\documentclass[11pt]{article}
\setlength{\textwidth}{6.9in}
\setlength{\textheight}{9in}
\setlength{\evensidemargin}{-0.4in}
\setlength{\oddsidemargin}{-0.4in}
\setlength{\topmargin}{-0.3in}
\newcommand{\eps}{\varepsilon}
\usepackage{amsmath,amssymb}
%\pagestyle{empty}
\begin{document}
%\pagestyle{empty}

\begin{center}
{\sc Sample Applied Nonlinear ODE Comprehensive Exam}
\end{center}


\vspace{0.1in}
\noindent {\sl Do any six problems. Clearly indicate in the table below which problems you want to be graded. If you do not select the problems we will grade the first 6 problems. Good luck!} \\
%\vspace{1mm}
\begin{center}
\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}
\end{center}

\vspace{1mm}

\begin{enumerate}
\item %p.84 2.27
\begin{enumerate}
\item Show that the phase paths of the Hamilton system $\dot{x}=-\partial H/\partial y, \dot{y} = \partial H/\partial x$ are given by $H(x,y)=constant.$ If $(x_0,y_0)$ is an equilibrium point, show that $(x_0,y_0)$ is stable according to the linear approximation if $H(x,y)$ has a maximum or a minimum at the point.  Assume that all the second derivatives of $H$ are nonzero at $x_0, y_0$.
\item %p.121 3.60
For the system $\dot{x} = y[16(2x^2	+ 2y^2-x)-1]$ and $\dot{y}=x-(2x^2	+ 2y^2-x)(16x-4)$, show that the system is Hamiltonian and obtain the Hamiltonian function $H(x,y)$. Obtain the equilibrium points and classify them.
\end{enumerate}

\vspace{5mm}
\item
\begin{enumerate}
\item %p.105 thm s 
For the system $\dot{x} = X(x,y),\ \dot{y} = Y(x,y)$, show that there are no closed paths in a simply connected region in which $\displaystyle \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y}$ is of one sign.
\item %p.119 modification of 3.43
Use part (a) and the directions of the phase paths to show that the system
$
\dot{x} = xy - y^2,\ \ \dot{y} = x^3y+\sin x
$
has no closed path in the entire $x,y$ plane.
\end{enumerate} 

\vspace{5mm}
\item %P. 151, 4.13
Use the method of equivalent linearization to find the amplitude and frequency of the limit cycle of the equation
$\ddot{x}+\varepsilon (|x|-1) \dot{x} + x +\varepsilon x^3 =0, \ 0 < \varepsilon <\!\!< 1$.  
State the equivalent linear equation.

\vspace{5mm}
\item %p.179, 5.4(ii), change the coefficient of x
Apply Lindstedt's method to the problem
\[
(1+\varepsilon \dot{x})\ddot{x} + 4x = 0,\  x(\varepsilon,0) = a,\  \dot{x}(\varepsilon,0) = 0,
\]
to obtain $2\pi$-periodic solutions to $O(\varepsilon)$, and to obtain the amplitude-frequency relation to $O(\varepsilon^2)$.


\vspace{5mm}
\item
\begin{enumerate}
\item %P261 Def 8.1 and P268 Def 8.2
Write down the definitions of Poincar\'{e} stability and Liapunov stability for plane autonomous systems.
\item %P301 8.8
Prove that Liapunov stability of a solution implies Poincar\'{e} stability for plane autonomous systems.  Construct an autonomous system as a counter-example and show that the converse is not true.
\end{enumerate}


%\vspace{5mm}
\newpage
\item
\begin{enumerate}
\item %theorem 8.15
Suppose that
\begin{enumerate}
\item[(i)] ${\bf A}$ is a constant $n\times n$ matrix whose eigenvalues have negative real parts;
\item[(ii)] For $t_0 \le t < \infty$, ${\bf C}(t)$ is continuous and $\displaystyle \int_{t_0}^t ||{\bf C}(t)||\ dt $ is bounded.
\end{enumerate}
Show that all solutions of the linear homogeneous system $\dot{\bf x} = \{{\bf A}+{\bf C}(t)\}{\bf x}$ are asymptotically stable.
\item %a shorter version of p.302, 8.27 with variations in f(t) and C(t)
Use the result in (a) to investigate the stability of the solutions of the linear system
\[
\dot{x} = (-2+te^{-t})  x + y+ t^2\sin(t), \    \dot{y} = -5 x + \sqrt{3}te^{-t} y  -\pi t.
\]
\end{enumerate}

\vspace*{5mm}
\item 
\begin{enumerate}
\item %theorem 10.11
Suppose that in a neighborhood $\mathcal N$ of the origin, the regular system $\dot{\boldsymbol x} = {\bf X}({\boldsymbol x})$ and the function $V({\boldsymbol x})$ satisfy
\begin{enumerate}
\item[(i)] {\bf X}({\bf 0}) = {\bf 0};
\item[(ii)] $V({\boldsymbol x})$ is continuous and positive definite;
\item[(iii)]  $\dot{V}({\boldsymbol x})$ is continuous and negative semidefinite.
\end{enumerate}
Show that the zero solution of the system is uniformly stable.
\item %p.377,  10.2
Show the zero solution of the system $\dot{x} = y-\sin^3x, \hspace{3mm} \dot{y} = -4x-\sin^3 y$ is uniformly stable  by using the function $V(x,y) = x^2+\alpha y^2$ with a suitable $\alpha$.
\end{enumerate}


\vspace*{5mm}
\item
\begin{enumerate}
\item %p.383 theorem 11.1
State the Poincar\'{e}-Bendixson theorem.  Do NOT prove the theorem.
\item % use theorem 11.1 to find periodic solution
Consider the system 
\[
\dot{r} = r(3-r^2) + \mu r \cos{\theta},\ \ \ \dot{\theta} = -1,
\]
 where $r^2 = x^2+y^2$ and $\mu > 0$ is a parameter.
Check the conditions of the Poincar\'e-Bendixson Theorem and determine for what values of $\mu$ there exists a periodic solution. Find the region where the closed path is in.  
\end{enumerate}


\end{enumerate}


\end{document}
