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.
[1][2] The method is named after Henri Poincaré,[3] and Anders Lindstedt.
[4] All efforts of geometers in the second half of this century have had as main objective the elimination of secular terms.The article gives several examples.
The theory can be found in Chapter 10 of Nonlinear Differential Equations and Dynamical Systems by Verhulst.
[5] The undamped, unforced Duffing equation is given by for t > 0, with 0 < ε ≪ 1.
[6] 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 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: We have the leading order ω0 = 1, because when
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 = 3/8.
Higher orders of accuracy can be obtained by continuing the perturbation analysis along this way.
This method can be continued indefinitely in the same way, where the order-n term
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.
Consider the van der Pol oscillator with equation
Then plug it into the second equation to obtain (after some trigonometric identities)
To eliminate the secular term, we must set both
increases from zero to a small positive constant, all circular orbits in phase space are destroyed, except the one at radius 2.
To eliminate the secular term, we set
The equation's solution would have two time-scales, one fast-varying on the order of
Now plug into the Mathieu equation and expand to obtain
The secular term coefficients in the third equation are
Setting them to zero, we find the equations of motion:
, the origin is a saddle point, so the amplitude of oscillation
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.
[9] For the van der Pol oscillator, we have
, and using the same method of eliminating the secular terms, we find
, which is exactly the desired asymptotic behavior.