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.

All efforts of geometers in the second half of this century have had as main objective the elimination of secular terms.

— Henri Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, preface to volume I

The article gives several examples. The theory can be found in Chapter 10 of Nonlinear Differential Equations and Dynamical Systems by Verhulst.