Noise-induced chaos and basin erosion in softening Duffing oscillator

It is common for many dynamical systems to have two or more attractors coexist and in such cases the basin boundary is fractal. The purpose of this paper is to study the noise-induced chaos and discuss the effect of noises on erosion of safe basin in the softening Duffing oscillator. The Melnikov approach is used to obtain the necessary condition for the rising of chaos, and the largest Lyapunov exponent is computed to identify the chaotic nature of the sample time series from the system. According to the Melnikov condition, the safe basins are simulated for both the deterministic and the stochastic cases of the system. It is shown that the external Gaussian white noise excitation is robust for inducing the chaos, while the external bounded noise is weak. Moreover, the erosion of the safe basin can be aggravated by both the Gaussian white and the bounded noise excitations, and fractal boundary can appear when the system is only excited by the random processes, which means noise-induced chaotic response is induced.     

Active control with delay of catastrophic motion and horseshoes chaos in a single well Duffing oscillator

In this paper, the control of escape and Melnikov chaos of an harmonically excited particle from a catastrophic (unbounded) single well φ⁴ potential is considered. In the linear limit, the range of the control gain parameter leading to good control is obtained and the effect of time delays on the control force is taken into account. The approximate critical external forcing amplitudes for catastrophe and chaos are obtained by using the energy and Melnikov methods. The control efficiency is found by analysing the behaviour of the external critical forcing amplitude of the controlled system as compared to that of the uncontrolled system.     

Inducing or suppressing chaos in a double-well Duffing oscillator by time delay feedback

The chaotic behavior of a double-well Duffing oscillator with both delayed displacement and velocity feedbacks under a harmonic excitation is investigated. By means of the Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. The analytical results reveal that for negative feedback the presence of time delay lowers the threshold and enlarges the possible chaotic domain in parameter space; while for positive feedback the presence of time delay enhances the threshold and reduces the possible chaotic domain in parameter space, which are further verified numerically through Poincare maps of the original system. Furthermore, the effect of the control gain parameters on the chaotic motion of the original system is studied in detail.     

Numerical simulation of the transition to chaos in a dissipative Duffing oscillator with two-frequency excitation

A mathematical modeling technique is proposed for oscillation chaotization in an essentially nonlinear dissipative Duffing oscillator with two-frequency excitation on an invariant torus in ?². The technique is based on the joint application of the parameter continuation method, Floquet stability criteria, bifurcation theory, and the Everhart high-accuracy numerical integration method. This approach is used for the numerical construction of subharmonic solutions in the case when the oscillator passes to chaos through a sequence of period-multiplying bifurcations. The value of a universal constant obtained earlier by the author while investigating oscillation chaotization in dissipative oscillators with single-frequency periodic excitation is confirmed.     

Effects on the bifurcation and chaos in forced Duffing oscillator due to nonlinear damping

In this communication, the two-well Duffing oscillator with non-linear damping term proportional to the power of velocity is considered. We mainly focus our attention on how the damping exponent affects the global dynamical behaviour of the oscillator. In particular, we obtain analytically the threshold condition for the occurrence of homoclinic bifurcation using Melnikov technique and compare the results with the computational results. We also identify the major route to chaos and the regions of the 2D parameter space (consists of external forcing amplitude and damping coefficient) corresponding to the various types of asymptotic dynamics under linear (viscous or friction like) and nonlinear (drag like) damping. We also attempt to analyze how the basins of attraction patterns change with the introduction of nonlinear damping. We also present our analysis for the physically less-interesting cases where damping is proportional to the 3rd and 4th power of velocity for the sake of generalizing our findings and establishing firm conclusion.     

Antiperiodic oscillations in a forced Duffing oscillator

Regularity has always been attributed to periodicity. However, there has been a spurt of interest in another unique type of regularity called anitperiodicity. In this paper we have presented results of antiperiodic oscillations obtained from a forced duffing equation with negative linear stiffness wherein the increase in the number of peaks in antiperiodic oscillation with the forcing strength has been observed. Similarity function has been used to identify the antiperiodic oscillation and further the bifurcation diagram has been plotted and stability analysis of the fixed points have been carried out to understand its dynamics. An analog electronic circuit governed by the forced Duffing equation has been designed and developed to investigate the dynamics of the antiperiodic oscillations. The circuit is quite robust and stable to enable the comparison of its analog output with the numerically simulated data. Power spectrum analysis obtained by fast Fourier transform has been corroborated using a nonlinear statistical technique called rescale range analysis method. By this technique we have estimated the Hurst exponents and detected the coherent frequencies present in the system.     

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

On the stochastic response of a fractionally-damped Duffing oscillator

A numerical method is presented to compute the response of a viscoelastic Duffing oscillator with fractional derivative damping, subjected to a stochastic input. The key idea involves an appropriate discretization of the fractional derivative, based on a preliminary change of variable, that allows to approximate the original system by an equivalent system with additional degrees of freedom, the number of which depends on the discretization of the fractional derivative. Unlike the original system that, due to the presence of the fractional derivative, is governed by non-ordinary differential equations, the equivalent system is governed by ordinary differential equations that can be readily handled by standard integration methods such as the Runge–Kutta method. In this manner, a significant reduction of computational effort is achieved with respect to the classical solution methods, where the fractional derivative is reverted to a Grunwald–Letnikov series expansion and numerical integration methods are applied in incremental form. The method applies for fractional damping of arbitrary order α (0 < α < 1) and yields very satisfactory results. With respect to its applications, it is worth remarking that the method may be considered for evaluating the dynamic response of a structural system under stochastic excitations such as earthquake and wind, or of a motorcycle equipped with viscoelastic devices on a stochastic road ground profile.     

Exact solution of the cubic-quintic Duffing oscillator

In this paper, we derive the exact solution of the cubic-quintic Duffing oscillator based on the use of Jacobi elliptic functions. We also showed that the exact angular frequency of this cubic-quintic Duffing equation is given in terms of the complete elliptic integral of the first kind.     

Temporal and spectral responses of a softening Duffing oscillator undergoing route-to-chaos

Because nonlinear responses are oftentimes transient and consist of complex amplitude and frequency modulations, linearization would inevitably obscure the temporal transition attributable to the nonlinear terms, thus also making all inherent nonlinear effects inconspicuous. It is shown that linearization of a softening Duffing oscillator underestimates the variation of the frequency response, thereby concealing the underlying evolution from bifurcation to chaos. In addition, Fourier analysis falls short of capturing the time evolution of the route-to-chaos and also misinterprets the corresponding response with fictitious frequencies. Instantaneous frequency along with the empirical mode decomposition is adopted to unravel the multi-components that underlie the bifurcation-to-chaos transition, while retaining the physical features of each component. Through considering time and frequency responses simultaneously, a better understanding of the particular Duffing oscillator is achieved.     

Controlling bifurcation and chaos of a plastic impact oscillator

A two-degree-of-freedom plastic impact oscillator is considered. Based on the analysis of sticking and non-sticking impact motions of the system, we introduce a three-dimensional impact Poincaré map with dynamical variables defined at the impact instants. The plastic impacts complicate the dynamic responses of the impact oscillator considerably. Consequently, the piecewise property and singularity are found to exist in the three-dimensional map. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after impact, and the singularity of the map is generated via the grazing contact of two masses and the instability of their corresponding periodic motions. The nonlinear dynamics of the plastic impact oscillator is analyzed by using the Poincaré map. The simulated results show that the dynamic behavior of this system is very complex under parameter variation, varying from different types of sticking or non-sticking periodic motions to chaos. Suppressing bifurcation and chaotic-impact motions is studied by using an external driving force, delay feedback and damping control law. The effectiveness of these methods is demonstrated by the presentation of examples of suppressing bifurcations and chaos for the plastic impact oscillator.     

T-controllable domains of the soft Duffing oscillator

In the time-optimal, origin-seeking control problem for both the hard and soft Duffing oscillator, anomalous features known as separatrices or lines of indifference are present. These are characterized by an ambiguity in the defined control policy. In this note, we show briefly the effect of the separatrix on the controllable domain for the soft spring.     

Bifurcation analysis of a simple 3D oscillator and chaos synchronization of its coupled systems

Tama?evi?ius et al. proposed a simple 3D chaotic oscillator for educational purpose. In fact the oscillator can be implemented very easily and it shows typical bifurcation scenario so that it is a suitable training object for introductory education for students. However, as far as we know, no concrete studies on bifurcations or applications on this oscillator have been investigated. In this paper, we make a thorough investigation on local bifurcations of periodic solutions in this oscillator by using a shooting method. Based on results of the analysis, we study chaos synchronization phenomena in diffusively coupled oscillators. Both bifurcation sets of periodic solutions and parameter regions of in-phase synchronized solutions are revealed. An experimental laboratory of chaos synchronization is also demonstrated.     

Synchronization of a periodically forced Duffing oscillator with a periodically excited pendulum

In this paper, the analytical conditions for a periodically forced Duffing oscillator synchronized with a chaotic pendulum are developed through the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domains are developed. For a better understanding of synchronization of two different dynamical systems, the partial and full synchronizations of the Duffing oscillator with the chaotic pendulum are presented for illustrations. The control parameter map is developed from the analytical conditions. Under special parameters, the two systems can be fully and partially synchronized. Since the forced pendulum has librational and rotational chaotic motions, the periodically forced Duffing oscillator can be synchronized only with the librational chaotic motions of the pendulum. It is impossible for the forced Duffing oscillator to be synchronized with the rotational chaotic motions.     

Surrogate test for noise-contaminated dynamics in the Duffing oscillator

To identify random signals from nonlinear system under stochastic background is very difficult, and standard dynamical methods are generally not applicable. The pseudo-periodic surrogate algorithm recently developed by Small is introduced to test the sample time series in the Duffing oscillator under the Gaussian white noise excitation. The correlation dimensions of the noisy periodic, noise-induced chaotic and random-dominant responses of the system are compared with their corresponding artificial data respectively. Meanwhile, the leading Lyapunov exponents by Rosenstein’s algorithm are also presented to validate the identification idea on the system’s sample time series.     

Bifurcations and chaos in Duffing equation with damping and external excitations

Duffng equation with damping and external excitations is investigated.By using Melnikov method and bifurcation theory,the criterions of existence of chaos under periodic perturbations are obtained.By using second-order averaging method,the criterions of existence of chaos in averaged system under quasi-periodic perturbations forff=nω+εσ,n=2,4,6(whereσis not rational toω)are investigated.However,the criterions of existence of chaos for n=1,3,5,7-20 can not be given.The numerical simulations verify the theoretical analysis,show the occurrence of chaos in the averaged system and original system under quasiperiodic perturbation for n=1,2,3,5,and expose some new complex dynamical behaviors which can not be given by theoretical analysis.In particular,the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations,and the period-doubling bifurcations to chaos has not been found under quasi-periodic perturbations.     

加性二值噪声激励下Duffing系统的随机分岔

研究了Duffing系统在加性二值噪声作用下的随机分岔现象.首先,根据二值噪声的统计特性,推导得到二值噪声状态间的跃迁概率,据此对二值噪声进行了数值模拟.其次,利用四阶Runge-Kutta(龙格-库塔)数值算法得到该系统位移和速率的稳态联合概率密度及位移的稳态概率密度.然后,通过对位移稳态概率密度单双峰结构变化的研究,发现加性二值噪声的状态和强度能够诱导系统产生随机分岔现象.最后,观察到随着系统非对称参数的逐渐变化,系统同样产生了随机分岔现象.     

Primary resonance of Duffing oscillator with fractional-order derivative

In this paper the primary resonance of Duffing oscillator with fractional-order derivative is researched by the averaging method. At first the approximately analytical solution and the amplitude-frequency equation are obtained. Additionally, the effect of the fractional-order derivative on the system dynamics is analyzed, and it is found that the fractional-order derivative could affect not only the viscous damping, but also the linear stiffness, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. This conclusion is remarkably different from the existing research results about nonlinear system with fractional-order derivative. Moreover, the comparisons of the amplitude-frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two parameters of the fractional-order derivative, i.e. the fractional coefficient and the fractional order, on the amplitude-frequency curves are investigated, which are different from the traditional integer-order Duffing oscillator.     

Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise-linearity

This paper presents an analysis of the dynamical behaviour of a non-symmetric oscillator with piecewise-linearity. The Chen–Langford (C–L) method is used to obtain the averaged system of the oscillator. Using this method, the local bifurcation and the stability of the steady-state solutions are studied. A Runge–Kutta method, Poincaré map and the largest Lyapunov’s exponent are used to detect the complex dynamical phenomena of the system. It is found that the system with piecewise-linearity exhibits periodic oscillations, period-doubling, period-3 solution and then chaos. When chaos is found, it is detected by examining the phase plane, bifurcation diagram and the largest Lyapunov’s exponent. The results obtained in this paper show that the vibration system with piecewise-linearity do exhibit quite similar dynamical behaviour to the discrete system given by the logistic map.