is much freedom in the choice of $\tilde \mu(t)$. For instance, picking $\tilde
  \mu(t) = 1$, we numerically find [$\epsilon \approx
    0.6482$]{}. The corresponding VHC is depicted in Figure \[fig:ellipse:virtual\_constraint\]. The condition, in Proposition \[prop:generator:periodic\_solutions\](ii), that $K^+/
  K^- < (\cos x_0)^{-1}$ is conservative. Indeed, with our choice of $\tilde \mu$ we have $K^+ =3/2$, $K^-=2/3$, and thus the condition is violated. Yet, we see in Figure \[fig:ellipse:virtual\_constraint\] that $ \Phi(s) \in (0,
  \pi/2)$ for all $s \in [\Re]_T$.

![VHC for the ellipse in Example \[ex:ellipse\].[]{data-label="fig:ellipse:virtual_constraint"}](FIGURES/ellipse_constraint_pend "fig:"){width=".495\textwidth"} ![VHC for the ellipse in Example \[ex:ellipse\].[]{data-label="fig:ellipse:virtual_constraint"}](FIGURES/ellipse_constraint_fun "fig:"){width=".495\textwidth"}

Motion on the constraint manifold {#sec:constrained_motion}
=================================

Having chosen $\tilde \delta(t)= \epsilon \tilde \mu(t)$ as in Proposition \[prop:generator:periodic\_solutions\](ii), and having obtained a VHC $\varphi= \Phi(s)$, the next step is to analyse the [*reduced dynamics*]{} of  on the constraint manifold $\Gamma = \{ (q,\dot q) \in \cX : \varphi = \Phi(s), \ \dot \varphi =
\Phi'(s) \dot s\}$. These are the zero dynamics of  with output function $H(\varphi,s)=\varphi - \Phi(s)$. To this end, we impose that $[(d/dt) (\Phi'(s) \dot s)] |_\Gamma = \ddot
\varphi|_\Gamma$. Expanding both sides of the equation above, using the expression of $\ddot \varphi$ in , identity , and the fact that $\delta \neq 