= 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:
$$
u_1''+u_1=\cos t,\qquad u_1=\tfrac12t\sin t
$$
is a particular solution, as direct <differentiation> verifies. Although $\epsilon u_1$ is small at fixed $t$, it can be order one at $t=O(\epsilon^{-1})$.
Such growth need not mean the exact solution grows. The exact bounded oscillation $\cos(\sqrt{1-\epsilon}\,t)$ expands as $\cos t+(\epsilon t/2)\sin t+\cdots$ 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.
Back to article page
