The three room
feedback project. The project must be handed in at the
beginning of the final exam. If you want to hand it in earlier, you
can do so during our last class on Friday.
The written exam at the time and location you can find in your
student center (see my.wisc.edu.) The
written exam will cover topics from the first and
second midterms as well as newer material. See below for a listing.
Exam topics
Uniqueness theorem, its proof.
Does the solution exist for all $t\gt 0$?
Bifurcation diagrams for equilibria of autonomous first order ODE
The variational equation.
The Poincaré-map. Definition, Theorems, and proofs.
The matrix exponential and solution of linear systems
Practice Problems from past exams and homework
Practice questions from the first midterm
Find all solutions of $\dot x = x^2-x$, $x(0)=0$.
Find all solutions of $\dot x = \sqrt{|x|}$, $x(0)=0$.
Find a Lipschitz constant for the function
$f(t, x) = x^2$ in the region $0\le t\le T$, $|x|\le M$.
Find a Lipschitz constant for the function
$f(t, x) = x(\sin(t)-x^2)$ in the region $0\le t\le 2\pi$, $|x|\le M$.
Let $x(t)$ and $y(t)$ be two solutions of
\[
\dot x= f(t, x)
\]
on the interval $0\le t \le T$. Assume that $f$ satisfies the Lipschitz
condition with Lipschitz constant $L$. Show that
\[
|x(t) - y(t)| \le e^{+Lt} |x(0) - y(0)|
\]
and also that
\[
|x(t) - y(t)| \ge e^{-Lt} |x(0) - y(0)|
\]
for $0\le t\le T$.
For each of the following initial value problems decide if the
solution exists for all $t\ge 0$, or only on a finite time interval
$0\le t\lt T$.
$\dot x = x(1-x)$, $x(0) = \frac12$
$\dot x = -x(1-x)$, $x(0) = \frac12$
$\dot x = -x(1-x)$, $x(0) = 2$
$\dot x = x^2(1-x^5)$, $x(0) = \frac12$
$\dot x = x(2+\sin(t)-x)$, $x(0) = 1$
Note that for some of these problems $x_*(t)=0$ and/or $x_\dagger(t)=1$ are a
special solution. You can use this fact together with the
uniqueness theorem to conclude that the solution you are looking
at has to stay in some region of the $(t,x)$ plane.
Find and solve the variational equation in the following cases.
$\dot x = x(1-x)$, $x(0)=\alpha$, at the given solution $\bar x(t) = 1$.
$\dot x = x(1-x^3)$, $x(0)=\alpha$, at the given solution $\bar x(t) = 1$.
$\dot x = x(1-x)^3$, $x(0)=\alpha$, at the given solution $\bar x(t) = 1$.
$\dot x = \alpha x$, $x(0)=1$, at the given solution $\bar x(t)=e^{t}$, $\alpha=1$.
$\dot x = -\alpha x^2$, $x(0)=1$, at the given solution $\bar x(t)=1/(1+t)$, $\alpha=1$.
A logistic equation with harvesting. A fish population in
a large lake, left to itself, would grow according to $\dot x = x^2(1-x)$.
In addition the natural birth and death in the population, fish are also
being caught at a rate $h(t)$ (the “harvesting rate”). Thus the
differential equation for the fish population which takes harvesting into
account is
\[
\dot x = x^2(1-x) - h(t).
\]
In the following questions you investigate the effect of a small amount of
harvesting on a steady fish population.
“Steady harvesting.”
$\dot x = x^2(1-x) - \alpha$, $x(0) = 1$, at the given solution $\bar
x(t) = 1$, and at $\alpha=0$.
“Periodic harvesting.” $\dot x = x^2(1-x) - \alpha\sin
\bigl(\frac{2\pi}{T} t\bigr)$, $x(0) = 1$, at the given solution $\bar
x(t) = 1$, and at $\alpha=0$. Here $T\gt0$ is a positive constant (it may
be simpler to abbreviate $k=2\pi/T$.)
Draw the bifurcation diagrams for the differential equation
$\dot x = f(a, x)$ for the following right hand sides $f$. In each
case find all bifurcation points and determine if they are
“standard bifurcations” or not.
$f(a, x) = x^2 - a$
$f(a, x) = x^2 - ax$
$f(a, x) = x^3 - ax$
$f(a, x) = x^4 - ax$
$f(a, x) = x^3-ax+1$
$f(a, x) = \sin(x)-ax$
$f(a, x) = ax-x^3+x^5$ (see also problem 3, page 46/47).
$f(x,a) = e^x-ax$
$f(x,a) = e^{ax}-x^2$
$f(x,a) = \ln(x) - ax$
Assume that $(x_0,a_0)$ is a fold point
in the bifurcation diagram for the differential equation $\dot x = f(x,a)$.
Thus $f=f_x=0$ and $f_a\ne 0$, $f_{xx}\ne 0$ at $(x_0,a_0)$. Let the
bifurcation set near $(x_0, a_0)$ be given by $a=a(x)$. Find an expression
for $a'''(x_0)$ in terms $f$ and its derivatives at $(x_0,a_0)$.
Practice questions from the second midterm
State the uniqueness theorem for ordinary differential equations of the form $\dot x = f(t, x)$.
State the definition for the Poincaré map of a differential equation $\dot x = f(t, x)$, where $f$ satisfies $f(t+T, x) = f(t, x)$ for all
$t,x$, and where the uniqueness theorem applies to $\dot x = f(t, x)$.
Let $\phi$ be the Poincaré map for the differential equation $\dot x = f(t, x)$. Assume that $\bar x(t)$ is a given solution with $\bar x(0)=a$.
What is $\phi(a)$?
Explain which linear differential equation you have to solve to compute $\phi'(a)$. Provide details.
Let $\phi$ be the Poincaré map for the differential equation $\dot x = f(t, x)$. Suppose $f(t+T, x) = f(t, x)$ for all $t,x$, and suppose that $\bar x(t)$ is a
periodic solution with period $3T$. Prove that $\bar x(t)$ is actually periodic with period $T$.
Let $\phi$ be the Poincaré map for the differential equation $\dot x = f(t, x)$ with $f(t+T, x) = f(t, x)$ for all $t,x$. Suppose that we are given a bounded
solution $\bar x(t)$, i.e. a solution for which there is a number $M$ such that $|\bar x(t)|\le M$ for all $t\ge 0$.
Show that $\lim_{n\to\infty} \bar x(nT)$ exists
Prove that the differential equation $\dot x = f(t, x)$ has a periodic solution.
Show that $x(nT) = \phi\Bigl(\dots \phi\bigl(x(0)\bigr)\dots\Bigr)$ for a solution $x(t)$ of a differential equation $\dot x=f(t, x)$ with $f(t+T,x) = f(t, x)$ (i.e.
prove the iteration theorem.)
Consider the matrix
\[
A =
\begin{pmatrix}
-1&1&0 \\ 0&-1&1 \\ 0&0&-1
\end{pmatrix}
\]
Suppose that for some $n\times n$ matrix $A$ and some vector
$\vv\in\R^n$ we know that $e^{tA}\vv = \vv$ for all $t\in\R$. Show
that $A\vv=0$. (Hint: differentiate the given identity $e^{tA}\vv=\vv$ with
respect to time.)
Find a $2\times 2$ matrix $A\neq 0$ such that $e^{2\pi A}\vv = \vv$ for all
$\vv\in\R^2$.
Let $A$ be an $n\times n$ matrix, and let $\vv\in\R^n$ be an eigenvector
of $A$, i.e. $A\vv=\lambda \vv$. Use the definition of the matrix
exponential in terms of a series to compute $e^{tA}\vv$. (Hint: how much is
$A^2\vv$? $A^n \vv$?)
Suppose that for some $n\times n$ matrix $A$ and some vector $\vv\in\R^n$
we know that $e^{tA}\vv = e^{-t}\vv$ for all $t\in\R$. Show that
$A\vv=-\vv$.
True or false?
Which (if any) of the following statements are true for
any pair of $n\times n$ matrices $A$ and $B$:
$e^{A}e^{B} = e^{B}e^{A}$;
$e^{A+B} = e^{A} e^{B}$.
$e^{-A} = \bigl(e^{A}\bigr)^{-1}$
$B^{-1}e^{A}B = e^{B^{-1}AB}$
Given two matrices $A$ and $B$, find a matrix $C$ such that $e^{-B} e^A e^B =
e^C$. Hint: note that $e^{-B} = \bigl(e^B\bigr)^{-1}$ and use the last
question above.
In this problem $\|A\|$ is the maximal row sum of the matrix $A$ (click here for the definition.)
Prove that for any vector $\vx\in\R^n$ and any matrix $A$ one has
\[
\|A\vx\|_\max
\leq \|A\|\; \|\vx\|_\max.
\]
Find a $2\times2$ matrix $A$ and a vector $\vx$ such that
\[
\|A\vx\|_\max \lt
\|A\|\; \|\vx\|_\max.
\]
Let $A$ be any $n\times n$ matrix. Show that there is a
vector $\vx$ such that
\[
\|A\vx\|_\max = \|A\|\,\|\vx\|_\max.
\]
Hint: if all the entries of $A$ are positive, then consider $\vx=(1, 1, \dots, 1)$.
Let $V$ be any invertible matrix. Which of the following are always true: