For a weakly perturbed harmonic oscillator, expand both its periodic solution and its angular frequency, using the stretched time . At each order, choose the frequency correction to remove fundamental-harmonic forcing that would produce a secular term. For a self-excited periodic orbit, cancellation of the independent sine and cosine terms can determine both the leading amplitude and the frequency correction. Actual existence follows when the associated Poincaré return map has a simple displacement zero, by the implicit function theorem.
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The Poincaré–Lindstedt method is a mathematical technique used to analyze and approximate solutions to nonlinear differential equations, particularly in the context of perturbation theory. It is named after Henri Poincaré and Karl Lindstedt, who contributed to the development of methods for understanding the behavior of dynamical systems. ### Overview: The method is typically applied to study oscillatory or periodic solutions of differential equations that have small parameters, often referred to as perturbations.