Secular term

A secular term is a perturbative correction whose growth makes an initially ordered asymptotic expansion fail on a longer scale. A forcing at the natural frequency of a harmonic oscillator gives the simplest example:is a particular solution, as direct differentiation verifies. Although $ϵu_1$ is small at fixed $t$, it can be order one at $t=O({ϵ}^{-1})$.

Such growth need not mean the exact solution grows. The exact bounded oscillation $\cos (\sqrt{1-ϵ}t)$ expands as $\cos t+(ϵt/2)\sin t+\cdots$ at fixed time: its secular term represents accumulating phase error. The Poincaré-Lindstedt method absorbs this into a corrected angular frequency, while the method of multiple scales evolves a slow phase or amplitude.