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.
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.
Example: the Duffing equation The undamped, unforced Duffing equation is given by
x ¨ + x + ε x 3 = 0 {\displaystyle {\ddot {x}}+x+\varepsilon \,x^{3}=0\,}
for t > 0, with 0 < ε ≪ 1. Consider initial conditions
x ( 0 ) = 1 , {\displaystyle x(0)=1,\,} x ˙ ( 0 ) = 0. {\displaystyle {\dot {x}}(0)=0.\,}
A perturbation-series solution of the form x(t) = x0(t) + ε x1(t) + ... is sought. The first two terms of the series are
x ( t ) = cos ( t ) + ε [ 1 32 ( cos ( 3 t ) − cos ( t ) ) − 3 8 t sin ( t ) ] + ⋯ . {\displaystyle x(t)=\cos(t)+\varepsilon \left[{\tfrac {1}{32}}\,\left(\cos(3t)-\cos(t)\right)-{\tfrac {3}{8}}\,t\,\sin(t)\right]+\cdots .\,}
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 t sin ( t ) {\displaystyle t\sin(t)} . 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:
τ = ω t , {\displaystyle \tau =\omega t,\,} where ω = ω 0 + ε ω 1 + ⋯ . {\displaystyle \omega =\omega _{0}+\varepsilon \omega _{1}+\cdots .\,}
We have the leading order ω 0 = 1 {\displaystyle \omega _{0}=1} , because when ε = 0 {\displaystyle \varepsilon =0} , the equation has solution x = cos ( t ) {\displaystyle x=\cos(t)} . Then the original problem becomes
ω 2 x ″ ( τ ) + x ( τ ) + ε x 3 ( τ ) = 0 {\displaystyle \omega ^{2}\,x''(\tau )+x(\tau )+\varepsilon \,x^{3}(\tau )=0\,}
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 {\displaystyle \varepsilon } are obtained:
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