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{\sc Sample Applied Nonlinear ODE Comprehensive Exam}
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\vspace{0.1in}
 {\sl Do any six problems. Clearly indicate in the table below which problems you want to be graded. If you do not select any problems we will grade the first 6 problems. Good luck!}

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\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Problems & 1 & 2& 3& 4 & 5& 6& 7& 8 \\
\hline
Select & \qquad \qquad & \qquad \qquad & \qquad \qquad 
       & \qquad \qquad & \qquad \qquad &\qquad \qquad 
       & \qquad \qquad & \qquad \qquad \\
 \hline
\end{tabular}
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\vspace{0.2in}
\begin{enumerate}
\item
\begin{enumerate}
\item %P98 Theorem 3.4
 Show that the index of the point at infinity, $I_{\infty}$, for the system $\dot{x} = X(x,y), \dot{y} = Y(x,y)$, having a finite number $n$ of equilibrium points with indices $I_i,\ i = 1,2,\dots,n$, is given by $\displaystyle I_{\infty} = 2-\sum_{i=1}^n I_i.$
%\item Calculate the index, $I_{\infty}$, of the point at infinity for this system directly from the definition.
\item %p. 97 ex 3.2
Find the index $I_{\infty}$ of the system $\dot{x} = 2xy,\ \dot{y} = x^2 -y^2$.
\end{enumerate}

\vspace{4mm}
\item %p.143 4.5 added one term at the end and ask for equivalent linear equation
Find the equivalent linear equation for $\ddot{x} + \varepsilon(|x|-1)\dot{x} + x - \varepsilon x = 0$, and obtain the resulting frequency and amplitude approximations of the limit cycle.

\vspace*{4mm}
\item %p.181 5.22
Use the Lindstedt's method to find the $O(1)$ and $O(\varepsilon)$ terms in the expansion of the periodic solutions of $\ddot{x}+x+\varepsilon x^2 =0$.  Also find the frequency-amplitude relation for the periodic solutions up to $O(\varepsilon^2)$.

\vspace*{4mm}
\item
\begin{enumerate}
\item %p.273 thm 8.1
Show that all solutions of the regular linear system $\dot{\boldsymbol x} = {\bf A}(t){\boldsymbol x} + {\boldsymbol f}(t)$ have the same Liapunov stability property, and the Liapunov stability of those solutions is the same as that of the zero solution of the homogeneous equation $\dot{\boldsymbol \xi} = {\bf A}(t){\boldsymbol \xi}$.
\item %p.283 thm 8.9
For the regular linear system $\dot{\boldsymbol x} = {\bf A}(t){\boldsymbol x}$ , show that all solutions are Liapunov stable on $t \ge t_0,\ t_0$ arbitrary, if and only if every solution is bounded as $t \rightarrow \infty$.
\end{enumerate}

\vspace*{4mm}
\item
\begin{enumerate}
\item %P298 Theorem 8.16
If ${\boldsymbol h}({\bf 0},t) = {\bf 0}$, $A$ is constant, and
\begin{enumerate}
\item[(i)] the solutions of $ \dot{\boldsymbol x} = A{\boldsymbol x}$ are asymptotically stable;
\item[(ii)] $\displaystyle \lim_{||{\boldsymbol x}||\rightarrow 0} \frac{||{\boldsymbol h}({\boldsymbol x},t)||}{||{\boldsymbol x}||} = 0$ uniformly in $t$, $0 \le t < \infty$;
\end{enumerate}
show that the zero solution ${\boldsymbol x}(t) = {\bf 0}$ for $t \ge 0$, is an asymptotically stable solution of the regular system $\dot{\boldsymbol x} = A{\boldsymbol x} + {\boldsymbol h}({\boldsymbol x},t)$.
\item %variation of P302 8.29 
Test the stability of the zero solution of the system
\[
\dot{x} = -3x+y+\frac{x^2}{1+t},\hspace{5mm} \dot{y} = 2x-y+\frac{xy}{1+t}.
\]
\end{enumerate}

\vspace*{4mm}
\item %theorem 10.15
Let the origin be an equilibrium point of the regular two-dimensional system $\dot{\boldsymbol x} = {\bf A}{\boldsymbol x} + {\boldsymbol h}({\boldsymbol x})$ where ${\boldsymbol h}({\boldsymbol x}) = \left[\begin{array}{c} h_1({\boldsymbol x})\\ h_2({\boldsymbol x})\end{array}\right]$, and $h_1({\boldsymbol x}), h_2({\boldsymbol x}) = O(|{\boldsymbol x}|^2)$ as $|{\boldsymbol x}| \rightarrow 0$.  Suppose the eigenvalues of ${\bf A}$ are both nonzero, real, and one of them is positive.  Show that the zero solution of the system is unstable.


\vspace*{4mm}
\item
\begin{enumerate}
\item %theorem 11.2
Show that the Li\'{e}nard equation  $\ddot{x} + f(x,\dot{x})\dot{x} + g(x) = 0$, or the equivalent system $\dot{x} =y,\ \dot{y} = -f(x,y)y-g(x)$,  has at least one periodic solution if $f$ and $g$ are continuous, and they satisfy the following conditions:
\begin{enumerate}
\item[(i)] there exists $a> 0$ such that $f(x,y) > 0$ when $x^2+y^2>a^2$;
\item[(ii)] $f(0,0) < 0$;
\item[(iii)] $g(0) = 0,\ g(x) > 0$ when $x>0$, and $g(x)<0$ when $x<0$;
\item[(iv)] $G(x) = \int_0^x g(u)\ du \rightarrow \infty$ as $x \rightarrow \infty$.
\end{enumerate}
\item %P.404 11.9
Apply the result in (a) to show that $\ddot{x} + (|x| + |\dot{x}| -1)\dot{x} +x|x| = 0$ has at least one periodic solution.
\end{enumerate}

\vspace*{4mm}
\item %p.117, 3.33
\begin{enumerate}
\item Let $\mathcal C$ be a closed path for the system $\dot{\boldsymbol x} = \bf{X}({\boldsymbol x})$, having $\mathcal D$ as its interior.  Show that
\[
\int\!\!\!\!\int_{\mathcal D} \mbox{div}({\bf X})\ dxdy =0.
\]
\item %p.117, 3.34
Assume that van der Pol's equation in the phase plane
\[
\dot{x} = y, \ \ \ \ \dot{y} = -\varepsilon(x^2-1)y-x
\]
has a  single closed path, which, for $\varepsilon$ is small, is approximately a circle, center the origin, of radius $a$.  Use the result in part (a) to show that approximately
\[
\int_{-a}^a\!\int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}} (x^2-1) \ dydx=0,
\]
and so deduce $a$.
\end{enumerate}




\end{enumerate}
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