secular term
A secular term is a piece of a perturbation series that grows without bound as time goes on — typically something like t times a sine — even though the true solution it is meant to approximate stays bounded forever. The name comes from 'saeculum', a long age, because in celestial mechanics such terms grow only over very long times yet eventually dominate and make the approximation meaningless. A secular term is a red flag that the naive expansion has been organised the wrong way.
Where it comes from: in a perturbation expansion of an oscillator, the correction at each order is driven by forcing built from the lower-order solution. If that forcing happens to oscillate at the system's own natural frequency, you are driving the oscillator on resonance, and the response is not a bounded oscillation but one whose amplitude grows linearly with time, the classic t cos(t) or t sin(t). For example, expanding the Duffing equation naively produces a term proportional to epsilon t sin t at first order; for fixed small epsilon this is fine briefly, but once t reaches order 1/epsilon the supposedly small correction is as large as the leading term and the approximation collapses.
Recognising and removing secular terms is the central technique of perturbation theory for oscillations. The Poincare-Lindstedt method removes them by adjusting the frequency; multiple-scale analysis removes them by letting the amplitude and phase drift on a slow time; both turn 'kill the secular term' into the very equation that governs the slow evolution. The honest point to internalise: a secular term is usually not a real physical instability — it is an artifact of expanding at the wrong frequency or ignoring slow modulation. The cure is to reorganise the expansion, not to conclude the system blows up.
Naively expanding y'' + y + epsilon y^3 = 0 gives, at first order, a forcing term proportional to cos t — exactly the resonant frequency — whose particular solution is proportional to t sin t, a secular term that grows without limit.
Resonant forcing at the natural frequency turns a bounded oscillation into a t-growing term — the signature of a secular term.
A secular term signals a flawed expansion, not necessarily a growing solution. The remedy is to demand the secular term vanish, which yields the slow-evolution equation; only rarely does genuine secular growth reflect a true physical instability.