Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

This paper aims to investigate the stochastic response of the van der Pol(VDP) oscillator with two kinds of fractional derivatives under Gaussian white noise excitation.First,the fractional VDP oscillator is replaced by an equivalent VDP oscillator without fractional derivative terms by using the generalized harmonic balance technique.Then,the stochastic averaging method is applied to the equivalent VDP oscillator to obtain the analytical solution.Finally,the analytical solutions are validated by numerical results from the Monte Carlo simulation of the original fractional VDP oscillator.The numerical results not only demonstrate the accuracy of the proposed approach but also show that the fractional order,the fractional coefficient and the intensity of Gaussian white noise play important roles in the responses of the fractional VDP oscillator.An interesting phenomenon we found is that the effects of the fractional order of two kinds of fractional derivative items on the fractional stochastic systems are totally contrary.     

强Van der Pol振子的谐波与修正算法

引入谐波平衡近似解的符号运算与同伦延拓法,获得了强Van der Pol振子的解析近似极限环与稳定响应.为提高其近似精度,给出了谐波解的修正算法,并与数值模拟比较.结果表明,该算法非常精确可靠,是研究强非线性动力学系统的有效分析方法.     

耦合的van der Pol振子的极限环幅值控制

设计反馈控制器，对一类耦合的van der Pol振子的极限环幅值进行控制. 用近似解析方法求出了控制系统的幅值的控制方程, 得到了控制参数与极限环幅值的函数关系, 使系统的振幅能按需求得到有效地调整. 通过数值计算绘制了在不同控制参数下, 系统响应的时间历程曲线和极限环. 近似解析方法计算得到的结果与数值计算进行了比较，两者是符合的. 这一方法也可以推广应用到其他耦合的van der Pol振子.     

Energy-based analysis of frequency entrainment described by van der Pol and phase-locked loop equations

This paper analyzes frequency entrainment described by van der Pol and phase-locked loop (PLL) equations. The PLL equation represents the dynamics of a PLL circuit that appear in typical phase-locking phenomena. These two equations describe frequency entrainment by a periodic force. The entrainment originates from two different types of limit cycles: libration for the van der Pol equation and rotation for the PLL one. To explore the relationship between the geometry of limit cycles and the mechanism of entrainment, we investigate the entrainment using an energy balance relation. This relation is equivalent to the energy conservation law of dynamical systems with dissipation and input terms. We show response curves for the dc component, harmonic amplitude, phase difference, and energy supplied by a periodic force. The obtained curves indicate that the entrainments for the two equations have different features of supplied energy, and that the entrainment for the PLL equation possibly has the same mechanism as does the regulation of the phase difference for the van der Pol equation.     

Compacton and solitary pattern solutions for nonlinear dispersive KdV-type equations involving Jumarie?s fractional derivative

In this Letter, the fractional variational iteration method using He?s polynomials is implemented to construct compacton solutions and solitary pattern solutions of nonlinear time-fractional dispersive KdV-type equations involving Jumarie?s modified Riemann-Liouville derivative. The method yields solutions in the forms of convergent series with easily calculable terms. The obtained results show that the considered method is quite effective, promising and convenient for solving fractional nonlinear dispersive equations. It is found that the time-fractional parameter significantly changes the soliton amplitude of the solitary waves.     

Periodic solution of the generalized Rayleigh equation

The periodic solutions of a strongly cubic nonlinear oscillator whose motion is described with the generalized Rayleigh equation are studied. Approximate analytic solving methods are introduced. A new method based on homotopy and averaging is developed to determine the limit cycle motion. The obtained analytical solutions are compared with those calculated by the elliptic harmonic balance method with generalized Fourier series and Jacobian elliptic functions. Three types of cubic nonlinearity are considered: the coefficients of the linear and cubic terms are positive, the coefficient of the linear term is positive and that of the cubic term is negative and the opposite case. Comparisons of the analytical solution and numerical solution, obtained by using the Runge-Kutta method, are illustrated with examples.     

Equivalent linearization finds nonzero frequency corrections beyond first order

We show that the equivalent linearization technique, when used properly, enables us to calculate frequency corrections of weakly nonlinear oscillators beyond the first order in nonlinearity. We illustrate the method by applying it to the conservative anharmonic oscillators and the nonconservative van der Pol oscillator that are respectively paradigmatic systems for modeling center-type oscillatory states and limit cycle type oscillatory states. The choice of these systems is also prompted by the fact that first order frequency corrections may vanish for both these types of oscillators, thereby rendering the calculation of the higher order corrections rather important. The method presented herein is very general in nature and, hence, in principle applicable to any arbitrary periodic oscillator.     

Direct quadrature method of moments solution of the Fokker–Planck equation

The direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the Fokker–Planck equation corresponding to stochastic nonlinear dynamical systems. The theoretical details of the solution procedure are first presented. The method is then used to solve Fokker–Planck equations for both 1D and 2D (noisy van der Pol oscillator) processes which possess nonlinear stochastic differential equations. Higher-order moments of the stationary solutions are computed and prove to be very accurate when compared to analytic (1D process) and Monte Carlo (2D process) solutions.     

van der Pol方程分析研究的新结果

依据等效小参量法原理,用主振荡包含两个谐波的符号分析算法求解van der Pol方程,在强非线性时获得的周期解准确度高于扰动法、KBM法、多尺度法、谐波平衡法及主振荡为一个谐波的等效小参量法,且求解过程简单,易于计算机编程．     

An open plus nonlinear closed loop control of chaotic oscillators

An open plus nonlinear closed loop control law is presented for chaotic oscillations described by a set of nonautonomous second-order ordinary differential equations.It is proven that the basins of entrainment are global when the right-hand sides of the equations are given by arbitrary polynomical functions.The forece Duffing oscillator and the forced van der Pol oscillator are treated as numerical examples to demonstrate the applications of the method.     

Solution procedure of residue harmonic balance method and its applications

This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems. In this solution procedure, no small parameter is assumed. The harmonic residue of balance equation is separated in two parts at each step. The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement. The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again. Three kinds of different differential equations involving general, fractional and delay ordinary differential systems are given as numerical examples respectively. Highly accurate limited cycle frequency and amplitude are captured. The results match well with the exact solutions or numerical solutions for a wide range of control parameters. Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems. Moreover, the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.     

Stationary and dynamic solutions of stochastic relaxation oscillators

Two different approaches are proposed to obtain explicit solutions for stochastic relaxation oscillator problems in the weak noise limit. The first method generalizes the idea of the cumulant expansion. It does not presuppose an analytical treatment of the deterministic motion. It is however restricted to the discussion of stationary situations. In the second method an adiabatic elimination of irrelevant variables allows for the computation of time dependent solutions. It can be carried through only if the deterministic limit cycle is known analytically. As special examples the stationary solutions of the stochastic van der Pol oscillator and time dependent solutions of a simple one dimensional model system have been obtained.This article is an excerpt from a dissertation presented at TH Darmstadt, Darmstädter Dissertation D17This work was performed within a program of the Sonderforschungsbereich 185 Darmstadt-Frankfurt, FRG     

Effect of noise and perturbations on limit cycle systems

This paper demonstrates that the influence of noise and of external perturbations on a nonlinear oscillator can vary strongly along the limit cycle and upon transition from limit cycle to stationary point behaviour. For this purpose we consider the role of noise on the Bonhoeffer-van der Pol model in a range of control parameters where the model exhibits a limit cycle, but the parameters are close to values corresponding to a stable stationary point. Our analysis is based on the van Kampen approximation for solutions of the Fokker-Planck equation in the limit of weak noise. We investigate first separately the effect of noise on motion tangential and normal to the limit cycle. The key result is that noise induces diffusion-like spread along the limit cycle, but quasistationary behaviour normal to the limit cycle. We then describe the coupled motion and show that noise acting in the normal direction can strongly enhance diffusion along the limit cycle. We finally argue that the variability of the system's response to noise can be exploited in populations of nonlinear oscillators in that weak coupling can induce synchronization as long as the single oscillators operate in a regime close to the transition between oscillatory and excitatory modes.     

Solution of a mixed parity nonlinear oscillator: Harmonic balance

The method of harmonic balance is used to calculate first-order approximations to the periodic solutions of a mixed parity nonlinear oscillator. First, the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction. Then the two auxiliary equations, where the restoring force functions are odd, are solved by using a first harmonic term (without a constant). Therefore, we obtain the two approximate solutions to the mixed parity nonlinear oscillator. One is expressed in terms of the exact amplitude in the negative direction, the other in terms of the approximate amplitude. These solutions are more accurate than the second approximate solution obtained by the Lindstedt–Poincaré method for large amplitudes.     

Response control for the externally excited van der Pol oscillator with non-local feedback

A non-local control force is introduced in such a way to obtain a third-order nonlinear differential equation (jerk dynamics) and to control nonlinear vibrations in an externally excited van der Pol oscillator. Two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of solutions are derived and subsequently the performance of the control strategy is investigated. Excitation amplitude–response and frequency–response curves are shown. In certain cases when the excitation amplitude is very low an approximate analytic solution corresponding to a modulated two-period quasi-periodic motion can be obtained for the uncontrolled system. Uncontrolled and controlled systems are compared and the appropriate choices for the feedback gains are found in order to reduce the amplitude peak of the response and to exclude the possibility of quasi-periodic motion. Numerical simulation confirms the validity of the new method.     

Nonlinear oscillations of the FitzHugh-Nagumo equations under combined external and two-frequency parametric excitations

The continuous FitzHugh-Nagumo (FHN for short) model is transformed into modified van der Pol oscillator with asymmetry under external and two-frequency parametric excitations. At the first, the dependence of the solutions on a combined external and two-frequency parametric stimulus forcing is investigated. By using the multiple scale method, ranges of applied current and/or parametric forcing in which nonlinear oscillations are observed are described. Second, when the multiple scale method cannot be used, we numerically prove that in the modified van der Pol oscillator with asymmetry under external and two-frequency parametric excitations, chaos and periodic solution depending on the combination between different frequencies of the model should appear. We also show that the amplitude of the oscillations can be reduced or increased. To do this, we perform the study of the FHN model by choosing a range of parameters exhibiting Hopf bifurcation and two qualitative different regimes in phase portrait.     

太阳强迫厄尔尼诺/南方涛动充电振子模型的Hopf分岔与混沌

本文研究了一类太阳强迫的厄尔尼诺/南方涛动(ENSO)充电振子数理模型,通过数学变换将此ENSO振子方程组变换为有周期强迫项的van der Pol-Duffing方程,利用谐波平衡法定性分析得到此ENSO系统发生Hopf分岔的条件并做简单数值模拟,结果发现随着强迫作用增大,11年周期太阳循环强迫的ENSO系统经历准周期、倍频锁相到混沌的过程.     

Spectral Analysis of Stochastic Phase Lockings and Stochastic Bifurcations in the Sinusoidally Forced van der Pol Oscillator with Additive Noise

Noise effects on the phase lockings and bifurcations in the sinusoidally forced van der Pol relaxation oscillator are investigated. Deterministic (noise-free) one-dimensional Poincaré mapping is extended to the iteration of the operator defined by a stochastic kernel function. Stochastic phase lockings and bifurcations are analyzed in terms of the density evolution by the operator. In particular, a new method which uses spectra (eigenvalues and eigenfunctions) of the operator to analyze stochastic bifurcations intensively is proposed.     

Determination of limit cycles for strongly nonlinear oscillators

An innovative approach to finding limit cycles is proposed and illustrated on the van der Pol equation. The technique developed in this Letter is similar to the Ritz's method in variational theory. The present theory can be applied to not only weakly nonlinear equations, but also strongly nonlinear ones, and the obtained results are valid for the whole solution domain.     

Coherence and stochastic resonance in a birhythmic van der Pol system

We consider the response to uncorrelated noise and harmonic excitation of a birhythmic van der Pol-type oscillator. This system, as opposed to the standard van der Pol oscillator, is characterized by two stable orbits. The noisy oscillator can be analytically mapped, with the technique of stochastic averaging, onto an ordinary bistable system with a bistable (quasi)potential. The birhythmic oscillator can also be numerically characterized through the diagnostics of coherent resonance and the signal-to-noise-ratio. The analysis shows the presence of noise-induced coherent states, influenced by the different time scales of the oscillator.