Sinusoidal synchronization of a Duffing oscillator with a chaotic pendulum

In this Letter, analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator are presented. From the analytical conditions, the invariant domain of such sinusoidal synchronization is determined, and the control parameter map of the synchronicity is achieved. Under specific parameters, numerical illustrations of the partial and full sinusoidal synchronizations of the controlled Duffing oscillator with the pendulum are carried out for a better understanding of such synchronization under specific function constraints. The methodology presented in this Letter is applicable to synchronizations with any specific function constraints.     

Role of asymmetries in the chaotic dynamics of the double-well Duffing oscillator

Duffing oscillator driven by a periodic force with three different forms of asymmetrical double-well potentials is considered. Three forms of asymmetry are introduced by varying the depth of the left-well alone, location of the minimum of the left-well alone and above both the potentials. Applying the Melnikov method, the threshold condition for the occurrence of horseshoe chaos is obtained. The parameter space has regions where transverse intersections of stable and unstable parts of left-well homoclinic orbits alone and right-well orbits alone occur which are not found in the symmetrical system. The analytical predictions are verified by numerical simulation. For a certain range of values of the control parameters there is no attractor in the left-well or in the right-well.     

Noise-induced reentrant transition of the stochastic Duffing oscillator

We derive the exact bifurcation diagram of the Duffing oscillator with parametric noise thanks to the analytical study of the associated Lyapunov exponent. When the fixed point is unstable for the underlying deterministic dynamics, we show that the system undergoes a noise-induced reentrant transition in a given range of parameters. The fixed point is stabilised when the amplitude of the noise belongs to a well-defined interval. Noisy oscillations are found outside that range, i.e., for both weaker and stronger noise.Received: 20 February 2004, Published online: 20 April 2004PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) - 05.45.-a Nonlinear dynamics and nonlinear dynamical systems     

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

The(un)forced(un)damped parametric pendulum oscillator(PPO) is analyzed analytically and numerically using some simple,effective,and more accurate techniques.In the first technique,the ansatz method is employed for analyzing the unforced damped PPO and for deriving some optimal and accurate analytical approximations in the form of angular Mathieu functions.In the second approach,some approximations to(un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz metho...     

Experiments on the bifurcation behaviour of a forced nonlinear pendulum

We investigate a mechanical system (forced nonlinear torsion pendulum). The state diagram is given as function of both the external driving frequency and the damping parameter. A bifurcation diagram is measured showing period doubling, chaos and periodic windows. The results are in qualitative agreement with the recent theory.     

The Duffing oscillator in the low-friction limit: Theory and analog simulation

By a joint use of theory and analog simulation the low-friction regime of the Duffing oscillator is explored. In the weak-temperature case it is shown that the low-friction regime, in turn, must be divided in two well-distinct subregimes. In the former one, characterized by the friction ranging from 2 ₀ ( ₀ is the harmonic frequency) up to a lower bound _T, the method of statistical linearization applies and a continued fraction procedure (CFP) generated by the Zwanzig-Mori projection techniques is shown to provide correct information for both the renormalized frequency of the oscillator and the corresponding line shape. The latter subregime, characterized by the friction ranging from = 0 to = _T , is fraught with the complete breakdown of the statistical linearization method. The CFP is shown to provide an incorrect picture of the line shape while suggesting a novel mean field approximation which is then proven analytically via an alternative method of calculation. This method consists of expressing the system in a form reminiscent of the model of Kubo's stochastic oscillator. By using this alternative approach we are in a position to account for the residual linewidth which is shown by the analog experiment to survive for 0.     

Duffing系统的主-超谐联合共振

非线性动力系统极易发生共振,在多频激励下可能发生联合共振或组合共振,目前关于非线性系统的主-超谐联合共振的研究少见报道.本文以Duffing系统为对象,研究系统在主-超谐联合共振时的周期运动和通往混沌的道路.应用多尺度法得到系统的近似解析解,并利用数值方法对解析解进行验证,结果吻合良好.基于Lyapunov第一方法得到...     

A complicated Duffing oscillator in the surface-electrode ion trap

The oscillation coupling and different nonlinear effects are observed in a single trapped ⁴⁰Ca⁺ ion confined in our home-built surface-electrode trap (SET). The coupling and the nonlinearity are originated from the high-order multipole potentials, such as hexapole and octopole potentials, due to different layouts and the fabrication asymmetry of the SET. We solve a complicated Duffing equation with coupled oscillation terms by the multiple-scale method, which fits the experimental values very well. Our investigation in the SET helps for exploring multi-dimensional nonlinearity using currently available techniques and for suppressing instability of qubits in quantum information processing with trapped ions.     

A Volterra series approximation to the coherence of the Duffing oscillator

Following previous work on computing approximate frequency response functions for the Duffing oscillator under white Gaussian excitation, an approximation is obtained here for the coherence function. A Padé approximation of order (1,1) is made for the asymmetric Duffing oscillator (i.e. with non-zero quadratic term), and an approximation of order (2,2) is made for the symmetric (no quadratic term) oscillator. The analytical results are shown to give excellent qualitative agreement with numerical simulation. However, in quantitative terms, the approximations underpredict the coherence distortion as is consistent with the low-order truncations of the Volterra series.     

The application of Duffing oscillator in characteristic signal detection of early fault

This paper presents a study of application of Duffing oscillator for extracting the features of early mechanical failure signal. By analysis of global solutions and bifurcation set of Duffing equation, we conclude that the bifurcation threshold, which corresponds to the maximum orbit outside the homoclinic orbit of Duffing equation, can be used to detect weak signal, such as the characteristic signal of early machinery fault. Therefore, the Duffing oscillator is suitable to be a model for weak signal detection, although some relative aspects for practical uses should be considered carefully. In this paper, the strategies for testing whether the weak signal is exists and how to estimate its amplitude are discussed in detail. Moreover, an example is presented here to demonstrate the utility of this method by analyzing the early rub-impact signal appearing in a rotor. The results show that the method is effective for early detection of fault described by specific periodic-motion components such as rub-impact fault of rotor in aeroengine.     

Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

This paper presents space-time numerical simulation and validation of analytical predictions for the finite-amplitude forced dynamics of suspended cables. The main goal is to complement analytical and numerical solutions, accomplishing overall quantitative/qualitative comparisons of nonlinear response characteristics. By relying on an approximate, kinematically non-condensed, planar modeling, a simply supported horizontal cable subject to a primary external resonance and a 1:1, or 1:1 vs. 2:1, internal resonance is analyzed. To obtain analytical solution, a second-order multiple scales approach is applied to a complete eigenfunction-based series of nonlinear ordinary-differential equations of cable damped forced motion. Accounting for both quadratic/cubic geometric nonlinearities and multiple modal contributions, local scenarios of cable uncoupled/coupled responses and associated stability are predicted, based on chosen reduced-order models. As a cross-checking tool, numerical simulation of the associated nonlinear partial-differential equations describing the dynamics of the actual infinite-dimensional system is carried out using a finite difference technique employing a hybrid explicit-implicit integration scheme. Based on system control parameters and initial conditions, cable amplitude, displacement and tension responses are numerically assessed, thoroughly validating the analytically predicted solutions as regards the actual existence, the meaningful role and the predominating internal resonance of coexisting/competing dynamics. Some methodological aspects are noticed, along with a discussion on the kinematically approximate versus exact, as well as planar versus non-planar, cable modeling.     

Approximation of the Duffing oscillator frequency response function using the FPK equation

Although a great deal of work has been carried out on nonlinear structural dynamic systems under random excitation, there has been a comparatively small amount of this work concentrating on the calculation of the quantities commonly measured in structural dynamic tests. Perhaps the most fundamental of these quantities is the frequency response function (FRF). A number of years ago, Yar and Hammond took an interesting approach to estimating the FRF of a Duffing oscillator system which was based on an approximate solution of the Fokker-Planck-Kolmogorov equation. Despite reproducing the general features of the statistical linearisation estimate, the approximation failed to show the presence of the poles at odd multiples of the primary resonance which are known to occur experimentally. The current paper simply extends the work of Yar and Hammond to a higher-order of approximation and is thus able to show the existence of a third ‘harmonic’ in the FRF. A comparison is made with previous work where an approximation to the FRF was computed using the Volterra series.     

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.     

一维受迫谐振子量子运动的经典特性

对经典一维受迫谐振子量子化,求解量子化后体系的时间演化算符.应用相空间准概率分布函数,研究了体系的量子特性.研究结果表明,初始为真空态,经过时间演化,系统波函数是一个二维高斯波包;波包中心的振幅和相位受到作用力的调制,成为调幅、调相波,波包中心的运动与经典受迫谐振子的运动形式相同.     

Multistability and chaotic beating of Duffing oscillators suspended on an elastic structure

We consider the dynamics of a number of externally excited chaotic oscillators suspended on an elastic structure. We show that for the given conditions of oscillations of the structure, initially uncorrelated chaotic oscillators become periodic and synchronous in clusters. In the periodic regime, we have observed multistability as two or four different attractors coexist in each cluster. A mismatch of the excitation frequency in the oscillators leads to the beating-like behaviour. We argue that the observed phenomena are generic in the parameter space and independent of the number of oscillators and their location on the elastic structure.     

受迫振子的对称性与守恒量

讨论受迫振子体系的对称性和守恒量,导出了其含时位势的具体形式,给出了相应的对称变换算符,并讨论了线谐振子体系的含时守恒量.     

Experiments on periodic and chaotic motions of a parametrically forced pendulum

An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic route to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.     

Experimental investigation of the response of a harmonically excited hard Duffing oscillator

A single degree-of-freedom torsional vibratory system, which constitutes a third-order dissipative dynamical system, has been fabricated as a mechanical analogue of hard Duffing equation with strong nonlinearity. The forced response of the system reveals complicated and chaotic motion at low frequency regime. Besides usual jump phenomenon, unpredictable jump phenomenon with two and three coexisting periodic attractors is also observed.      

Fluxon dynamics on the circular Josephson oscillator

A hamiltonian perturbation theory is developed for the perturbed sine-Gordon equation with periodic boundary conditions modelling the Josephson ring oscillator. Stationary fluxon velocities are determined as function of length, loss and bias parameters.     

The crossover from classical to quantum behavior in Duffing oscillator based on quantum state diffusion

The classical Duffing oscillator is a dissipative chaotic system, and so there is not a definite Hamiltonian. We quantize the Duffing oscillator on the basis of quantum state diffusion (QSD) which is a formulation for open quantum systems and a useful tool for analyzing nonlinear problems and classical limits. We can define a ‘Lyapunov exponent’, which corresponds to the classical one in the proper limit, and investigate the behavior of the system by varying the Planck constant effectively. We show that there exists a critical stage, where the behavior of the system crosses over from classical to quantum one.