The Duffing equation

In order to compare calculated and experimentally observed phase portraits it is necessary to know very exactly all the coefficients of the describing nonlinear differential Equation 14.3. Therefore, different methods of determination of the nonlinear coefficient in the Duffing equation have been compared. In the paraelectric phase the value of the nonlinear dielectric coefficient B is determined by measuring the shift of the resonance frequency in dependence on the amplitude of the excitation ( [1], [5]). In the ferroelectric phase three different methods are used in order to determine B. Firstly, the coefficient B is calculated in the framework of the Landau theory from the coefficient of the high temperature phase (e.g. [4]). This means B = const, and B has the same values above and below the phase transition. Secondly, the shift of the resonance frequency of the resonator in the ferroelectric phase as a function of the driving field is used in order to determine the coefficient B. The amplitude of the exciting field is smaller than the coercive field and does not produce polarization reversal during the measurements of the shift of the resonance frequency. In the third method the coefficient B was determined by the values of the spontaneous polarization... 

The Duffing Equation 14.4 seems to be a model in order to describe the nonlinear behavior of the resonant system. A better agreement between experimentally recorded and calculated phase portraits can be obtained by consideration of nonlinear effects of higher order in the dielectric properties and of nonlinear losses (e.g. [6], [7]). In order to construct the effective thermodynamic potential near the structural phase transition the phase portraits were recorded at different temperatures above and below the phase transition. The coefficients in the Duffing Equation 14.4 were derived by the fitted computer simulation. Figure 14.6 shows the effective thermodynamic potential of a TGS-crystal with the transition from a one minimum potential to a double-well potential. So the tools of the nonlinear dynamics provide a new approach to the study of structural phase transitions. 

It also makes sense that r = 0. The Duffing equation is a conservative system and for all e sufficiently small, it has a nonlinear center at the origin (Exercise 6.5.13), Since all orbits close to the origin are periodic, there can be no long-term change in amplitude, consistent with r = 0. ... 

Poincare-Lindstedt method) This exercise guides you through an improved version of perturbation theory known as the Poincare-Lindstedt method. Consider the Duffing equation x + x + e.v = 0, where 0phase plane analysis that the true solution x(r, e) is periodic our goal is to find an approximate formula for x(z, ) that is valid for all t. The key idea is to regard the frequency co as unknown in advance, and to solve for it by demanding that x(z,e) contains no secular terms. 

We shall now show an application of the centre manifold theorem to the system without marked time hierarchy (this is the Duffing equation) ... 

Owing to a transformation to new variables, the Duffing equation (5.52) is of the form (5.29). We will thus look for an (approximate) equation for the centre manifold in the form... 

Inserting 14.3 into 14.1 provides the so-called Duffing equation (e.g. [1])... 

In order to illustrate the efficiency of the new produced methods, the author applied them to the well-known undamped Duffing equation with Dooren s parameters. The numerical results show that the Numerov method fitted with the Fourier components is much more stable, accurate and efficient than the one with no Fourier component. The accuracy of the fitted method with the first three Fourier components can attain 10 for a remarkable range of step sizes, which is much higher than the one of the traditional Numerov method, with eight orders for step size of tc/2.011. 

The problem of small divisors is related to another well known problem that shows up in classical perturbation theory, namely the problem of secular terms. Let us illustrate the problem with a very simple example. We consider the Duffing s equation... 

The elimination of secular terms from the power series expansion of the solution is achieved by the method of Lindstedt. The underlying idea is to pick a fixed frequency p, and to look for a quasi-periodic solution with basic frequencies /i and v. This is actually the same thing as looking for a quasi-periodic orbit on an invariant 2-dimensional torus. The process of solution is the following. Write the Duffing s equation as... 

FIGURE 4.15 Chaotic behaviour of the Duffing s equations (Parker and Chua, 1989). 

By solving Equations (3.81)-(3.83), the response variance of the Duffing oscillator can be approximated by ... 

From the analysis the author concluded that the properties of the above two cases are the same i.e. they are both of algebraic order two, they have both phase-lag of order eighth and they have the same interval of periodicity (0,1). The author applied the new proposed methods to several well known problems of the literature (Duffing equation, two-body problem etc.) in order to show the efficiency of the new methods. 

The problem of estimating the response of a randomly excited Duffing oscillator is considered. The governing equation of motion of a Duffing oscillator is given by... 

Simple dynamical systems have proved valuable as models of certain classes of physical systems in many branches of science and engineering. In mechanics and electrical engineering Duffing s and van der Pol s equations have played important roles and in physical chemistry and chemical engineering much has been learned from the study of simple, even artificially simple, systems. In calling them simple we mean to imply that their formulation is as elementary as possible their behaviour may be far from simple. Models should have the two characteristics of feasibility and actuality. By the first we mean that a favourable case can be made for the proposed reaction, perhaps by some further elaboration of mechanism but within the framework of accepted kinetic principles. Thus irreversible reactions are acceptable provided that they can be obtained as the limit of a consistent reversible set. By actuality we mean that they are set in an actual context, as taking place in a stirred tank, on a catalytic surface or in a porous medium. It is not usually necessary to assume the reaction to take place in a closed system with certain components held constant presumably by being in excess. 

This case is, again, unidentifiable. The updated PDF is plotted together with the previous one in Figure 3.16 and the trajectories of the peaks in the (A i, K3) plane have different slopes. By Equation (3.79), the equivaient linear system has a stiffness Ki + 3a K3 so different Duffing oscillators with K - - = K (a constant) are associated with the same equivalent linear... 

For the investigation of stationary waves in steady flow, all partial derivatives with respect to time are removed from equations (1) - (4) and (7) - (9). With the help of equations (1) and (5), the continuity, momentum and energy equations (2) - (4) may then be recast as a set of three simultaneous equations for dug/dx, dTg/dx and dp/dx. Equations (1) and (7) - (9) furnish expressions for duf/dx, dn/dx, dTf/dx and dm/dx. The resulting set of seven simultaneous first order differential equations can then be integrated numerically using a fourth order Runge-Kutta procedure. 

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