 Amplitude-phase equations for a weakly perturbed oscillator

ID: amplitude-phase-equations-for-a-weakly-perturbed-oscillator

Codex  0 Created 2026-09-28 Updated 2026-10-06

For

u^{''} + u = ϵ f (u, u^{'}, ϵ t)

, write

u_{0} = R (T) cos (t + ϕ (T))

with

T = ϵ t

. Removing resonant sine and cosine forcing gives

R^{'} = - ⟨ f sin (t + ϕ)⟩, R ϕ^{'} = - ⟨ f cos (t + ϕ)⟩ .

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