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{ \bf AMES 207:}\hspace{4.08in} Prof. M. Krsti\'{c}\\[0.5cm]
{\sc Nonlinear Systems}\hspace{3.7in} Winter 1998\\[0.5cm]
{\Large\bf FINAL EXAM}

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Take home. Open books and notes.\\
Total points: 35\\
Due Saturday, March 14, 1998, at 12:00 noon in Professor Krsti\'{c}'s office.\\[-0.2cm]
\begin{center} {\bf Late submissions will not be accepted. Collaboration not allowed.}\end{center}

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{\large\bf 1.} (8 pts) Consider the system
\[
\dot{x}_i = x_{i+1} -c_i x_i - k_i s_i(x) x_i + w_i(x) d\,,
\]
$i=1\,, \ldots \,,n\,, x_{n+1}=0$, where $c_i$, $k_i>0$ and $|w_i(x)|\le s_i(x)$. Show that the system is ISS w.r.t. $d$. What is the type of the gain function (linear, quadratic, exponential, $\ldots$)?

\vspace{0.5cm}
{\large\bf 2.} (9pts) Using the center manifold theorem, determine whether the origin of the following system is asymptotically stable:
\begin{eqnarray}
\dot{x}_1 &=& -x_2+x_1x_3\nonumber\\
\dot{x}_2 &=& x_1 + x_2x_3\nonumber\\
\dot{x}_3 &=& -x_3 - \left( x_1^2 + x_2^2 \right) + x_3^2\,.\nonumber
\end{eqnarray}

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{\large\bf 3.} (9 pts) Consider the system
\begin{eqnarray}
\dot{x}_1 &=& x_2\nonumber\\
\dot{x}_2 &=& -x_2 - \mbox{sin} \omega t \left( \left( x_1 + \mbox{sin} \omega t \right)^2 + x_3 \right) \nonumber\\
\dot{x}_3 &=& -x_3^n - \left( x_1 + \mbox{sin} \omega t \right)^2 + \frac 12\,.\nonumber
\end{eqnarray}

a) For $n=1$, show that for sufficiently large $\omega$ there exists an exponentially stable periodic orbit in an $O\left( \frac 1{\omega} \right)$ -- neighborhood of the origin.

b) What can you claim for $n=3$?

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{\large\bf 4.} (9pts) Show that, for sufficiently small $\epsilon$, the origin of the system
\begin{eqnarray}
\dot{x} &=& x^2 + z + \mbox{cos}(\epsilon y) -1\nonumber\\
\epsilon \dot{y} &=& -y + x^2 - x\nonumber\\
\epsilon^2 \dot{z} &=& -z + \mbox{sin}y + \epsilon x^3\nonumber
\end{eqnarray}
is exponentially stable. (Hint: treat $\mu = \epsilon^2$ as a seperate small parameter.) Since the system has three (rather than two) time scales, it has three levels of invariant manifolds -- slow, medium, and fast. Without going into high accuracy, give the approximate expressions for these manifolds and discuss the trajectories of the system.
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