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 Ito equation is deduced. We obtained the Fokker–Planck–Kolmogorov equation connected to the averaged Ito equation and solved it to yield the approximate stationary response of the system. The analytical solution is confirmed by using Monte Carlo simulation.

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.

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.

Impulsive Control for T-S Fuzzy Model Based Chaotic Systems Consisting of Van der Pol Oscillators Coupled to Linear Oscillators

This paper proposes an impulsive control scheme for chaotic systems consisting of Van der Pol oscillators coupled to linear oscillators （VDPL） based on their Takagi-Sugeno （T-S） fuzzy models. A T-S fuzzy model is utilized to represent the chaotic VDPL system. By using comparison method, a general asymptotical stability criterion by means of linear matrix inequality （LMI） is derived for the T-S fuzzy model of VDPL system with impulsive effects. The simulation results demonstrate the effectiveness of the proposed scheme.

Phase synchronization and synchronization frequency of two-coupled van der Pol oscillators with delayed coupling

In this paper, phase synchronization and the frequency of two synchronized van der Pol oscillators with delay coupling are studied. The dynamics of such a system are obtained using the describing function method, and the necessary conditions for phase synchronization are also achieved. Finding the vicinity of the synchronization frequency is the major advantage of the describing function method over other traditional methods. The equations obtained based on this method justify the phenomenon of the synchronization of coupled oscillators on a frequency either higher, between, or lower than the highest, in between, or lowest natural frequency of the aggregate oscillators. Several numerical examples simulate the different cases versus the various synchronization frequency delays.

Time Delay Induced Oscillation: An Example on a Class of <i>n</i>Coupled Van Der Pol Oscillators Model with Delays

In this paper, a class of n coupled van der Pol oscillator model with delays is considered. By employing an analysis approach, some sufficient conditions to guarantee the existence of stability and oscillations for themodel are obtained. Examples are provided to demonstrate the results.

Application of the Homotopy Perturbation Method to Nonlinear Heat Conduction and Fractional Van der Pol Damped Nonlinear Oscillator

In this paper, a powerful analytical method, called He’s homotopy perturbation method is applied to obtaining the approximate periodic solutions for some nonlinear differential equations in mathematical physics via Van der Pol damped non-linear oscillators and heat transfer. Illustrative examples reveal that this method is very effective and convenient for solving nonlinear differential equations. Comparison of the obtained results with those of the exact solution, reveals that homotopy perturbation method leads to accurate solutions.

Application of Van der Pol oscillator screening for peripheral arterial disease in patients with diabetes mellitus

This paper proposes a Van der Pol (VDP) oscillator screening for peripheral arterial disease (PAD) in patients with diabetes mellitus. The long-term elevated blood sugar levels produce a high risk of peripheral neuropathy and peripheral vascular disease, especially in the foot of a diabetic. Early detection is important, in order to prevent foot problems for diabetic patients with PAD. Photoplethysmography (PPG) is a non-invasive method for the detection of blood volume changes in peripheral arteries. Because of changes in the resistance-compliance, the rise time and transit time for the PPG signals increase and change in their shape are highly correlated with PAD severity. In this study, the Burg autoregressive (AR) method is used to determine the characteristic frequencies of the right-and left-side PPG signals. For bilateral frequency spectra, the VDP oscillator uses asynchronous signals. This produces damped sinusoidal responses and the oscillation overshoot (OS) is an approximating function only of the damped factor. This index is used to estimate the degree of PAD, including normal the condition and diabetic patients with PAD. The results show that the proposed method is efficient and more accurate in the estimation of PAD.

Collective Behavior of a Ring of Van Der Pol Oscillators under Gaussian and Random Coupling Schemes

The collective behavior of a ring of coupled identical van der Pol oscillators is numerically studied in this work. Constant, gaussian and random distributions of the coupling parameter along the ring are considered. Three values of the oscillators constant are assumed in order to cover from quasilinear to nonlinear dynamic performance. Single and multiple coupled frequencies are obtained using power spectra of the long term time series. Phase portraits are obtained from numerical simulations, and the coupled behavior is analyzed, compared and discussed.

Nonlinear Control of Chaotic Forced Duffing and Van der Pol Oscillators

This paper discusses a novel technique and implementation to perform nonlinear control for two different forced model state oscillators and actuators. The paper starts by discussing the Duffing oscillator which features a second order non-linear differential equation describing complex motion whereas the second model is the Van der Pol oscillator with non-linear damping. A first order actuator is added to both models to expand on the chaotic behavior of the oscillators. In order to control the system without comprising linearization, Lyapunov non-linear control was used. A control Lyapunov function was tailored to the system. This led to improved maneuverability of the controller and the performance of the overall system. The controller was found to be highly efficient in system tracking and had swift response time. Simulations were performed on both the uncontrolled and controlled cases. Both simulation results ultimately confirmed the effectiveness of the proposed controller.

Application of Homotopy Perturbation Method and Parameter Expanding Method to Fractional Van der Pol Damped Nonlinear Oscillator

In this study, homotopy perturbation method and parameter expanding method are applied to the motion equations of two nonlinear oscillators. Our results show that both the (HPM) and (PEM) yield the same results for the nonlinear problems. In comparison with the exact solution, the results show that these methods are very convenient for solving nonlinear equations and also can be used for strong nonlinear oscillators.

PRINCIPAL RESPONSE OF VAN DER POL-DUFFING OSCILLATOR UNDER COMBINED DETERMINISTIC AND RANDOM PARAMETRIC EXCITATION

The principal resonance of Van der Pol-Duffing oscillator to combined deterministic and random parametric excitations is investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied. Jumps were shown to occur under some conditions. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analysis were verified by numerical results.

参数激励Duffing-Van der Pol振子的动力学响应及反馈控制

研究了Duffing_VanderPol振子的主参数共振响应及其时滞反馈控制问题·依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关·通过调整反馈增益和时滞,可以使不稳定的零解变得稳定·非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生·数值仿真的结果验证了理论分析的正确性·

Duffing-van der Pol振子的时滞反馈控制研究

研究了Duffing-van der Pol振子在一类时滞反馈控制下零解的稳定性问题以及极限环的振幅和稳定性问题。依平均法和对时滞反馈控制项泰劳展开的截断得到的平均方程表明,零解的稳定性除与原方程中线性项的系数有关外,只与线性反馈有关,与非线性反馈无关。通过调整线性反馈的增益和时滞,可以使不稳定的零解变得稳定。零解发生Hopf分岔导致的周期解的振幅除与原方程中非线性项的系数有关外,与线性反馈和非线性反馈均有关。通过调整反馈增益和时滞,不仅可以控制极限环的振幅,还可以抑制极限环的产生。此外,根据平均方程还容易发现反馈时滞对系统动力学行为的影响具有周期性。数值仿真的结果验证了理论分析的正确性。

Van der Pol振荡器的模拟仿真

利用运算放大器和乘法器进行电路设计,对早期利用电子管实现的Van der Pol振荡器利用现代集成电路加以实现。文中还利用OrCAD PSpice对设计的电路进行了模拟,得到了Van der Pol振荡器输出信号的波形图,并利用文本文件作为OrCAD PSpice和Matlab之间的接口,将OrCAD PSpice仿真得到的波形在Matlab中进行处理,得到Van der Pol振荡器两个状态变量的相图,并以此说明了Van der Pol振荡器所具有的丰富的非线性动力学特性。

谐和与随机噪声联合作用下VanderPol-Duffing振子的参数主共振

研究了VanderPol_Duffing振子在谐和与随机噪声联合激励下的参数主共振响应和稳定性问题· 用多尺度法分离了系统的快变项 ,并求出了系统的最大Liapunov指数和稳态概率密度函数 ,还分析了失稳、分叉和跳跃现象 ,讨论了系统的阻尼项、非线性项。

具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

研究了具有时滞耦合的n个van der Pol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2∶5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2∶5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.

一类时变分岔问题与Duffing-Van der Pol振子

用量级平衡方法分析一类时变分岔问题，得出正反两个方向的分岔转迁滞后且具有对称性．将Ｄｕｆｆｉｎｇ－Ｖａｎ ｄｅｒ Ｐｏｌ振动方程转化为对该时变分岔的反馈控制问题，得出在适当的条件下， Ｄｕｆｆｉｎｇ－Ｖａｎ ｄｅｒ Ｐｏｌ振子的相图是一动态滞后环．

多频激励下vander Pol系统主参数-组合共振

应用多尺度法分析了 van der Pol系统受参数激励和多频强迫激励联合作用下的主参数 -组合共振 ,求得了稳态响应的分岔方程 ,应用奇异性理论进行分析 ,得到了系统稳态响应的转迁集和分岔图 ,并分析了原系统参数对普适开折参数的影响。研究表明 ,该系统的稳态响应为一叉型分岔 ,激励幅值 F1 ,F2 和阻尼 μ对普适开折参数的影响很大 ,通过调整 F1 ,F2 和 μ可以很方便地控制解的分岔特性。

基于Duffing与Van der Pol强耦合系统的随机共振研究及轴承故障诊断

微弱信号检测是一种重要的技术,已经被广泛用于许多领域。与众多微弱信号处理方法不同的是,随机共振(SR)可以利用噪声实现对特征信号的检测。Van der Pol振子具有丰富的动力学性质,被广泛运用在混沌检测中。基于此,将Duffing与Van der Pol振子进行线性耦合,并添加微分项作为耦合系统的反馈以加强耦合强度,构建一种Duffing与Van der Pol强耦合系统,实验发现,该系统具有丰富的SR现象,且通过广义时间尺度变换,可以实现任意特征频率下的微弱信号检测。实验显示,强耦合系统比一般耦合系统具有更好的稳定性;在三值噪声背景下,该系统的输出平均信噪比增益(MSNRI)随着三值噪声状态值的变化呈现不同形式的对称分布;在被控系统中,随着阻尼系数的逐渐增强,输出响应将变得逐渐平滑;另外,在控制系统中,当阻尼系数很小时,系统输出将会出现二次共振现象,随着阻尼系数的增大,次级共振峰将逐渐减小直至消失,一级共振峰将逐渐增强,这是阱间共振完全取代了阱内,阱间联合共振的缘故。最后,借助遗传算法(GA)自适应参数寻优,与传统双稳以及耦合Duffing系统对比发现,所提系统具有更高的输出MSNRI。经测试,该系统对实际轴承故障信号也具有良好的检测效果,并且与传统双稳系统相比能更显著的消除边频信号干扰。