Equation Duffing

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

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. 

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. 

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... 

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. 

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