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With $\sigma =\omega (ϵ)t$, $\tau =ϵt$, the multiscale chain rule gives $d/dt=(1+ϵ{\Omega }_1+\cdots ){\partial }_{\sigma }+ϵ{\partial }_{\tau }$. Expanding $F(u_{i,0}+ϵu_{i,1})=F(u_{i,0})+ϵDF(u_{i,0})u_{i,1}+\cdots$ gives at order oneAt order $ϵ$, moving the linearized reference dynamics to the left givesThe component sum runs over the $m$ state variables; the source uses $n$ in its first displayed sum and $m$ later. All functions for the solvability calculation are periodic in $\sigma$ with period one.

Differentiating the autonomous cycle equation $U^{\prime }=F(U)$ gives $U^{\prime \prime }=DF(U)U^{\prime }$, so $U^{\prime }$ lies in the periodic null space of $\hat{L}$. For a hyperbolic attracting limit cycle, its tangent direction has Floquet multiplier one and every transverse multiplier has modulus below one. Hence it is the only periodic direction, proving the required one-dimensional null space.

Strictly, mere nonlinear stability does not ensure this spectral assertion. In polar coordinates $\dot{r}=-(r-1{)}^3$, $\dot{\phi }=2\pi$ has an attracting unit-period cycle but neutral radial linearization at $r=1$, supplying a second periodic solution of the variational equation. Thus the intended phase reduction assumes the usual simple unit Floquet multiplier, or equivalently hyperbolic transverse attraction; this assumption is made explicit here.