\documentclass[11pt]{article}
\setlength{\textwidth}{6.9in}
\setlength{\textheight}{8.6in}
\setlength{\evensidemargin}{-0.4in}
\setlength{\oddsidemargin}{-0.4in}
\setlength{\topmargin}{-0.6in}
\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}
 {\sl Do any 6 problems. Clearly indicate which ones you want to be graded. Good luck!}
\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}


\begin{enumerate}

\item %P.115 3.17 (4th ed)
Consider $\dot{x}=y$ and $\dot{y}=f(x,\lambda)$, where $f$ and $f'$ are continuous.
\begin{enumerate}
\item
Show that the index $I_\Gamma$ of any simple closed curve $\Gamma$ that encloses all
equilibrium  points can only be $1$, $-1$ or zero.
\item
Show that at a bifurcation point the sum of the indices of the equilibrium points resulting from the bifurcation is unchanged.
\item
By using the results in (a) and (b), deduce that the system $\dot{x}=y$, $\dot{y}= -\lambda x+x^3$ has a saddle point at $(0,0)$
when $\lambda<0$, which bifurcates into a center and two saddle points
as $\lambda$ becomes positive.
\end{enumerate}


\vspace*{4mm}
\item %P141 example 4.12
Find the equivalent linear equation and the frequency-amplitude relation for the equation \\$\ddot{x} + \mathrm{sgn}(x) = 0$, where $\mathrm{sgn}(x) = \left\{\begin{array}{rl} 1 & \mbox{ when }x >0 \\ 0 & \mbox{ when }x = 0\\ -1 & \mbox{ when }x < 0\end{array}\right.$.

\vspace*{4mm}
\item %P180 5.13
Apply the Lindstedt method to van der Pol's equation $\ddot{x} + \varepsilon(x^2-1)\dot{x} + x = 0$, $|\varepsilon| \ll 1$.  Determine the frequency of the limit cycle to the order of $\varepsilon^2$.

\vspace*{4mm}
\item % Page 367, Theorem 10.13
  Let $ \dot{\bf x} = {\bf X} ({\bf x}) $ be a regular autonomous system of
  dimension $ n $, with $ {\bf X} ({\bf 0}) = {\bf 0} $. Suppose there exists a
  function $ U ({\bf x}) $ such that in some neighborhood $ \| {\bf x} \| \leq k $, where $k>0$,
\begin{enumerate}
\item[(i)] $ U ({\bf x}) $ and its partial derivatives are continuous;
\item[(ii)] $ U({\bf 0}) = 0 $;
\item[(iii)] $ \dot{U} ({\bf x}) $ is positive definite for the given system;
\item[(iv)] in every neighborhood of the origin there exists at least one point $ {\bf x} $ at which $ U({\bf x}) > 0 $.
\end{enumerate}
Show that the zero solution of the system $ \dot{\bf x} = {\bf X} ({\bf x}) $ is unstable.

\vspace*{4mm}
\item %p.115, 3.14
\begin{enumerate}
\item Suppose that, for the two plane systems $\dot{\boldsymbol x}_1 = {\bf X}_1({\boldsymbol x}_1), \dot{\boldsymbol x}_2 = {\bf X}_2({\boldsymbol x}_2)$, and for a given closed curve $ \Gamma $, there is no point on $ \Gamma $ at which $ {\bf X}_1$ and $ {\bf X}_2$ are opposite in direction.  Show that the index of $ \Gamma $ is the same for both systems.
\item The system $ \dot{x} = y, \ \dot{y} = x $ has a saddle point at the origin.  Show that the index of the origin for the system $ \dot{x} = y + c x^2y, \ \dot{y} = x - c x y^2 $
  is likewise $ -1 $ for $c \neq 0 $.
\end{enumerate}



\vspace*{4mm}
\item
\begin{enumerate}
\item %P359 Theorem 10.14
Let the origin be the equilibrium point of the system $\dot{x} = ax + by + h_1(x,y), \dot{y} = cx+ dy + h_2(x,y)$ where $a, b, c, d$ are constants, and $ h_1, h_2$ are differentiable and their first derivatives are continuous. Moreover
\[ h_1(x,y)= O(x^2 + y^2), \quad  h_2(x,y)= O(x^2 + y^2), \quad {\rm as} \ x^2 + y^2\rightarrow 0.\]
Then show that the zero solution of the system is asymptotically stable when its linear approximation is asymptotically stable.
\item %P363 Example 10.8
Test the stability of the zero solution of the van der Paol's equation
\[
\ddot{x}+\beta(x^2-1)\dot{x}+x = 0,
\]
when $\beta < 0$.
\end{enumerate}


\vspace*{4mm}
\item
\begin{enumerate}
\item %P378 10.11
The $n$-dimensional system $\dot{\boldsymbol x} = -\mathrm{grad}\ W({\boldsymbol x})$ has an isolated equilibrium point at ${\boldsymbol x } = 0$.  Show that the zero solution is asymptotically stable if $W$ has a local minimum at ${\boldsymbol x} = 0$.  Given a condition for instability of zero solution and justify.
\item %P378 10.7
Determine the stability of the zero solution of the system
\[
\dot{x} = x^2 - y^2, \hspace{7mm} \dot{y} = -2xy.
\]
\end{enumerate}



\vspace*{4mm}
\item %P420 Theorem 12.1
Given the equations ${\dot x} = \mu x + y - x f(r), {\dot y} = -x +\mu y - y f(r)$, where $r=\sqrt{x^2 + y^2}$, $f(r), f'(r)$ are continuous for $r\ge 0$, $f(0)=0$, $f'(r)>0$ for $r>0$, and $f(r) \rightarrow \infty$ as $r \rightarrow \infty$. The origin is the only equilibrium point. Then prove the followings:
\begin{enumerate}
\item[(i)] for $\mu<0$ the origin is a stable spiral covering the whole plane;
\item[(ii)] for $\mu=0$ the origin is a stable spiral;
\item[(iii)] for $\mu>0$ there is a stable limit cycle whose radius increases from zero as $\mu$ increases from zero.
\end{enumerate}


\end{enumerate}
\end{document}