Regular perturbation and secular terms

Consider an oscillator with squared frequency $1+\varepsilon$, initial displacement one, and initial velocity zero. Assume an expansion in powers of the parameter on a fixed bounded time interval, substitute it into the equation, and collect coefficients. The leading solution is the unit-frequency cosine. The first correction solves a forced linear equation whose forcing is that same cosine. Its homogeneous initial data are zero because the original initial conditions do not depend on the parameter.

The forcing resonates with the leading oscillator, producing a correction $-t\sin t/2$. This coefficient grows with time even though the exact solution has constant amplitude. It is entirely consistent with a valid expansion for each fixed bounded time interval. A remainder estimate on such an interval may have a constant depending on its length. It does not imply that the same estimate remains valid when the interval itself grows as the inverse parameter. Initial conditions and the intended domain must therefore accompany the coefficient equations.

The exact solution is the cosine with frequency $\sqrt{1+\varepsilon}$. Expanding that frequency and then the cosine at fixed time reproduces the secular correction. Its size is proportional to the product of parameter and time. When time reaches the inverse-parameter scale, this first correction is no longer uniformly small relative to the leading oscillation's amplitude. Near zeros of the leading cosine, relative error is especially unhelpful; an absolute norm over time is a clearer accuracy measure.

Instead keep the first frequency correction inside the cosine. This resums the leading phase drift into a bounded oscillatory expression. A simple exact estimate justifies the improvement here: the cosine function is globally Lipschitz with constant one, so its difference is bounded by the frequency error times time. Taylor's theorem bounds the difference between the exact square-root frequency and its first-order approximation by $\varepsilon^2/8$ for nonnegative parameter. This gives a genuine long-time estimate rather than a formal hope.

For a general differential equation, solving coefficient equations produces an approximation with a residual. Converting a small residual into a small solution error requires a stability estimate and matching initial or boundary data. A forcing residual can accumulate over long times or be amplified by unstable dynamics. Thus a residual of second order does not automatically imply a second-order solution error over an interval that grows with the inverse parameter. The exact oscillator estimate above works because its phase and stability are explicitly available.

Poincaré–Lindstedt theory systematically adjusts the time coordinate for periodic solutions, while multiple scales introduces separate fast and slow clocks. Both methods target information revealed by secular behavior. The appropriate method depends on whether frequency, amplitude, or another slowly changing quantity matters. Removing resonant forcing is a formal construction principle; proving a uniform approximation on a chosen long interval still needs hypotheses controlling the actual solution. The next lessons apply the same distinction to rapid spatial changes and slowly evolving amplitudes.