\magnification=\magstephalf
\def\equaldef{{\buildrel \rm def \over =}}
\def\rr{{\rm I}\kern-.2em{\rm R}}
\def\vitem#1{\vskip 0.2true cm\item{{\bf #1.}}}

\vitem{1}  Let $I$ be an open subinterval of the real line.   Let $f:I\to\rr$ be a 
continuous  function such that $f(x)\neq 0$ for all $x\in  I$.   Prove that the 
autonomous O.D.E.  $\dot x=f(x)$, on $D=\rr\times I$,  has the uniqueness property.

\vitem{2} Give an example of an autonomous O.D.E $\dot x=f(x)$, with $x\in\rr^2$, 
such that the map $f:\rr^2\to\rr^2$ is continuous, $f(x)\neq 0$ for all $x\in\rr^2$, 
and the uniqueness property fails.   

\vitem{3} Let $n$ be a positive integer, and suppose that we are given 
an open subset $\Omega$ 
of $\rr^{2n}$.   Use $q_1,\ldots,q_n,p_1,\ldots,p_n$ ---rather than 
$x_1,\ldots,x_{2n}$---  to label the coordinate functions.   The {\it Poisson bracket}
of two functions $H,K:\Omega\to\rr$ of class $C^1$ is the function 
$\{H,K\}:\Omega\to\rr$  defined by 
$$\{H,K\}=\sum_{j=1}^n\Bigl({\partial H\over\partial q_i}\cdot
{\partial K\over\partial p_i}-
{\partial H\over\partial p_i}\cdot{\partial K\over\partial q_i}\Bigr)\,.$$
To a function $H:\Omega\to\rr$ of class $C^1$ we associate the {\it Hamiltonian
system}
$$\dot q_i={\partial H\over\partial p_i}\,,\,\,\,\,
\dot p_i=-{\partial H\over\partial q_i}\,,\,\,\,i=1,\ldots,n\,.$$
Prove that, if $H$ and $K$ are $C^1$ functions on $\Omega$, then the following
three conditions are equivalent:

\itemitem{(3.i)}  $K$ is constant along all trajectories of the Hamiltonian system 
defined by $H$,

\itemitem{(3.ii)}  $H$ is constant along all trajectories of the Hamiltonian system 
defined by $K$, 

\itemitem{(3.iii)} $\{H,K\}\equiv 0$.

\item{} (This is known as {\it Noether's theorem}.)

\vitem{4}  Let $f:\rr^n\to\rr^n$ be continuous. Suppose that $f$ satisfies the 
following  {\it growth condition}: 

{\parindent=44pt

\medbreak

\itemitem{(G+)}  {\sl there exists a constant $C>0$ such that}
$$\langle x,f(x)\rangle\le C(1+||x||^2)\,\,\,{\rm for\,\,\,all}\,\,\,
x\in\rr^n\,.$$

\item{(4.i)} Prove: for every $\bar x\in\rr^n$ there exists a solution $\xi$ 
of the initial value problem $\dot x=f(x)$, $x(0)=\bar x$, such that 
$[0,+\infty)\subseteq{\rm Domain}(\xi)$.

\item{(4.ii)}  Give an example of a continuous $f$  satisfying (G+)
and of an $\bar x$ such that the initial value problem of 4.i does {\it not}
have a solution $\xi$ defined on the whole real line. 

}

\vitem{5}  Let $f:\rr^n\to\rr^n$ be continuous. Suppose that $f$ satisfies the 
following  {\it growth condition}: 

{\parindent=44pt

\medbreak
\itemitem{(G)}  {\sl there exists a constant $C>0$ such that}
$$\Bigl|\langle x,f(x)\rangle\Bigr|\le C(1+||x||^2)\,\,\,{\rm for\,\,\,all}\,\,\,
x\in\rr^n\,.$$

\item{(5.i)} Prove: for every $\bar x\in\rr^n$ there exists a solution $\xi$ 
of the initial value problem $\dot x=f(x)$, $x(0)=\bar x$, such that 
${\rm Domain}(\xi)=\rr$.   (You are allowed to use the result of 4.)

}

\vitem{6} Let $f:\rr^n\to\rr^n$ be continuous. Suppose that $f$ satisfies the 
{\it linear growth condition}: 

{\parindent=44pt

\medbreak
\item{(LG)}  {\sl there exists a constant $C>0$ such that}
$$\|f(x)\|\le C(1+||x||)\,\,\,{\rm for\,\,\,all}\,\,\,x\in\rr^n\,.$$

\medbreak

\item{} Prove: for every $\bar x\in\rr^n$ there exists a solution $\xi$ 
of the initial value problem $\dot x=f(x)$, $x(0)=\bar x$, such that 
${\rm Domain}(\xi)=\rr$.  (You are allowed to use the results of 4 and 5.)

}

\vitem{7}  Let $f:\rr\to\rr$ be given by
$$f(x)=x^2\sin{{1\over x}}\,\,\,{\rm for\,\,\,}x\neq 0\,;\,\,\,
f(0)=0\,.$$

{\parindent=54pt

\medbreak

\item{(7.i)} Verify that the function $f$ is locally Lipschitz.

\item{(7.ii)} Is $f$ globally Lipshitz? 

\item{(7.iii)} Prove that for every $\bar x\in\rr$ the initial value problem 
$\dot x=f(x)$, $x(0)=\bar x$ has a solution $\xi$ defined on the whole real line
$\rr$.  

\item{(7.iv)} The solution $\xi$ whose existence was proved in 7.iii is unique,
because of 7.i.   Call it $\xi_{\bar x}$.   Prove that for every $\bar x$ 
the limits 
$$L_+(\bar x)=\lim_{t\to+\infty}\xi_{\bar x}(t)\,,\,\,\,
L_-(\bar x)=\lim_{t\to-\infty}\xi_{\bar x}(t)\,,$$
exist.  (They may be finite or infinite.) Determine $L_+(\bar x)$,
$L_-(\bar x)$ explicitly as functions of $\bar x$.

}

\vitem{8}  Consider the two-dimensional autonomous system 
$$\dot x=y^3\,,\,\,\,\,\,\dot y=-x^3\,.$$

{\parindent=60pt

\medbreak



\item{(8.i)} Prove that all the maximal solutions of this system are defined
on the whole real line. 

\item{(8.ii)}  If $\xi:\rr\to\rr^2$ is an arbitrary maximal solution, prove 
that $\xi$ is {\it periodic}, i.e. there exists a $T>0$ such that
$\xi(t+T)=\xi(t)$ for all $t\in\rr$.   

\item{(8.iii)}  Try to get some information about the period $T$ as a function of 
the initial condition $\xi(0)$. (This is not a precise ``yes-no'' question.
I am asking you to get as much information as you can. For example, make a 
crude estimate of $T$ for some values of $\xi(0)$, such as $(0.001,0)$,
$(1,0)$, $(2,0)$, $(10,0)$, $(300,0)$.  Determine, if you can, the asymptotic
behavior of $T$, as a function of $\bar x$, for $\xi(0)=(\bar x,0)$, 
as $\bar x\downarrow 0$ and as $\bar x\to+\infty$.  Get an exact formula for 
this function if you can.)

}

\vitem{9}  Same as Problem 8, for the system

$$\dot x=y^5+x^4y^3\,,\,\,\,\,\,\dot y=-x^5-x^3y^4\,.$$

\vitem{10}  For an autonomous O.D.E. system $\dot x=f(x)$, defined
on $\rr^n$, and with a locally Lipschitz right-hand side,
we say that a point $\bar x$ is an {\it asymptotically stable
equilibrium} if there exists a neighborhood $U$ of $\bar x$ such 
that $\Phi^f(t,x)$ is defined for all $x\in U$ and all $t\ge 0$, and 
$$\lim_{t\to+\infty}\Phi^f(t,x)=\bar x$$
for every $x\in U$, uniformly with respect to $x$.  (Note: 
``$\Phi^f(t,x)$'' means ``$\Phi^f(t,0,x)$''.)

\item{} Prove that for a Hamiltonian system there do not exist asymptotically
stable equilibria.

\vitem{11}  Let $D$ be open in ${\bf C}^{n+1}$, and suppose 
$f:D\to{\bf C}^n$ is analytic.  (That is, $f=(f_1,\ldots,f_n)$,
where the $f_j$ are analytic complex-valued functions on $D$.

{\parindent=54pt

\medbreak

\item{(a)} {\it Prove or disprove.} Given any $(\hat z,\hat w)\in D$,
there is a maximal solution of the initial value problem
$${dw\over dz}=f(z,w)\,,\,\,\,w(\hat z)=\hat w\,.$$ (The definition of
``maximal solution'' is similar to the real case: a {\it maximal solution}
is a solution $z\to w(z)$, defined on an open connected subset $I$ of the
complex plane, such that there does not exist an open connected subset
$\tilde I$ with the property that $I\subseteq\tilde I$, $I\neq \tilde I$,
and there is a solution $\tilde w$, defined on $\tilde I$ and 
such that $\tilde w(z)=w(z)$ for $z\in I$. 

\item{(b)} {\it Prove or disprove.} Given any $(\hat z,\hat w)\in D$,
there is a unique maximal solution of the initial value problem
of (a).

\item{(c)} {\it Prove or disprove.} If $w:I\to{\bf C}^n$ is a maximal
solution, and $\hat z\in\partial I$ (that is, $\hat z\in{\rm Closure}(I)$ 
but $\hat z\notin I$), then there does not a sequence $\{z_j\}$ 
of points in $I$ with the property that (i)~$z_j\to\hat z$,
(ii)~$\hat w=\lim_{j\to\infty}w(z_j)$ exists, and 
(iii)~$(\hat z,\hat w)\in D$. 

\item{(d)} Discuss whether or not it is possible to define the {\it flow
map} $(z,\hat z,\hat w)\to\Phi^f(z,\hat z,\hat w)$ as in the real 
case. 

\item{(e)} Suppose we change the definition of ``solution,''  by imposing
the extra requirement that the domain $I$ of definition of a solution
of ${dw\over dz}=f(z,w)\,,\,\,\,w(\hat z)=\hat w$ must be {\it star-shaped}
with respect to $\hat z$.  (This means that, whenever $z\in I$, it follows
that the segment joining $\hat z$ and $z$ is entirely contained in $I$.) 
{\it Prove or disprove.} Given any $(\hat z,\hat w)\in D$,
there is a unique maximal solution of the initial value problem
of (a).

}

\vitem{12}  Let $D$ be open in $\rr^{n+1}$, and suppose 
$f:D\to\rr^n$ is real-analytic.  (That is, $f=(f_1,\ldots,f_n)$,
where the $f_j$ are real-analytic functions on $D$.  Then in particular
the O.D.E. satisfies all the hypotheses of the existence, uniquenes,
and continuous  dependence theorems proved in class, so we know that the
flow map $(t,s,x)\to\Phi^f(t,s,x)\equaldef\xi^f_{max,s,x}(t)$ is well
defined, and that ${\rm Domain}(\Phi^f)$ is open in
$\rr\times\rr\times\rr^n$ and $\Phi^f$ is continuous there. 
{\it Prove} that $\Phi^f$ is real-analytic. 


\vitem{13}  Consider the initial value problem 
$$
\dot x={\alpha x\over t}+f(t)\,,\,\,\,x(0)=0\,,$$
where $t\to f(t)$ is a given continuous function on some interval 
$[-a,a]$.

{\parindent=34pt

\medbreak

\item{(a)} {\sl Prove} that this initial value problem
has unique solution of class $C^1$ if $|\alpha|<1$. 

\item{(b)} Give an {\sl a priori} bound for this solution 
in terms of $||f||_{sup}\,$, $a$ and $\alpha$. 

\item{(c)} What happens if $|\alpha|\ge 1$?

\item{(d)} What would happen if we only asked for absolutely continuous 
rather than $C^1$ solutions?   

}

\vitem{14}  Consider an ordinary differential equation in $\rr^n$, given 
by 
$$
\dot x=Ax+f(t,x)\,,\leqno{(14.*)}
$$
where $f$ is a continuous map from $\rr\times\rr^n$ to $\rr^n$ that
satisfies a bound
$$
||f(t,x)||\le\delta\,,\,\,\,\,{\rm for\,\,\,all}\,\,\,x\in\rr^n\,
$$
for some $\delta>0$, and the matrix $A$ is Hurwicz.    

\medbreak

{\parindent=43pt

\medbreak\item{(a)} {\sl Prove} that
if $\xi$ is any maximal solution of (14.*) then 
$\xi(t)$ is defined for all $t\in\rr$.

\item{(b)} {\sl Prove} that there exists a constant $c$ such that 
$$
\sup\Big\{||\xi(t)-e^{tA}\xi(0)||:t\ge 0\Big\}\le c
$$
for all maximal trajectories $\xi$ of (14.*).

\item{(c)} What can you say about the constant $c$ of (b)?
In particular, for a fixed Hurwicz matrix $A$:

\itemitem{(c.1)} can $c$ be chosen to depend only on $\delta$, and not on
$f$?

\itemitem{(c.2)}  how does it depend on $\delta$ (e.g. is it 
proportional to $\delta$, or to $\delta^2$, or to $\sqrt{\delta}$, or to 
some other function of $\delta$)?

}

\vitem{15}  Consider the second-order ordinary differential equation 
$$
\ddot x+(1+f(t))x=0\,,\leqno{(15.*)}
$$
on an interval $I=[a,+\infty[$, where $f:I\to\rr$ is a continuous 
function such that 
$$
\int_a^{+\infty}|f(t)|\,dt<\infty\,.
$$

\medbreak
\item{} {\sl Prove} that every nonzero solution $\xi$ of (15.*) is such that
there are functions $t\to r(t)$, $t\to\theta(t)$, with the property that 

{\parindent=43pt

\item{(a)} the formulas
$$
\xi(t)=r(t)\cos(t+\theta(t))\,\,\,\,{\rm and}\,\,\,\,
\dot\xi(t)=-r(t)\sin(t+\theta(t))\,,
$$ 
hold for sufficiently large $t$,

\item{(b)} the limits $r(\infty)=\lim_{t\to+\infty}r(t)$
and $\theta(\infty)=\lim_{t\to+\infty}\theta(t)$ exist, and 
$r(\infty)\neq 0$.

}

\medbreak\item{} HINT: rewrite (15.*) as a $2$-dimensional  first-order 
linear time-varying system $\dot X=A(t)X$. Express $A(t)$ as $A_0+B(t)$,
with $A_0$ constant and $B$ integrable. Compute $e^{tA_0}$.  Solve $\dot
X=A(t)X$ by the method of variation of constants, i.e.  write
$X(t)=e^{tA_0}Y(t)$, and find an equation for $Y(t)$.  Look for solutions
$t\to Y_v(t)$ of this equation such that $\lim_{t\to+\infty}Y_v(t)$ is a
nonzero constant vector $v$. For a given $v$, rewrite the equation for
$Y_v(t)$ as an integral equation, involving integrals from $t$ to
$\infty$. Using the contraction mapping (Picard) theorem, prove existence
and uniquenes of a solution of this equation.  Then express the vector
$Y_v(t)$ in polar coordinates and use standard addition formulas for
trigonometric functions.


\end




