\in [\Re]_T) \ \Phi'(s) + a(s) \cos \Phi(s) \neq 0.$$

Letting $H(\varphi,s):=\varphi - \Phi(s)$, condition  is the requirement that system  with output $H(\varphi,s)$ has a well-defined uniform relative degree 2 on the set $\{(q,\dot q)\in \cX: \varphi =
\Phi(s)\}$. The associated zero dynamics manifold is precisely $\Gamma$, and it is controlled invariant.

The foregoing lemma inspires the following observation. Instead of guessing a relation $\varphi = \Phi(s)$ and checking whether it is a VHC for , as in Lemma \[lem:feasibility\], we will assign a nonzero right-hand side to , view the resulting identity as an ODE, and [*generate*]{} VHCs by solving the ODE. More precisely, recall the notation $\tilde h:=h \circ \pi$, and consider the scalar $T$-periodic differential equation $$\label{eq:constraint_generator}
\dot x = - \tilde a(t)\cos x + \tilde \delta(t),$$ where $\tilde \delta:\Re \to \Re \backslash \{0\}$ is a $T$-periodic function to be assigned. Then, $T$-periodic solutions of  give rise to VHCs.

\[lem:vcg\] Suppose that $x(t)$ is a $T$-periodic solution of , where $\tilde \delta :\Re \to \Re
\backslash \{0\}$ is smooth and $T$-periodic. Then, the relation $\varphi = \Phi(s)$, with $\Phi:=x \circ \pi^{-1}:[\Re]_T \to \Re$, is a VHC for .

Let $x(t)$ be a $T$-periodic solution of . Then, 