The duffing equation wiley online books wiley online library. The necessary condition for the chaos in the sense of smale horseshoes is established based on the melnikov method, and then the chaotic threshold curve is obtained. The duffing equation introduction we have already seen that chaotic behavior can emerge in a system as simple as the logistic map. Discretization of forced duffing system with fractional.
Forced duffing oscillator file exchange matlab central. Both oscillators are good examples of periodically forced oscillators with. In this paper, the necessary condition for the chaotic motion of a duffing oscillator with the fractionalorder derivative under harmonic excitation is investigated. The electromechanical coupling governing equations of helmholtzduffing energy harvester is established.
In that case the route to chaos is called perioddoubling. The duffing oscillator moves in a double well potential, sometimes characterized as nonlinear elasticity, with sinusoidal external forcing. A new complex duffing oscillator used in complex signal detection. Figure 1 shows the great effect of the fractional order on the frequency of the oscillator. A new type of responses called as periodicchaotic motion is found by numerical simulations in a duffing oscillator with a slowly periodically. With the help of a parallel computer we perform a systematic computation of lyapunov exponents for a duffing oscillator driven externally by a force proportional to cos. Multiple resonance and antiresonance in coupled duffing. In that case the behavior of the system is sensitive to the initial condition. We have derived the frequencyresponse equations for a system composed of twocoupled oscillators by using a theoretical approach. Analysis of a duffing oscillator that exhibits hysteresis.
If tr 0, then both eigenvalues are positive and the solution becomes unbounded as t goes to in. Rand nonlinear vibrations 5 if det 0andtr2 4 det, then there are still two real eigenvalues, but both have the same sign as the trace tr. Dimensional analysis is used to reduce the number of dimensionless quantities in the duf. Forced harmonic vibration of a duffing oscillator with. Duffing oscillator, named after georg duffing, is a famous damped and forced nonlinear dynamical system. This paper discusses a novel technique and implementation to perform nonlinear control for two different forced model state oscillators and actuators. Duffing oscillator or duffing equation is one of the most significant and classical nonlinear ordinary differential equations in view of its diverse applications in science and engineering. Experiments with the duffing oscillator from nonlinear dynamics and chaos by j. In this paper we are interested in studying the effect of the fractionalorder damping in the forced duffing oscillator before and after applying a discretization process to it. Harmonic and subharmonic solutions are determined numerically for the forced oscillations of a system governed by duffings equation, and the stability of these oscillations is. Approximate solution for nonlinear duffing oscillator with damping.
Some numerical methods for obtaining harmomc duffings. Nonlinear oscillators and their behaviour brings together the results of a wealth of disseminated research literature on the duffing equation, a key engineering model with a vast number of applications in science and engineering, summarizing the findings of this research. The system undergoes local bifurcations when the linear stiffness and damping are changed. In this paper, we investigates the effect of potential function asymmetries on the dynamic behaviors under harmonic excitation. Analogue electrical circuit for simulation of the duffing. When the periodic force that drives the system is large, chaotic behavior emerges and the phase space diagram is a strange attractor. The duffing oscillator is a common model for nonlinear phenomena in science and engineering. View duffing oscillator research papers on academia. Duffing equation has become a famous example of nonlinear system widely being utilized in sciences, engineering and biology 1. This content was uploaded by our users and we assume good faith they have the permission to share this book. Duffing oscillators for secure communication sciencedirect. The duffing oscillator, considered in this paper, has three parameters, a, p, and q that are used to implement the suggested secure communication system.
Negative linear stiffness and negative linear damping can produce divergence buckling and flutter self. The duffing equation or duffing oscillator, named after georg duffing 18611944, is a nonlinear secondorder differential equation used to model certain damped and driven oscillators. In order to study the stability of periodic responses of the forced duffing oscillator, local stability analysis is carried out on the equations describing the slow time. Response curve of the duffing oscillator 4 for both varying excitation amplitude and frequency. Experiments with duffing oscillator ucsd mathematics. The general solution is a linear combination of the two eigensolutions, and for large time the. All of these cases can exhibit chaos with the right parameter values.
In this paper, we use the modified differential transform method to obtain the approximate solution of a nonlinear duffing oscillator with a damping effect under different initial conditions. In this chapter, the duffing oscillator without external excitation is studied. The usual duff point out that a stochastic forcing is capable of re ing oscillator has the equation of motion moving hysteresis in the duffing oscillator. It first became popular for studying anharmonic oscillations and, later, chaotic nonlinear dynamics in the wake of early studies by the engineer georg duffing 1. To improve the ability of energy harvester, a helmholtzduffing oscillator with nonlinear capacitance is considered, which has an asymmetric characteristic. As in previously reported designs,1,5 the oscillator itself consists of a steel strip in our case, an ordinary 12in. Section 1section 2section 3section 4section 5section 6section 7 outline 1 introduction. Approximate solution for nonlinear duffing oscillator with. It forms a clear dynamic analog of the general torquefree motion of an arbitrary rigid body, meaning it covers most of the arbitrary rigid body dynamics. Pdf a simple electronic circuit is described which can be used in the student laboratory to demonstrate and study nonlinear effects and chaos.
The largest lyapunov exponents are provided, and some other. Pdf on jan 10, 2017, j sunday and others published the duffing oscillator. Solutions to the oscillator equation can exhibit extreme nonlinear dynamics, including limit cycles, strange attractors, and. It is an example of a dynamical system that exhibits chaotic behavior.
We study the dissipative quantum duffing oscillator in the deep quantum regime with two different approaches. The case with k1 0 is called duffings twowell oscillator and models a ball rolling along a trough having two dips with a hump in between. The first is based on the exact floquet states of the linear oscillator and the nonlinearity is treated perturbatively. Multiple scale method and perturbative solution for lo.
We investigate the resonance behaviour in a system composed by ncoupled duffing oscillators where only the first oscillator is driven by a periodic force, assuming a nearest neighbour coupling. Finally, numerical simulations using matlab are carried out to investigate the dynamic behavior such as bifurcation, chaos, and chaotic. The forced duffing oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics. Generation and evolution of chaos in doublewell duffing. Moreover, the solutions of the nonlinear duffing oscillator with the damping effect are obtained using the fourthorder.
Lecture notes on nonlinear vibrations cornell university. Each chapter is written by an expert contributor in the field of nonlinear dynamics and addresses a. Numerical integration for poincare sections of chaotic duffing oscillator dynamics. The duffing oscillator is one of the prototype systems of nonlinear dynamics. Duffing oscillator defined by a system of differential equation 7 with no damping and with no external excitation force has one stationary point x,y 0,0 as a stable center in the case of a strong spring. Applications and computational simulations find, read and. We address two aspects of the dynamics of the forced duffing oscillator which are relevant to the technology of micromechanical devices and, at the. Amplitudefrequency relationship to a fractional duffing oscillator. Fourthorder rungekutta may not adequately handle coupled nonlinear quantum oscillators. New periodicchaotic attractors in slowfast duffing. The duffing oscillator displays chaotic motion for. In this program helps to find the phase portraits of the duffing oscillator as well as to save the data file from which we have to plot in present.
The duffing map also called as holmes map is a discretetime dynamical system. Note that only forward paths where the frequency and amplitude are increasing are studied. In practice one would like to understand the route to chaos in systems described by partial differential equations, such as flow in a randomly stirred fluid. A 33g brass weight can be clamped to the ruler at any height. This paper attempts to study some applications of duffing. For weak nonlinearities and weak damping, a perturbation method is used to obtain an analytical approximation for the primary resonance response. Fixed points and their stability are discussed for the discrete system obtained. Here we chose the parameters so as to see chaos thanks to work of ueda in 1980. It well describes the nonlinear oscillator dynamics away from resonance. Double jump broadband energy harvesting in a helmholtz. Theoretical analysis of largeorbit tperiodic solution for the doublewell duffing oscillator. The duffing oscillator was chosen because studying it gives a better view of how rigid bodies act. So motion in the doublewell duffing oscillator and its theoretical analysis. A quantitative study on detection and estimation of weak signals by using chaotic duffing oscillators.
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