationary equilibria, specifically for van der Pol oscillators and Lorenz oscillators. Through the observation of the normalized amplitude changing with the coupling strength, the sudden and irreversible transition appears in both systems, which has a close relation to the mutual repulsion on coupled oscillators. Whereas, for coupled van der Pol oscillators, three typical transition scenarios are found by varying the weight ratio of these two couplings, while the Lorenz system shows only one transition mode no matter how the weight ratio changes. Besides, in the cases of explosive transitions, the coexistence areas of oscillatory and death states also reveal a distinct manifestation for periodic and chaotic systems. The details of theoretical critical transition points on the first-order phase transition are also obtained. Our results pave a new way to control the explosive phenomenon, which is crucial to explain the sudden oscillation quenching and the coexistence of oscillatory and stationary states in biological as well as chemical systems.

Boosting synchronization in chaotic systems: Combining past and present interactions

A classical interconnection for inducing synchronization in dynamical systems is static diffusive coupling, i.e., a coupling without dynamics that considers the output or state differences of neighboring systems. However, this type of coupling has some well-known limitations: 1) for some chaotic systems and certain input-output combinations, it is impossible to synchronize a pair or a network of identical systems, 2) the interval of coupling strength values for which synchronization is achieved is limited. Here, we show that these limitations may be removed, or at least considerably relaxed, by using a coupling with memory, i.e., a coupling composed by a standard diffusive-like term plus a delayed version of itself. As a particular example, we consider a pair of Rössler systems in a master-slave configuration, where we analyze the local stability of the synchronous solution using the linearized error dynamics in combination with the D-decomposition method. Also, analytic conditions for tuning the coupling strength values corresponding to the non-delayed and delayed terms of the overall coupling are provided. Furthermore, the applicability of the proposed coupling for boosting synchronization in other chaotic systems is also discussed. In particular, we show how the proposed coupling with memory is applicable for synchronizing a pair of uni-directionally coupled van der Pol oscillators with sinusoidal excitation and a pair of Hindmarsh-Rose neurons. Ultimately, the results presented in this paper suggest that the synchronizability of the coupled systems is enhanced when the present and past interactions are taken into account.

Identification of van der Pol oscillator network parameters

The problem of state observation and parameters identification of an oscillatory system consisting of coupled van der Pol oscillators is considered. The unknowns are: velocity of oscillations and parameters that characterize the threshold values for displacements of network's oscillators at which the damping forces change sign. An invariant relations method for simultaneous of the state and parameters estimation is used. Such approach is based on dynamical extension of original system and synthesis of appropriate invariant relations, from which the unknowns can be expressed as a function of the known quantities on the trajectories of extended system during the observed motion. The stability property is formally checked considering the oscillatory behavior of the system. On the first step the corresponding observation and identification problems are solved for one of autonomous van der Pol oscillator, further, the results obtained are extended to a system of interconnected oscillators. The simulation results confirm efficiency of the proposed scheme of nonlinear observer and identifier design for network of oscillators.

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.