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 Nonlinear Differential Equations and Dynamical Systems by Verhulst.

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)+εx1(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. 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:

1. where

We have the leading order. Then the original problem becomes

Now search for a solution of the form x(τ)=x0(τ)+εx1(τ)+... 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

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 is a small positive number. Perform substitution to the second order: where increases from zero to a small positive constant, all circular orbits in phase space.

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Example: Mathieu equation

This is an example of parametric resonance.

Consider the Mathieu equationSetting them to zero, we find the equations of motion:

grows unboundedly.

In other words, when the angular frequency) of the original oscillator, the oscillation grows unboundedly, like a child swinging on a swing pumping all the way to the moon.

Shohat expansion

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, which is exactly the desired asymptotic behavior. This is the idea behind Shohat expansion.

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References and notes