Duffing’s equation

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). 

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. 

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. 

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