Duffing System Research Articles

Identification of Nonlinear Systems with Additive External Action

The problem of identification of nonautonomous nonlinear systems of ordinary differential equations is considered. An algorithm for reconstructing a system under harmonic external excitation is proposed. This algorithm is optimal from the point of view of computing time. An example of the solution of the system identification problem for the generalized Duffing system in the regime of deterministic chaos is considered.

A new kind of metal detector based on chaotic oscillator

The sensitivity of a metal detector greatly depends on the identification ability to weak signals from the probe. In order to improve the sensitivity of metal detectors, this paper applies the Duffing chaotic oscillator to metal detectors based on its characteristic which is very sensitive to weak periodic signals. To make a suitable Duffing system for detectors, this paper computes two Lyapunov characteristics exponents of the Duffing oscillator, which help to obtain the threshold of the Duffing system in the critical state accurately and give quantitative criteria for chaos. Meanwhile, a corresponding simulation model of the chaotic oscillator is made by the Simulink tool box of Matlab. Simulation results shows that Duffing oscillator is very sensitive to sinusoidal signals in high frequency cases. And experimental results show that the measurable diameter of metal particles is about 1.5mm. It indicates that this new method can feasibly and effectively improve the metal detector sensitivity.

Probabilistic response and analysis for a vibro-impact system driven by real noise

In this paper, a new stochastic averaging method is proposed to derive the stationary probabilistic response of vibro-impact system subject to real noise excitation, and the effects of real noise and the restitution coefficient on the response are discussed in detail. According to the size relationship between the system energy and the potential energy of the impact position $$\varDelta $$ , the motions of the unperturbed vibro-impact system are divided into two types: the free motion with no energy loss and the impact motion with energy loss. Based on those relationships, a new method that combines a transformation and the Fourier expansion scheme by introducing, the mean drift and diffusion coefficients of the averaged Ito stochastic differential equation for the two types of motions are obtained, which are depended on the bandwidth and the noise density of real noise. The probability density function of stationary response of vibro-impact system is derived through solving the associated Fokker–Planck–Kolmogorov equation. Finally, the effects of the restitution coefficient and the noise intensity of real noise in different bandwidths are detailedly analyzed through an example. The main results on probabilistic response analysis of vibro-impact Duffing system are obtained to illustrate the proposed stochastic averaging method, and Monte Carlo simulation method is also conducted to show the effectiveness of the proposed method.

Approximately analytical technique for random response of LuGre friction system

This manuscript concentrates on the random response evaluation of dynamic friction system, in which the dynamic friction is described by LuGre friction law while the random excitation by Gaussian white noise. The definitive solution can be obtained by solving two coupled equations: a second-order stochastic differential equation with respect to the system displacement and an auxiliary first-order nonsmooth differential equation with respect to the internal friction state. The auxiliary differential equation is first integrated to derive the approximate relation between the friction force and current system states under the assumption of slowly varying amplitude, and then the original problem with definitive solution is approximated by a modified second-order stochastic differential equation with respect to the system displacement and amplitude. With the application of the equivalent non-linearization technique, an equivalent nonlinear system is constructed and solved to obtain the analytical expression of the stationary probability density of system displacement and velocity, which is the approximately analytical solution for the modified system and the original frictional system. Numerical results on the Duffing system with LuGre friction validate the effectiveness and accuracy of the proposed approximately analytical technique.

Location identification of closed crack based on Duffing oscillator transient transition

Abstract The existence of a closed micro-crack in plates can be detected by using the nonlinear harmonic characteristics of the Lamb wave. However, its location identification is difficult. By considering the transient nonlinear Lamb under the noise interference, we proposed a location identification method for the closed crack based on the quantitative measurement of Duffing oscillator transient transfer in the phase space. The sliding short-time window was used to create a window truncation of to-be-detected signal. And then, the periodic extension processing for transient nonlinear Lamb wave was performed to ensure that the Duffing oscillator has adequate response time to reach a steady state. The transient autocorrelation method was used to reduce the occurrence of missed harmonic detection due to the random variable phase of nonlinear Lamb wave. Moreover, to overcome the deficiency in the quantitative analysis of Duffing system state by phase trajectory diagram and eliminate the misjudgment caused by harmonic frequency component contained in broadband noise, logic operation method of oscillator state transition function based on circular zone partition was adopted to establish the mapping relation between the oscillator transition state and the nonlinear harmonic time domain information. Final state transition discriminant function of Duffing oscillator was used as basis for identifying the reflected and transmitted harmonics from the crack. Chirplet time-frequency analysis was conducted to identify the mode of generated harmonics and determine the propagation speed. Through these steps, accurate position identification of the closed crack was achieved.

Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing

We study the Duffing equation with parametric excitation and single external forcing and obtain abundant dynamical behaviors of bifurcations and chaos. The criteria of chaos of the Duffing equation under periodic perturbation are obtained through the Melnikov method. And the existence of chaos of the averaged system of the Duffing equation under the quasi-periodic perturbation [Formula: see text] (where [Formula: see text] is not rational relative to [Formula: see text]) and [Formula: see text] is shown, but the existence of chaos of averaged system of the Duffing equation cannot be proved when [Formula: see text],[Formula: see text]7–15, whereas the occurrence of chaos in the original system can be shown by numerical simulation. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit some new complex dynamical behaviors, including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent diagrams, phase portraits and Poincaré maps. We find a large chaotic region with some solitary period parameter points, a large period and quasi-period region with some solitary chaotic parameter points, period-doubling to chaos and chaos to inverse period-doubling, nondense curvilinear chaotic attractor, nonattracting chaotic motion, nonchaotic attracting set, fragmental chaotic attractors. Almost chaotic motion and almost nonchaotic motion appear through adjusting the parameters of the Duffing system, which can be taken as a strategy of chaotic control or a strategy of chaotic motion to nonchaotic motion.

Predicting the bounds of large chaotic systems using low-dimensional manifolds.

Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D) embedded in high-dimensional (high-D) phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely.

Detection of micro-cracks using nonlinear lamb waves based on the Duffing-Holmes system

To solve the problem that weak nonlinear Lamb waves caused by micro-cracks are often buried in background noises, a Duffing-Holmes system was used to enhance the weak nonlinear Lamb waves and the crack size was quantitatively characterized using Lyapunov exponent(LE). The frequency and amplitude critical threshold of the external driving force were determined according to the center frequency of excitation signal and the Poincaré map of the Duffing-Holmes oscillator, respectively. After periodic extending and filtering, the Lamb waves were input to the Duffing-Holmes system, and then the nonlinear second harmonic was identified according to the phase trajectory and Poincaré map. Based on the phase space reconstruction of Duffing system output, the maximal LE was calculated to characterize the magnitude of the second harmonic. Based on S0 Lamb wave mode, the simulations on models with different micro-crack sizes showed that the Duffing-Holmes system could accurately identify the structural micro-cracks under strong noise interference. Compared with the traditional acoustic nonlinearity parameter β′, the damage index defined here had a better linear relationship with the crack size, and the micro-cracks could be accurately quantified. The proposed method has obvious advantages for the detection of micro-crack defects under noise interference, and can greatly improve the sensitivity of nonlinear Lamb waves detection.

Semi‐global cooperative output regulation of a class of nonlinear uncertain multi‐agent systems under switching networks

SummaryIn this paper, we consider the semi‐global cooperative output regulation problem for a class of nonlinear uncertain multi‐agent systems under switching networks. At first, we study the nonadaptive case when the exosystem has no parametric uncertainties and construct a common Lyapunov function to achieve the output regulation for general switching connected networks. Next, we study the case when the exosystem contains some parametric uncertainties. To solve the problem, we establish a stability result for a class of time‐varying system, which is then used in the design of distributed adaptive internal model‐based control. Then we construct multiple Lyapunov functions for the switching signal with its average dwell time lower bounded by a given constant. Throughout the paper, we treat the closed‐loop multi‐agent system from the viewpoint of singular perturbation. In fact, the singular perturbation‐based method provides an effective tool to handle the multi‐agent system under switching networks. Finally, we give numerical simulations based on Duffing systems and flexible manipulator systems to illustrate the effectiveness of our method. Copyright © 2017 John Wiley & Sons, Ltd.

Onset of colored-noise-induced chaos in the generalized Duffing system

The effects of colored noise, red noise and green noise, on the onset of chaos are investigated theoretically and confirmed numerically in the generalized Duffing system with a fractional-order deflection. Analytical predictions concerning the chaotic thresholds in the parameter space are derived by using the stochastic Melnikov method combined with the mean-square criterion. To qualitatively confirm the analytical results, numerical simulations obtained from the mean largest Lyapunov exponent are used as test beds. We show that colored noise can induce chaos, and the effects for the case of red noise on the onset of chaos differ from those for the case of green noise. The most noteworthy result of this work is the formula, which relates the chaotic thresholds among red, green and white noise, holds for noise-induced chaos in the Duffing system. We also show that Gaussian white noise can induce chaos more easily than colored noise.

Homoclinic and heteroclinic chaos in nonlinear systems driven by trichotomous noise**Project supported by the National Natural Science Foundation of China (Grant No. 11672231), the Natural Science Foundation of Shaanxi Province, China (Grant No. 2016JM1010), the Fundamental Research Funds for the Central Universities, China (Grant No. 3102015ZY073), and the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical

The homoclinic and heteroclinic chaos in nonlinear systems subjected to trichotomous noise excitation are studied. The Duffing system and the Josephson-junction system are taken for example to calculate the corresponding amplitude thresholds for the onset of chaos on the basis of the stochastic Melnikov process with the mean-square criterion. It is shown that the amplitude threshold for the onset of chaos can be adjusted by changing the internal parameters of trichotomous noise, thereby inducing or suppressing chaotic behaviors in the two systems driven by trichotomous noise. The effects of trichotomous noise on the systems are verified by vanishing the mean largest Lyapunov exponent and demonstrated by phase diagrams and time histories.

Stationary response of stochastically excited nonlinear systems with continuous-time Markov jump

An approximate method for predicting the stationary response of stochastically excited nonlinear systems with continuous-time Markov jump is proposed. By using the stochastic averaging method, the original system is reduced to one governed by a 1D averaged Ito equation for the total energy with the Markov jump process as parameter. A Fokker-Planck- Kolmogorov (FPK) equation is then deduced, from which the approximate stationary probability density of the response of the original system is obtained for different jump rules. To illustrate the effectiveness of the proposed method, a stochastically excited Markov jump Duffing system is worked out in detail.

PARAMETER IDENTIFICATION OF NONLINEAR VIBRATION SYSTEMS BASED ON THE HILBERT-HUANG TRANSFORM

Parameters of time-varying damping systems and nonlinear vibration systems, such as the Duffing and Van der Pol vibration systems, were identified by the Hilbert-Huang transform (HHT) in this study. The nonlinear vibration signal was first decomposed into free vibration and forced vibration components by the empirical mode decomposition. Subsequently, the amplitude envelope and the instantaneous frequency of the decomposed components were computed by the empirical envelop method. Damping ratios and other parameters of the nonlinear vibration systems were finally identified by using the least square method. Compared with the results from the wavelet analysis, the proposed method is proven to be more effective through numerical simulations of three typical nonlinear vibration systems and is featured by a higher level of accuracy.

Period-Doubling Bifurcation of Stochastic Fractional-Order Duffing System via Chebyshev Polynomial Approximation

Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.

Chaotic Motion in Forced Duffing System Subject to Linear and Nonlinear Damping

This paper investigates the chaotic motion in forced Duffing oscillator due to linear and nonlinear damping by using Melnikov technique. In particular, the critical value of the forcing amplitude of the nonlinear system is calculated by Melnikov technique. Further, the top Lyapunov exponent of the nonlinear system is evaluated by Wolf’s algorithm to determine whether the chaotic phenomenon of the nonlinear system actually occurs. It is concluded that the chaotic motion of the nonlinear system occurs when the forcing amplitude exceeds the critical value, and the linear and nonlinear damping can generate pronounced effects on the chaotic behavior of the forced Duffing oscillator.

Sequential Parameter Identification of Fractional-Order Duffing System Based on Differential Evolution Algorithm

Using the dynamic properties of fractional-order Duffing system, a sequential parameter identification method based on differential evolution optimization algorithm is proposed for the fractional-order Duffing system. Due to the step by step parameter identification method, the dimension of parameter identification of each step is greatly reduced and the search capability of the differential evolution algorithm has been greatly improved. The simulation results show that the proposed method has higher convergence reliability and accuracy of identification and also has high robustness in the presence of measurement noise.

Attractive Domain of Nonlinear Systems with Time-delayed Feedback Control and Time-delay Effects

The dynamics of system with time delay resides in an infinite dimensional state space. It is difficult to determine the domain of attraction of the steady state response in the infinite dimensional state space. A definition of the domain of attraction in the physical space is introduced, which is intuitive, and furthermore, it is numerically computable under the hypothesis that the time delay is short. The effect of time delay on the domain of attraction of the popular Duffing system is considered. It is found that the time delay has significant influence on the structure of domain of attraction. The width of attractive domain can be enlarged with a small time delay, however, the domain of attraction shrinks when the time delay increases and approach to the stability boundary.

Response of Duffing Oscillator with Time Delay Subjected to Combined Harmonic and Random Excitations

This paper aims to investigate the stationary probability density functions of the Duffing oscillator with time delay subjected to combined harmonic and white noise excitation by the method of stochastic averaging and equivalent linearization. By the transformation based on the fundamental matrix of the degenerate Duffing system, the paper shows that the displacement and the velocity with time delay in the Duffing oscillator can be computed approximately in non-time delay terms. Hence, the stochastic system with time delay is transformed into the corresponding stochastic non-time delay equation in Ito sense. The approximate stationary probability density function of the original system can be found by combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function. The response of Duffing oscillator is investigated. The analytical results are verified by numerical simulation results.

Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings

This paper investigates bursting dynamics of the Duffing system with multiple-frequency external forcings, in which novel bursting patterns (i.e., the so-called turnover-of-hysteresis-induced bursting patterns) can be observed. Typically, distinct oscillations are observed in the quasi-static processes of these bursting patterns. We show that the oscillations appear in the quasi-static processes because the equilibrium hysteresis curve of the fast subsystem becomes the one with twists and turns, and this forms a new route to bursting, which we call turnover of hysteresis. Besides, we investigate the effects of forcing frequencies and amplitudes on the turnover-of-hysteresis-induced bursting. Our study shows that the three frequency components observed in the bursting are decided by the forcing frequencies, while the transition of the bursting depends on the forcing amplitudes.

A frequency up-converting harvester based on internal resonance in 2-DOF nonlinear systems

This paper reports the design and experimental testing of a novel frequency up- converting piezoelectric energy harvester. The harvester is firstly approximated as a 2-degree- of-freedom cubic nonlinear system instead of the general Duffing systems. A 1:3 internal resonance innovatively applied in the frequency up-conversion approach is thoroughly investigated. Finally, the theoretical dynamic model confirmed by the experimental results clearly shows the effect of the frequency up-conversion.