Poincare-Lindstedt method

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. 

Two comments (1) This exercise shows that the Duffing oscillator has a frequency that depends on amplitude w = 1-i- a -I-O( ), in agreement with (7.6.57). (2) The Poincare-Lindstedt method is good for approximating periodic solutions, but that s all it can do if you want to explore transients or nonperiodic solutions, you can t use this method. Use two-timing or averaging theory instead. 

Using the Poincare-Lindstedt method, show that the frequency of the limit cycle for the van der Pol oscillator x-l- (x - l)x-l-x = 0 is given by... 

Computer algebra) Using Mathematica, Maple, or some other computer algebra package, apply the Poincare-Lindstedt method to the problem X + X - ex = 0, with x(0) = a, and x(0) = 0. Find the frequency O) of periodic solutions, up to and including the ( ( ) term. 

The reasoning above is shaky. See Drazin (1992, pp. 188-190) for a proper analysis via the Poincare-Lindstedt method. 

A numerical tool, sensitivity analysis, which can be used to study the effects of parameter perturbations on systems of dynamical equations is briefly described. A straightforward application of the methods of sensitivity analysis to ordinary differential equation models for oscillating reactions is found to yield results which are difficult to physically interpret. In this work it is shown that the standard sensitivity analysis of equations with periodic solutions yields an expansion that contains secular terms. A Lindstedt-Poincare approach is taken instead, and it is found that physically meaningful sensitivity information can be extracted from the straightforward sensitivity analysis results, in some cases. In the other cases, it is found that structural stability/instability can be assessed with this modification of sensitivity analysis. Illustration is given for the Lotka-Volterra oscillator. 

By applying the method of Lindstedt and Poincare [10] which essentially renormalizes the time variable t so that it depends on the aj, one can recast the straightforward expansion as one in which tne expansion coefficients contain no secular terms. The linear expansion coefficient in this modified approach is found to be identical with the first term of equation (6) (see ref. [6]for details). The modified expansion in equation (7) is now uniformly valid for all times t and the expansion coefficients are found to be periodic functions with easily interpretable physical meaning. 

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