Complex order van der Pol oscillator

In this paper a complex-order van der Pol oscillator is considered. The complex derivative D^a±jbD^{\alpha\pm\jmath\beta}, with α,β∈R ⁺ is a generalization of the concept of integer derivative, where α=1, β=0. By applying the concept of complex derivative, we obtain a high-dimensional parameter space. Amplitude and period values of the periodic solutions of the two versions of the complex-order van der Pol oscillator are studied for variation of these parameters. Fourier transforms of the periodic solutions of the two oscillators are also analyzed.     

A dual state variable model of the van der Pol oscillator

A new, “dual” state variable (DSV) formulation is used to construct a model of the van der Pol oscillator. The model is valid for small degrees of non-linearity, and results are superior to those from a common perturbation technique, especially as non-linearity begins to increase. The DSV formulation utilizes a unique state space, and behavior in this space is illustrated for a wider range of non-linearity.     

A modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator

In this paper, a modified averaging scheme is presented for a class of time-delayed vibration systems with slow variables. The new scheme is a combination of the averaging techniques proposed by Hale and by Lehman and Weibel, respectively. The averaged equation obtained from the modified scheme is simple enough but it retains the required information for the local nonlinear dynamics around an equilibrium. As an application of the present method, the delay value for which a secondary Hopf bifurcation occurs is successfully located for a delayed van der Pol oscillator.     

Stochastic response of fractional-order van der Pol oscillator

We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system based on the principle of minimum mean-square error. Through stochastic averaging, an averaged Itô equation is deduced. We obtained the Fokker-Planck-Kolmogorov equation connected to the averaged Itô equation and solved it to yield the approximate stationary response of the system. The analytical solution is confirmed by using Monte Carlo simulation.     

Primary resonance of fractional-order van der Pol oscillator

In this paper the primary resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied analytically and numerically. At first the approximately analytical solution is obtained by the averaging method, and it is found that the fractional-order derivative could affect the dynamical properties of VDP oscillator, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude–frequency equation for steady-state solution is established, and the corresponding stability condition is also presented based on Lyapunov theory. Then, the comparisons of several different amplitude–frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two fractional parameters, i.e., the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated for some typical excitation amplitudes, which are different from the traditional integer-order VDP oscillator.     

Nonlinear dynamics of self-, parametric,and externally excited oscillator with time delay: van der Pol versus Rayleigh models

The regular and chaotic vibrations of a nonlinear structure subjected to self-, parametric, and external excitations acting simultaneously are analysed in this study. Moreover, a time delay input is added to the model to control the system response. The frequency-locking phenomenon and transition to quasi-periodic oscillations via Hopf bifurcation of the second kind (Neimark–Sacker bifurcation) are determined analytically by the multiple time scales method up to the second-order perturbation. Approximate solutions of the quasi-periodic motion are determined by a second application of the multiple time scales method for the slow flow, and then, slow–slow motion is obtained. The similarities and differences between the van der Pol and Rayleigh models are demonstrated for regular, periodic, and quasi-periodic oscillations, as well as for chaotic oscillations. The control of the structural response, and modifications of the resonance curves and bifurcation points by the time delay signal are presented for selected cases.

     

Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback

We investigate the primary resonance of an externally excited van der Pol oscillator under state feedback control with a time delay. By means of the asymptotic perturbation method, two slow-flow equations on the amplitude and phase of the oscillator are obtained and external excitation-response and frequency-response curves are shown. We discuss how vibration control and high amplitude response suppression can be performed with appropriate time delay and feedback gains. Moreover, energy considerations are used in order to investigate existence and characteristics of limit cycles of the slow-flow equations. A limit cycle corresponds to a two-period modulated motion for the van der Pol oscillator. We demonstrate that appropriate choices for the feedback gains and the time delay can exclude the possibility of modulated motion and reduce the amplitude peak of the primary resonance. Analytical results are verified with numerical simulations.     

Fast parametrically excited van der Pol oscillator with time delay state feedback

We investigate the effect of a fast vertical parametric excitation on self-excited vibrations in a delayed van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic in the vicinity of the trivial equilibrium. Then, we apply the multiple scales method on this slow dynamic to derive a second-order slow flow system describing the modulation of slow dynamic. In particular we analyze the slow flow to obtain the effect of a fast excitation on the regions in parameter space where self-excited vibrations can be eliminated. We have shown that in the case where the time delay and the feedback gains are imposed, fast vertical parametric excitation can be an alternative to suppress undesirable self-excited vibrations in a delayed van der Pol oscillator.     

Analog circuit implementation and synchronization of a system consisting of a van der Pol oscillator linearly coupled to a Duffing oscillator

This paper deals with the analog circuit implementation and synchronization of a model consisting of a van der Pol oscillator coupled to a Duffing oscillator. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of the systems solutions (i.e., elastic coupling). The primary motivation of our investigations lays in the fact that coupled attractors of different types might serve as a good model for real systems in nature (e.g., electromechanical, physical, biological, or economic systems). The stability of fixed points is examined. The bifurcation structures of the system are analyzed with particular emphasis on the effects of nonlinearity. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results shows a very good agreement. By exploiting recent results on adaptive control theory, a controller is designed that enables both synchronization of two unidirectionally coupled systems and the estimation of unknown parameters of the drive system.     

Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control

Analytical derivations and numerical calculations are employed to gain insight into the parametric resonance of a stochastically driven van der Pol oscillator with delayed feedback. This model is the prototype of a self-excited system operating with a combination of narrow-band noise excitation and two time delayed feedback control. A slow dynamical system describing the amplitude and phase of resonance, as well as the lowest-order approximate solution of this oscillator is firstly obtained by the technique of multiple scales. Then the explicit asymptotic formula for the largest Lyapunov exponent is derived. The influences of system parameters, such as magnitude of random excitation, tuning frequency, gains of feedback and time delays, on the almost-sure stability of the steady-state trivial solution are discussed under the direction of the signal of largest Lyanupov exponent. The non-trivial steady-state solution of mean square response of this system is studied by moment method. The results reveal the phenomenon of multiple solutions and time delays induced stabilization or unstabilization, moreover, an appropriate modulation between the two time delays in feedback control may be acted as a simple and efficient switch to adjust control performance from the viewpoint of vibration control. Finally, theoretical analysis turns to a validation through numerical calculations, and good agreements can be found between the numerical results and the analytical ones.     

Dynamical analysis of Mathieu equation with two kinds of van der Pol fractional-order terms

In this paper the dynamics of Mathieu equation with two kinds of van der Pol (VDP) fractional-order terms is investigated. The approximately analytical solution is obtained by the averaging method. The steady-state solution, existence conditions and stability condition for the steady-state solution are presented, and it is found that the two kinds of VDP fractional coefficients and fractional orders remarkably affect the steady-state solution, which is characterized by the additional damping coefficient (ADC) and additional stiffness coefficient (ASC). The comparisons between the analytical and numerical solutions verify the correctness and satisfactory precision of the approximately analytical solution. The presented typical amplitude–frequency curves illustrate the important effects of two kinds of VDP fractional-order terms on system dynamics. The application of two VDP fractional-order terms in vibration control is discussed. At last, the detailed results are summarized and the conclusions are made.     

Uniformly valid solution of limit cycle of the Duffing–van der Pol equation

The limit cycle of the Duffing–van der Pol equation is studied. By considering the product of the frequency ω of the limit cycle and the coefficient ε as an independent parameter μ=εω, an equivalent equation is obtained and then solved by Liao’s homotopy analysis method. The frequency ω is deduced as a function of μ and δ. This function provides us with an algebraic equation for ω, according to which we have an analytical approximation for the frequency. Numerical examples show that the attained approximation is very accurate. More importantly, the results are uniformly valid for all positive values of ε.     

RESPONSE OF PARAMETRICALLY EXCITED DUFFING-VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.     

Exit cycling for the Van der Pol oscillator and quasipotential calculations

We discuss the phenomenon of cycling for noise induced escape to a unstable periodic orbit. The presence of cycling is shown to follow from qualitative properties of two quasipotential functions. A method of numerically evaluating these quasipotential functions is described, and applied to the Van der Pol oscillator as an example. Figures resulting from these calculations reveal that nonconvergent cycling of exit measures does occur for the Van der Pol example.     

Study of mixed-mode oscillations in a parametrically excited van der Pol system

This paper investigates the effects of slowly varying parametric excitation on the dynamics of van der Pol system. Periodic bifurcation delay behaviors are exhibited when the parametric excitation slowly passes through Hopf bifurcation value of the controlled van der Pol system. The first bifurcation delay behavior relies on initial conditions, while the bifurcation delay behaviors that follow the first one are immune to initial conditions. These bifurcation delay behaviors result in a hysteresis loop between the spiking attractor and the rest state, which is responsible for the generation of mixed-mode oscillations. Then an approximate calculation for the number of spikes in each cluster of repetitive spiking of mixed-mode oscillations is explored based on bifurcation delay behaviors. Theoretical results agree well with numerical simulations.     

Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback

In this paper, we consider the traditional Van der Pol oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity, which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of center manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.     

The Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling

We investigate the dynamics of a system of twovan der Pol oscillators with delayed velocity coupling.We use the method of averaging to reduce the problem to the studyof a slow-flow in three dimensions.We study the steady state solutions of this slow-flow, with specialattention given to the bifurcations accompanying their change innumber and stability. We compare these stability results with numericalintegration of the original equations and show that the two sets of resultsare in excellent agreement under certain parameter restrictions.Our interest in this system is due to its relevance to coupled laseroscillators.     

The Response of a Parametrically Excited van der Pol Oscillator to a Time Delay State Feedback

We investigate the parametric resonance of a van der Pol oscillator under state feedback control with a time delay. Using the asymptotic perturbation method, we obtain two slow-flow equations on the amplitude and phase ofthe oscillator. Their fixed points correspond to a periodic motion forthe starting system and we show parametric excitation-response andfrequency-response curves. We analyze the effect of time delay andfeedback gains from the viewpoint of vibration control and use energyconsiderations to study the existence and characteristics of limit cycles of the slow-flow equations. A limit cycle corresponds to a two-periodmodulated motion for the van der Pol oscillator. Analytical results areverified with numerical simulations. In order to exclude the possibilityof quasi-periodic motion and to reduce the amplitude peak of theparametric resonance, we find the appropriate choices for the feedbackgains and the time delay.