Poincaré–Lindstedt method
In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms—terms growing without bound—arising in the straightforward application of perturbation theory to weakly nonlinear problems with finite oscillatory solutions. The method is named after Henri Poincaré, and Anders Lindstedt. The article gives several examples. The theory can be found in Chapter 10 of. Example: the Duffing equation
The undamped, unforced Duffing equation is given by

for t > 0, with 0 < ε ≪ 1. Consider initial conditions

  

A perturbation-series solution of the form x(t) = x0(t) + ε x1(t) + …  is sought. The first two terms of the series are

This approximation grows without bound in time, which is inconsistent with the physical system that the equation models. The term responsible for this unbounded growth, called the secular term, is . The Poincaré–Lindstedt method allows for the creation of an approximation that is accurate for all time, as follows. In addition to expressing the solution itself as an asymptotic series, form another series with which to scale time t:

  where  

We have the leading order ω0 = 1, because when , the equation has solution . Then the original problem becomes

Now search for a solution of the form x(τ) = x0(τ) + ε x1(τ) + … . The following solutions for the zeroth and first order problem in ε are obtained:

So the secular term can be removed through the choice: ω1 = . Higher orders of accuracy can be obtained by continuing the perturbation analysis along this way. As of now, the approximation—correct up to first order in ε—is

Example: the van der Pol oscillator 
We solve the van der Pol oscillator only up to order 2. This method can be continued indefinitely in the same way, where the order-n term  consists of a harmonic term , plus some super-harmonic terms . The coefficients of the super-harmonic terms are solved directly, and the coefficients of the harmonic term are determined by expanding down to order-(n+1), and eliminating its secular term. See chapter 10 of  for a derivation up to order 3, and  for a computer derivation up to order 164. Consider the van der Pol oscillator with equationwhere  is a small positive number. Perform substitution to the second order:  where  which yields the equationNow plug in , and we have three equations, for the orders  respectively:The first equation has general solution . Pick origin of time such that . Then plug it into the second equation to obtain (after some trigonometric identities)To eliminate the secular term, we must set both  coefficients to zero, thus we have yielding . In particular, we found that when  increases from zero to a small positive constant, all circular orbits in phase space are destroyed, except the one at radius 2. Now solving  yields . We can always absorb  term into , so we can WLOG have just . Now plug into the second equation to obtainTo eliminate the secular term, we set . Thus we find that . Example: Mathieu equation 
This is an example of parametric resonance. Consider the Mathieu equation , where  is a constant, and  is small. The equation's solution would have two time-scales, one fast-varying on the order of , and another slow-varying on the order of . So expand the solution as Now plug into the Mathieu equation and expand to obtainAs before, we have the solutionsThe secular term coefficients in the third equation are Setting them to zero, we find the equations of motion:

Its determinant is , and so when , the origin is a saddle point, so the amplitude of oscillation  grows unboundedly. In other words, when the angular frequency (in this case, ) in the parameter is sufficiently close to the angular frequency (in this case, ) of the original oscillator, the oscillation grows unboundedly, like a child swinging on a swing pumping all the way to the moon. Shohat expansion 

For the van der Pol oscillator, we have  for large , so as  becomes large, the serial expansion of  in terms of  diverges and we would need to keep more and more terms of it to keep  bounded. This suggests to us a parametrization that is bounded:Then, using serial expansions  and , and using the same method of eliminating the secular terms, we find . Because , the expansion  allows us to take a finite number of terms for the series on the right, and it would converge to a finite value at  limit. Then we would have , which is exactly the desired asymptotic behavior. This is the idea behind Shohat expansion. The exact asymptotic constant is , which as we can see is approached by . References and notes 

Perturbation theory