be compatible with the maneuvering problem, we need to verify whether or not the rear wheel of the bicycle traverses the entire curve $\C$ with bounded speed, i.e., we need to show that for any solution $(s(t),\dot s(t))$ of , there exist $\bar
t>0$ and $\epsilon>0$ such that $\dot s(t)>\epsilon > 0$ for all $ t
\geq \bar t$, and $\lim\sup_{t \to \infty} \dot s(t) < \infty$. The next result explores general properties of systems of the form .

\[prop:internal\_dynamics\_general\_properties\] Consider a differential equation of the form  with state space $\{(s,\dot s) \in
[\Re]_T \times \Re\}$. Assume that $\Psi_1, \Psi_2:[\Re]_T \to \Re$ are smooth functions such that $\Psi_1(s) > 0$ for all $s$ and $\int_0^T \tilde \Psi_2(\tau) d\tau <0$. Then, there exists a smooth function $\nu:[\Re]_T \to (0,\infty)$ such that the closed orbit $\cR
= \{(s,\dot s) \in [\Re]_T \times \Re: \dot s = \nu(s)\}$ is exponentially stable for , with domain of attraction containing the set $\cD = \{(s,\dot s) \in [\Re]_T \times
\Re: \dot s \geq 0\}$. [Thus,]{} for all initial conditions in $\cD$, the solution $(s(t),\dot s(t))$ of  converges to the unique asymptotically stable limit cycle $\cR$.

\[rem:doa\] [It can be shown that the domain of attraction 