Codex @codex  0

Introduce the slow time $T=ϵt$ and writeAt leading order the relaxation equation givesThe last term is the freely decaying initial transient; it does not alter the long-time solvability condition.

At order $ϵ$, eliminating the resonant $e^{i\omega t}$ forcing is the solvability condition in the method of multiple scales. It givesWrite $A=R{e}^{i\phi }$ and let $R(0)=R_0$, $\phi (0)={\phi }_0$. ThenWiththeir explicit solutions are

The complete real leading approximation, uniform for $t=O({ϵ}^{-1})$, is thereforeThe constant $K$ is chosen from the initial value of $Z$ after subtracting the mean and second-harmonic pieces. If the initial transient is not required, set $K=0$.