Bifurcations of a parametrically excited oscillator with strong nonlinearity

A parametrically excited oscillator with strong nonlinearity, including van der Pol and Duffing types, is studied for static bifurcations. The applicable range of the modified Lindstedt－Poincaré method is extended to 1/2 subharmonic resonance systems. The bifurcation equation of a strongly nonlinear oscillator, which is transformed into a small parameter system, is determined by the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analysed.     

对称双弹簧振子受迫、有阻尼横振动的混沌行为

对受周期外力驱动的对称双弹簧振子进行了研究,建立了系统的动力学方程,用线性稳定性分析方法讨论了平衡点附近邻域的稳定性,利用数值计算并结合多种分析方法,求解非线性方程和判断解的性质.通过改变系统参数,画出时域图、相图及分岔图等.计算分析和数值实验发现,这个简单的力学系统存在十分丰富的动力学行为(分岔、混沌).理论分析和数值实验结果一致.     

谐和与噪声联合作用下Duffing振子的安全盆分叉与混沌

研究了软弹簧Duffing振子在确定性谐和外力和有界随机噪声联合作用下，系统安全盆的侵蚀和混沌现象.推导出系统的随机Melnikov过程，根据Melnikov过程在均方意义上出现简单零点的条件给出了系统出现混沌的临界值，然后用数值模拟方法计算了系统的安全盆分叉点.结果表明，由于随机扰动的影响，系统的安全盆分叉点发生了偏移，并且使得混沌容易发生. 关键词： Duffing振子 安全盆 分叉 混沌     

A conjecture on route to chaos in a hard Duffing oscillator by homoclinic entanglement

Response of a weakly damped hard Duffing oscillator, which apparently does not admit any homoclinic entanglement, is analysed. A possibility of homoclinic entanglement is conjectured that may help to understand onset of chaotic behaviour under simple harmonic excitation.     

Analytical approximations to a generalized forced damped complex Duffing oscillator:multiple scales method and KBM approach

In this investigation, some different approaches are implemented for analyzing a generalized forced damped complex Duffing oscillator, including the hybrid homotopy perturbation method(H-HPM), which is sometimes called the Krylov-Bogoliubov-Mitropolsky(KBM) method and the multiple scales method(MSM). All mentioned methods are applied to obtain some accurate and stable approximations to the proposed problem without decoupling the original problem. All obtained approximations are discussed graphic...     

Effect of rectified and modulated sine forces on chaos in Duffing oscillator

The effect of rectified and modulated sine forces on the onset of horseshoe chaos is studied both analytically and numerically in the Duffing oscillator. With single force analytical threshold condition for the onset of horseshoe chaos is obtained using the Melnikov method. The Melnikov threshold curve is drawn in a parameter space. For the rectified sine wave, onset of cross-well asymptotic chaos is observed just above the Melnikov threshold curve. For the modulus of sine wave long time transient motion followed by a periodic attractor is realized. The possibility of controlling of chaos by the addition of second modulated force is then analyzed. Parametric regimes where suppression of horseshoe chaos occurs are predicted analytically and verified numerically. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.     

有界噪声与谐和激励联合作用下Duffing-Rayleigh振子的Melnikov混沌

研究了有界噪声与谐和激励作用下的Duffing-Rayleigh振子的动力学行为.首先运用随机Melnikov过程方法得到系统出现混沌的条件,结果表明随着非线性阻尼参数的增加系统会从混沌运动到周期运动,随着Wiener过程强度参数的增加,系统由混沌进入周期的临界幅值会先递增后不变.最后,用两类数值方法即最大Lyapunov指数与Poincare截面验证了上述结果. 关键词： 有界噪声 随机Melnikov过程 混沌运动 周期运动     

有界噪声和谐和激励联合作用下一类非线性系统的混沌研究

研究一类有界噪声和谐和激励联合作用下的非线性系统，首先用多尺度方法将该系统约化，针对约化后的平均系统，利用随机Melnikov过程方法结合均方值准则导出随机系统可能产生混沌运动的临界条件，结果表明在一定的参数范围内，随着Weiner过程强度参数值的增大，混沌的临界激励幅值先递减继而递增. 同时，用两类数值方法即最大Lyapunov指数法和Poincare截面法验证了解析结果. 关键词： 有界噪声 多尺度 随机Melnikov过程 混沌     

Duffing振子系统周期解的唯一性与精确周期信号的获取方法

研究Duffing振子系统的周期解的唯一性与精确周期信号的获取方法.应用定性分析方法,获得了一类Duffing振子系统具有唯一周期解的必要条件,同时也得到了一类更广泛的非线性周期系统的周期解的唯一性.在一定条件下,给出了Duffing振子系统精确周期信号的获取方法。     

阻尼谐振子到简谐振子的变换

在牛顿力学、拉格朗日力学和哈密顿力学3种形式中,利用变量变换将阻尼谐振子变换成简谐振子.     

Quantum chaos production in anharmonic oscillator under train of Gaussian pulses

The regimes of quantum chaos for anharmonic oscillator under train of Gaussian pulses are studied depending on the pulse parameters. The presented results are derived in the semiclassical limit within the framework of Poincaré section and Lyapunov exponent. In the quantum limit the regimes of chaos are considered on the basis of the Wigner function. Relations between system parameters are found when chaotic dynamics appears at mesoscopic level of oscillator excitation numbers.     

Mixed-mode oscillations and chaos from a simple second-order oscillator under weak periodic perturbation

In this study, we propose a remarkably simple oscillator that exhibits extremely complicated behaviors. The second-order nonautonomous differential equation discussed in this Letter is considered to be one of the simplest dynamics that can produce mixed-mode oscillations (MMOs) and chaos. Our model uses a Bonhoeffer-van der Pol (BVP) oscillator under weak periodic perturbation. The parameter set of the BVP equation is chosen such that a focus and a relaxation oscillation coexist when no perturbation is applied. Under weak periodic perturbation, various types of MMOs and chaos with remarkably complicated waveforms are observed.     

Stochastic resonance of a damped oscillator with frequency fluctuation driven by a periodic external force

<正>Considering a damped linear oscillator model subjected to a white noise with an inherent angular frequency and a periodic external driving force,we derive the analytic expression of the first moment of output response,and study the stochastic resonance phenomenon in a system.The results show that the output response of this system behaves as a simple harmonic vibration,of which the frequency is the same as the external driving frequency,and the variations of amplitude with the driving frequency and the inherent frequency present a bona fide stochastic resonance.     

多频谐和与噪声作用下Flickering振子的安全盆侵蚀与混沌

研究了催化反应Flickering振子在多频率确定性谐和外力和有界随机噪声联合作用下,系统安全盆的侵蚀和混沌现象.将Melnikov方法推广到包含有限多个频率外力和随机噪声联合作用的情形,推导出了系统的随机Melnikov过程,根据Melnikov过程在均方意义上出现简单零点的条件给出了系统出现混沌的临界值,然后用数值模拟方法计算了系统的安全盆分岔点.结果表明,由于随机扰动的影响,系统的安全盆分岔点发生了偏移,并且使得混沌容易发生.同时证明,激励频率数目的增加扩大了参数空间上的混沌区域,也使得安全盆分岔提 关键词： 多频率激励 Flickering振子 安全盆 混沌     

Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator

This paper investigates the possibility of controlling horseshoe and asymptotic chaos in the Duffing-van der Pol oscillator by both periodic parametric perturbation and addition of second periodic force. Using Melnikov method the effect of weak perturbations on horseshoe chaos is studied. Parametric regimes where suppression of horseshoe occurs are predicted. Analytical predictions are demonstrated through direct numerical simulations. Starting from asymptotic chaos we show the recovery of periodic motion for a range of values of amplitude and frequency of the periodic perturbations. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.     

Pitchfork bifurcation and vibrational resonance in a fractional-order Duffing oscillator

The pitchfork bifurcation and vibrational resonance are studied in a fractional-order Duffing oscillator with delayed feedback and excited by two harmonic signals. Using an approximation method, the bifurcation behaviours and resonance patterns are predicted. Supercritical and subcritical pitchfork bifurcations can be induced by the fractional-order damping, the exciting high-frequency signal and the delayed time. The fractional-order damping mainly determines the pattern of the vibrational resonance. There is a bifurcation point of the fractional order which, in the case of double-well potential, transforms vibrational resonance pattern from a single resonance to a double resonance, while in the case of single-well potential, transforms vibrational resonance from no resonance to a single resonance. The delayed time influences the location of the vibrational resonance and the bifurcation point of the fractional order. Pitchfork bifurcation is the necessary condition for the double resonance. The theoretical predictions are in good agreement with the numerical simulations.