In
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating
periodic solutions to
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
s, when regular perturbation approaches fail. The method removes
secular terms—terms growing without bound—arising in the straightforward application of
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
to weakly
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
problems with finite oscillatory solutions.
The method is named after
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, and
Anders Lindstedt.
Example: the Duffing equation
The undamped, unforced
Duffing equation is given by
:
for ''t'' > 0, with 0 < ''ε'' ≪ 1.
[J. David Logan. ''Applied Mathematics'', Second Edition, John Wiley & Sons, 1997. .]
Consider initial conditions
:
A
perturbation-series solution of the form ''x''(''t'') = ''x''
0(''t'') + ''ε'' ''x''
1(''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
"The Equation" is the eighth episode of the first season of the American science fiction drama television series '' Fringe''. The episode follows the Fringe team's investigation into the kidnapping of a young musical prodigy (Charlie Tahan) w ...
models.
[The Duffing equation has an invariant energy = constant, as can be seen by multiplying the Duffing equation with and integrating with respect to time ''t''. For the example considered, from its initial conditions, is found: ''E'' = ½ + ¼ ''ε''.] 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 In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
, form another series with which to scale time ''t'':
:
where
For convenience, take ''ω''
0 = 1 because the
leading order The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.math.ucdavis.edu/~ ...
of the solution's
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
is 1. Then the original problem becomes
:
with the same initial conditions. Now search for a solution of the form ''x''(''τ'') = ''x''
0(''τ'') + ''ε'' ''x''
1(''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: ''ω''
1 =
. 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
:
References and notes
{{DEFAULTSORT:Poincare-Lindstedt method
Perturbation theory