Hopf bifurcation in delayed van der Pol oscillators

In this paper, we consider a classical van der Pol equation with a general delayed feedback. Firstly, by analyzing the associated characteristic equation, we derive a set of parameter values where the Hopf bifurcation occurs. Secondly, in the case of the standard Hopf bifurcation, the stability of bifurcating periodic solutions and bifurcation direction are determined by applying the normal form theorem and the center manifold theorem. Finally, a generalized Hopf bifurcation corresponding to non-semisimple double imaginary eigenvalues (case of 1:1 resonance) is analyzed by using a normal form approach.     

Synchronization of four coupled van der Pol oscillators

It is possible that self-excited vibrations in turbomachine blades synchronize due to elastic coupling through the shaft. The synchronization of four coupled van der Pol oscillators is presented here as a simplified model. For quasilinear oscillations, a stability condition is derived from an analysis based on linearizing the original equation around an unperturbed limit cycle and transforming it into Hill’s equation. For the nonlinear case, numerical simulations show the existence of two well-defined regions of phase relationships in parameter space in which a multiplicity of periodic attractors is embedded. The size of these regions strongly depends on the values of the oscillator and coupling constants. For the coupling constant below a critical value, there exists a region in which a diversity of phase-shift attractors is present, whereas for values above the critical value an in-phase attractor is predominant. It is observed that the presence of an anti-phase attractor in the subcritical region is associated with sudden changes in the period of the coupled oscillators. The convergence of the coupled system to a particular periodic attractor is explored using several initial conditions. The study is extended to non-identical oscillators, and it is found that there is synchronization even over a wide range of difference among the oscillator constants.     

Dynamics of two strongly coupled van der pol oscillators

A perturbation method is used to study the steady state behavior of two Van der Pol oscillators with strong linear diffusive coupling. It is shown that a bifurcation occurs which results in a transition from phase-locked periodic motions to quasi-periodic motions as the coupling is decreased or the detuning is increased. The analytical results are compared with a numerically generated solution.     

Dynamics of two delay coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol oscillators with delayed position and velocity coupling is studied by the method of averaging together with truncation of Taylor expansions. According to the slow-flow equations, the dynamics of 1:1 internal resonance is more complex than that of non-1:1 internal resonance. For 1:1 internal resonance, the stability and the number of periodic solutions vary with different time delay for given coupling coefficients. The condition necessary for saddle-node and Hopf bifurcations for symmetric modes, namely in-phase and out-of-phase modes, are determined. The numerical results, obtained from direct integration of the original equation, are found to be in good agreement with analytical predictions.     

Synchronization in ensembles of delay-coupled nonidentical neuronlike oscillators

Nonlinear Dynamics - We study both numerically and experimentally the synchronization in an ensemble of nonidentical neuronlike oscillators described by the FitzHugh–Nagumo equations. The...     

Anticipating spike synchronization in nonidentical chaotic neurons

Anticipating synchronization is investigated in nonidentical chaotic systems unidirectionally coupled in a master-slave configuration without a time-delay feedback. We show that if the parameters of chaotic master and slave systems are mismatched in such a way that the mean frequency of a free slave system is greater than the mean frequency of a master system, then the phase synchronization regime can be achieved with the advanced phase of the slave system. In chaotic neural systems, this leads to the anticipating spike synchronization: unidirectionally coupled neurons synchronize in such a way that the slave neuron anticipates the chaotic spikes of the master neuron. We demonstrate our findings with coupled Rössler systems as well as with two different models of coupled neurons, namely, the Hindmarsh–Rose neurons and the adaptive exponential integrate-and-fire neurons.     

Harmonically excited generalized van der Pol oscillators: Entrainment phenomenon

Harmonically excited generalized van der Pol oscillators with power-form non-linearities in the restoring and damping-like force are investigated from the viewpoint of the occurrence of harmonic entrainment. Locked periodic motion is obtained by adjusting the averaging method. The influence of the powers of the restoring and damping-like force on the occurrence of this phenomenon is examined.     

Dynamical analysis of two coupled parametrically excited van der Pol oscillators

The dynamical behavior of two coupled parametrically excited van der pol oscillators is investigated in this paper. Based on the averaged equations, the transition boundaries are sought to divide the parameter space into a set of regions, which correspond to different types of solutions. Two types of periodic solutions may bifurcate from the initial equilibrium. The periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. The instabilities of both the quasi-periodic solutions and the 3D tori may directly lead to chaos with the variation of the parameters. Two symmetric chaotic attractors are observed and for certain values of the parameters, the two attractors may interact with each other to form another enlarged chaotic attractor.     

A pair of van der Pol oscillators coupled by fractional derivatives

We consider the stability of the in-phase and out-of-phase modes of a pair of fractionally-coupled van der Pol oscillators: 1 2 where D ^?? x is the order ?? derivative of x(t), and 0<??<1. We use a two-variable perturbation method on the system??s corresponding variational equations to derive expressions for the transition curves separating regions of stability from instability in the ??, ?? parameter plane. The perturbation results are validated with numerics and through direct comparison with known results in the limiting cases of ??=0 and ??=1, where the fractional coupling reduces to position coupling and velocity coupling, respectively.     

Response statistics of van der Pol oscillators excited by white noise

The joint probability density function of the state space vector of a white noise exoited van der Pol oscillator satisfics a Fokker-Planck-Kolmogorov (FPK) equation. The paper describes a numerical procedure for solving the transient FPK equation based on the path integral solution (PIS) technique. It is shown that by combining the PIS with a cubic B-spline interpolation method, numerical solution algorithms can be implemented giving solutions of the FPK equation that can be made accurate down to very low probability levels. The method is illustrated by application to two specific examples of a van der Pol oscillator.     

Constructing invariant tori for two weakly coupled van der Pol oscillators

An algorithm is developed for the construction of an invariant torus of a weakly coupled autonomous oscillator. The system is put into angular standard form. The determining equations are found by averaging and are solved for the approximate amplitudes of the torus. A perturbation series is then constructed about the approximate amplitudes with unknown coefficients as periodic functions of the angular variables. A sequence of solvable partial differential equations is developed for determining the coefficients. The algorithm is applied to a system of nonlinearly coupled van der Pol equations and the first order coefficients are generated in a straightforward manner. The approximation shows both good numerical accuracy and reproducibility of the periodicities of the van der Pol system. A comparitive analysis of integrating the van der Pol system with integrating the phase equations from the angular standard form on the approximate torus shows numerical errors of the order of the perturbation parameter =0.05 for integrations of up to 10,000 steps. Applying FFT to the numerical periodicities generated by integrating the van der Pol system near the tours reveals the same predominant frequencies found in the perturbation coefficients. Finally an expected rotation number is found by integrating the phase equations on the approximate torus.Contribution of the National Institute of Standards and Technology, a Federal agency.     

Effects of gradient coupling on amplitude death in nonidentical oscillators

The effects of the gradient coupling on the amplitude death in an array and a ring of diffusively coupled nonidentical oscillators are explored, respectively. The gradient coupling plays a significant role on the amplitude death dynamics, however, it is strongly related to the boundary conditions of the coupled system. With the increment of the gradient coupling, the domain of the amplitude death is monotonically enlarged in an array of coupled oscillators. However, for a ring of coupled oscillators, it is firstly enlarged and then decreased as the gradient coupling increases. The domain of the amplitude death in parameter space is analytically predicted for a small number of gradiently coupled oscillators.     

The analysis of the square array of van der pol oscillators using the averaged potential

This paper describes an approach to finding stable oscillations in van der Pol oscillators with many degrees of freedom. We summarize a new concept of the “averaged potential” which is derived from “mixed potential” defined by Brayton and Moser. The averaged potential is the time average of the losses (dissipation function) in the system. It is shown that the averaged equations of a system take the form of the gradient system of the averaged potential. Hence, the stable oscillations of this system correspond to the minimal points of the averaged potential. Therefore, finding the stable oscillations is reduced to constructing the averaged potential and finding its minimal points. This method is successfully applied to the analysis of a square array of van der Pol oscillators coupled by inductors. It is shown that the triple and quadruple mode oscillations can be stably excited as well as simple and double mode oscillations.     

Finite-time synchronization of complex networks with nonidentical discontinuous nodes

In this paper, we study the finite-time synchronization problem for linearly coupled complex networks with discontinuous nonidentical nodes. Firstly, new conditions for general discontinuous chaotic systems is proposed, which is easy to be verified. Secondly, a set of new controllers are designed such that the considered model can be finite-timely synchronized onto any target node with discontinuous functions. Based on a finite-time stability theorem for equations with discontinuous right-hand and inequality techniques, several sufficient conditions are obtained to ensure the synchronization goal. Results of this paper are general, and they extend and improve existing results on both continuous and discontinuous complex networks. Finally, numerical example, in which a BA scale-free network with discontinuous Sprott and Chua circuits is finite-timely synchronized onto discontinuous Chen system, is given to show the effectiveness of our new results.     

Stability of Subsonic Planar Phase Boundaries in a van der Waals Fluid

We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional van der Waals fluid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic, and thus undercompressive. The lacking information lies in an additional jump condition, which may be derived from the viscosity-capillarity criterion. This condition is rather simple in the case of reversible phase transitions, since it reduces to a generalized equal area rule. In a previous work, I proved that reversible planar phase boundaries are weakly linearly stable, in the sense introduced by Majda for shock fronts. This means that they satisfy a generalized Lopatinsky condition but not a uniform one. The aim of this paper is to point out the influence of viscosity on the stability analysis, in order to deal with the more realistic case of dissipative phase transitions. The main difficulty lies in the additional jump condition, which is no longer explicit and depends on the (unknown) internal structure of the interface. We overcome it by using bifurcation arguments on the nondimensional parameter measuring the competition between viscosity and capillarity. We show by perturbation that the positivity of this parameter stabilizes the phase transitions. As a conclusion, we find that dissipative planar phase boundaries are uniformly linearly stable, in the sense of the uniform Lopatinsky condition. Accepted December 14, 1998     

Mode analysis of a system of mutually coupled van der Pol oscillators with coupling delay

A system of mutually coupled van der Pol oscillators containing fifth-order conductance characteristic, with the coupling delay, are analyzed by using the non-linear mode analysis. In particular, it has been demonstrated that zero state, two single modes, and one double mode are stable only for sufficiently small τ.The analytical results have been verified by using the digital simulation.     

Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes

A fractional-order weighted complex network consists of a number of nodes, which are the fractional-order chaotic systems, and weighted connections between the nodes. In this paper, we investigate generalized chaotic synchronization of the general fractional-order weighted complex dynamical networks with nonidentical nodes. The well-studied integer-order complex networks are the special cases of the fractional-order ones. Based on the stability theory of linear fraction-order systems, the nonlinear controllers are designed to make the fractional-order complex dynamical networks with distinct nodes asymptotically synchronize onto any smooth goal dynamics. Numerical simulations are provided to verify the theoretical results. It is worth noting that the synchronization effect sensitively depends on both the fractional order ?? and the feedback gain k _i . Moreover, generalized synchronization of the fractional-order weighted networks can still be achieved effectively with the existence of noise perturbation.