Generalized Van Der Pol Oscillator Research Articles

Effect of colored noise on precursors of thermoacoustic instability in model gas turbine combustors

In this work, we numerically investigate the dynamics of a prototypical thermoacoustic system, the generalized Van der Pol oscillator, in the presence of additive noise of varying color (correlation time) and intensity while the system undergoes supercritical and subcritical Hopf bifurcation. We specifically investigate the influence of noise color on trends in the coherence factor and the Hurst exponent in the subthreshold region to assess their reliability as instability precursors. The Hurst exponent is found reliable only for correlation times much larger than the time scale of the instability while the coherence factor is found to be reliable for the entire range of noise color investigated. These inferences are found to hold for both supercritical and subcritical bifurcation cases.

Study of aerodynamic damping of a crosswind-excited circular cylinder and its application to full-scale flexible structures

The accurate assessment of aerodynamic damping is essential when estimating the impact of aero-elastic effects on crosswind-excited line-like structures. This study aims to determine the crosswind aerodynamic damping of a circular cylinder by curve-fitting to the response probability density function. The generalized Van der Pol oscillator model is employed to depict the nonlinearity associated with amplitude-dependent aerodynamic damping. The initial section of this study presents the theoretical foundation of the identified approach used in this research. Subsequently, the efficacy of the identification methodology is validated by means of a stochastic simulation of the crosswind response of a circular cylinder. Moreover, identification of nonlinear aerodynamic damping is further accomplished by the utilization of an aero-elastic model wind tunnel test. The identified nonlinear aerodynamic damping is then utilized to determine the response statistical quantities for a total of 16 chimneys. The determined aerodynamic damping provides a satisfactory assessment of the peak response, which is crucial for design considerations.

Analysis of the stationary probability density of a generalized and bistable van der Pol system excited by colored noise

The stochastic P-bifurcation behavior of bi-stability in a generalized van der Pol oscillator with the fractional damping under colored noise and thermal excitation is investigated. Firstly, using the principle of minimal mean square error and lin-earization method, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order van der Pol system. Secondly, the system amplitude stationary probability density function is obtained by the stochastic averaging, and then based on the singularity theory, the critical parametric con-ditions for the system amplitude stochastic P-bifurcation are found. Finally, the types of the stationary probability density function of the system amplitude are qualitatively analyzed in each area divided by the transition set curves. The con-sistency between the analytical results and the numerical results acquired from Monte-Carlo simulation also testifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

New Stability Results for Periodic Solutions of Generalized Van der Pol Oscillator via Second Bogolyubov’s Theorem

A certain class of nonlinear differential equations representing a generalized Van der Pol oscillator is proposed in which we study the behavior of the existing solution. After using the appropriate variables, the first Levinson’s change converts the equations into a system with two equations, and the second converts these systems into a Lipschitzian system. Our main result is obtained by applying the Second Bogolubov’s Theorem. We established some integrals, which are used to compute the average function of this system and arrive at a new general condition for the existence of an asymptotically stable unique periodic solution. One of the well-known results regarding asymptotic stability appears, owing to the Second Bogolubov’s Theorem, and the advantage of this method is that it can be applied not only in the periodic dynamical systems, but also in non-almost periodic dynamical systems.

Hopf Bifurcation, Approximate Periodic Solutions and Multistability of Some Nonautonomous Delayed Differential Equations

Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields.

ANALYSIS OF APPROXIMATE SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS IN THEIR GENERAL FORM WITH GENERALIZED ORDER

In this paper, we dedicate our study on the approximate solutions of van der Pol equation in their general form. First, we prove the approximate analytic solutions to this equation by different perturbation methods, simple perturbation method (SPM), Lindstedt-Poincaré method (PLM) and Averaging method (AM). Then we compare these approximations with each other and with the exact solution. Second, we introduce a new form of generalized Van Der Pol oscillator with fractional-order derivatives. Which is analyzed through phase portraits, Poincaré maps and analytic solutions, we use numerical simulation to illustrate the behavior of the fractional order system.

Limit cycle periods in damage detection

This paper introduces a damage detection scheme based on changes in the period of limit cycles (LCs) realized by nonlinear output feedback. The parameterized feedback law is selected to make the dynamic equilibrium of the actuated coordinate approximate that of a generalized van der Pol oscillator and potential model-related difficulties are avoided by fixing the parameters from experimental results. It is found that not all sensor/actuator arrangements are suitable for generating a LC in closed loop, so a methodology to determine which arrangements are viable is presented. Numerical results show that LC periods are adequately stable with respect to noise and system changes from environmental effects and have sensitivity to localized damage that is often orders of magnitude larger than those of the modal frequencies.

Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator

Abstract In this paper, nonlinear dynamics study of a RLC series circuit modeled by a generalized Van der Pol oscillator is investigated. After establishing a new general class of nonlinear ordinary differential equation, a forced Van der Pol oscillator subjected to an inertial nonlinearity is derived. According to the external excitation strength, harmonic, subharmonic and superharmonic oscillatory states are obtained using the multiple time scales method. Bifurcation diagrams displayed by the model for each system parameter are performed numerically through the fourth-order Runge–Kutta algorithm.

Stochastic transition behaviors in a generalized Van der Pol oscillator with fractional damping under Gaussian white noise

The stochastic P-bifurcation behavior of bi-stability in a generalized Van der Pol oscillator with a fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary probability density function is obtained by stochastic averaging. Then according to the singularity theory, the critical parametric conditions for the sys-tem amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary probability density function curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte-Carlo simulation verifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

The Dynamics Behavior of Coupled Generalized van der Pol Oscillator with Distributed Order

In this paper, we presented different behaviors such as chaotic and hyperchaotic of the generalized van der Pol oscillator with distributed order. We introduced the parameter intervals of these behaviors by computing the Lyapunov exponents of the oscillator, which is a good test for classifying the dynamical systems’ solutions. The active control approach with the Laplace transform technique was used to realize the antisynchronization and control of the proposed oscillator. Finally, numerical investigations have been carried out on the dynamics of the proposed oscillator to verify the reliability of our analytical results.

Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

This paper investigates the stochastic P-bifurcation behavior of bistability in a generalized Van der Pol system with fractional time-delay feedback under additive and multiplicative Gaussian white noise excitations. First, using the minimal mean square error principle, the fractional derivative is found to be equivalent to a linear combination of damping and restoring forces, and the original system is transformed into an equivalent integer-order system. Second, the stationary probability density function of system amplitude is obtained by stochastic averaging, and according to singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the different types of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each region divided by the transition set curves. The consistency between the analytical solutions and the Monte Carlo simulation results verifies the theoretical analysis in this paper. The method used in this paper can directly guide the design of the fractional-order controller to adjust the response of the system.

Interaction of high-frequency and low-frequency oscillations in the synchronized oscillator

Theoretical analysis of the oscillation synchronization under conditions of simultaneous low-frequency and high-frequency external impact on the oscillator has been conducted using an example of the generalized Van der Pol oscillator. It was shown that the interaction of high-frequency and low-frequency oscillations resulted in the appearance of additional regions of synchronization as a result of three-frequency interaction between the frequencies of such interaction and the natural frequency of oscillator. For the case of three-frequency interaction, characteristics of synchronous oscillations were determined and compared with those for the case of primary tone synchronization of oscillator.

An Application of the Optimal Homotopy Asymptotic Method to Generalized Van der Pol Oscillator

In this paper we illustrate the application of an alternative of the Optimal Homotopy Asymptotic Method (OHAM) to nonlinear generalized van der Pol oscillator. The obtained results proved a very fast convergence and validate this approach, which is found to be reliable and easy to use. Two numerical examples are developed in order to emphasize the accuracy and efficiency of the proposed approach.

Time-delayed feedback control of coherence resonance near subcritical Hopf bifurcation: theory versus experiment.

Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation, we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay, either suppression or enhancement of coherence resonance can be achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit.

Oscillators with a power-form restoring force and fractional derivative damping: Application of averaging

In this paper free oscillators with a power-form restoring force and with a fractional derivative damping term are considered. An analytical approach based on the averaging method is adjusted to derive analytical expressions for the amplitude and phase of oscillations. Effects of the fractional-order derivative on the amplitude and frequency of oscillations are discussed in several examples, including a generalized van der Pol oscillator, purely nonlinear oscillators and a linear oscillator.

On the motion of a generalized van der Pol oscillator

A generalized van der Pol oscillator is considered, with positive real power nonlinearities in the restoring and damping force, including fractional powers. An analytical approach based on the Krylov–Bogoliubov method is adjusted to derive analytical expressions for the amplitude of a limit cycle for small values of the damping coefficient. These expressions are also derived for some integer power nonlinearities in the equation of motion and the results obtained compared with the existing results from the literature. Relaxation oscillations are studied for larger values of the damping coefficient. Matched asymptotic expansions are used and the influence of the powers of the restoring and damping force on the period of these oscillations is investigated. It is shown that not only can the period increase with the damping power, but it can also have a decreasing trend for some cases and the condition for this to hold is obtained.

A hyperbolic perturbation method for determining homoclinic solution of certain strongly nonlinear autonomous oscillators

A hyperbolic perturbation method is presented for determining the homoclinic solution of certain strongly nonlinear autonomous oscillators of the form x ¨ + c 1 x + c 2 x 2 = ε f ( μ , x , x ˙ ) in which hyperbolic functions can be employed instead of the usual periodic functions in the perturbation procedure. The generalized van der Pol oscillator in which f ( μ , x , x ˙ ) = ( μ + μ 1 x - μ 2 x 2 ) x ˙ is studied. To illustrate the accuracy of the present method, its predictions are compared with those of Runge-Kutta method.

Bifurcations of a generalized van der Pol oscillator with strong nonlinearity

A generalized van der Pol oscillator with parametric excitation is studied for its bifurcation. On the basis of the MLP method, we enable a strongly nonlinear system to be transformed into a small parameter system. The bifurcation response equation of a 1/2 subharmonic resonance system is determined by the multiple scales method. According to the singularity theory, the bifurcation of equilibrium points is analyzed. The stability of the zero solution is researched by the eigenvalues of the variational matrix and the bifurcation sets are constructed in various regions of the parameter plane.

A Generalized KBM Method for Strongly Nonlinear Oscillators with Slowly Varying Parameters

A generalized Krylov-Bogoliubov-Mitropolsky (KBM) method is extended for the study of strongly nonlinear oscillators with slowly varying parameters. The asymptotic amplitude and phase are derived and then the asymptotic solutions of arbitrary order are obtained theoretically. Cubic nonlinear oscillators with polynomial damping are studied in detail. Three examples are considered: a generalized Van der Pol oscillator, a Rayleigh equation and a pendulum with variable length. Comparisons are also made with numerical solutions to show the efficiency and accuracy of the present method.

AMPLITUDE CONTROL OF LIMIT CYCLE IN VAN DER POL SYSTEM

The feedback controllers are designed to modify the amplitude of limit cycles in van der Pol oscillator and generalized van der Pol oscillator. Bifurcation control equations of weakly nonlinear systems are obtained by using the method of multiple scales. Gain-amplitude curves of controlled systems are drawn. Based on numerical study, the brief results of controlling amplitude of limit cycle are given for strongly nonlinear system.