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\def\Rl{\mathbb{R}}
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\begin{document}

\centerline{{\sc Problem set 4}}
\centerline{Math 207A, Fall 2018}
\centerline{Due: Fri., Oct. 26}


\bigskip\noindent
{\bf 1.} The ODE for a linear oscillator with displacement $x(t)$ is
\[
\ddot{x} + \delta \dot{x} + \omega^2 x = 0,
\] 
where the damping coefficient $\delta \ge 0$ and the frequency $\omega > 0$ are constants. Write this ODE as a $2\times 2$ first order linear system for $(x,y)$
with $y = \dot{x}$. Sketch the phase plane for the following cases: (a) $\delta = 0$ (undamped); (b) $0<\delta < 2\omega$ (underdamped);
(c) $\delta = 2\omega$ (critically damped); (d) $\delta > 2\omega$ (overdamped). Classify the equilibrium $(x,y)=(0,0)$ in each case.
In which cases is the equilibrium hyperbolic?


\bigskip\noindent
{\bf 2.} (a) An $n\times n$ matrix $N$ is said to be nilpotent if $N^k = 0$ for some $k\in \Nl$. Compute $e^{tA}$ if
$A = \lambda I + N$ where $N$ is nilpotent. Justify all your steps.

\smallskip\noindent
(b) Compute $e^{tA}$ if $A$ is the $3\times 3$ Jordan block
\[
A = \left(\begin{array}{ccc} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 &\lambda \end{array}\right).
\]
(c) Consider the $3\times 3$ linear system $x_t = Ax$ where $A$ is the matrix in (b) and $\lambda\in \Rl$. Describe the stable, unstable, and center subspaces
and the stability of $x=0$ in the cases (i) $\lambda < 0$; (ii) $\lambda = 0$; (iii) $\lambda > 0$.



\bigskip\noindent
{\bf 3.} Suppose that $A : \Rl \to \Rl^{n\times n}$ is a continuous, simultaneously diagonalizable, matrix valued function, meaning that
there exists a constant matrix $P$ and a diagonal matrix valued function $\Lambda : \Rl \to \Rl^{n\times n}$ with
\[
\Lambda(t) = \diag\left(\lambda_1(t),\dots, \lambda_n(t)\right)
\]
such that
$A(t) = P \Lambda(t) P^{-1}$.

\smallskip\noindent
(a) Show that $A(s)$ and $A(t)$ commute for any $s, t\in \Rl$.

\smallskip\noindent
(b) Show that the solution of the initial value problem
\[
x_t = A(t) x,\qquad x(0) = x_0
\]
is given by
\[
x(t) = e^{\int_0^t A(s)\, ds} x_0.
\]


\bigskip\noindent
{\bf 4.} Let
\[
A(t) = \left(\begin{array}{cc}  1 & 2t \\ 0 & -1 \end{array}\right).
\]
(a) Compute the fundamental matrix $\Phi(t;0)$ that satisfies
\[
\Phi_t = A(t) \Phi,\qquad \Phi(0;0) = I.
\]
(b) Compute
\[
E(t) = e^{\int_0^t A(s)\, ds}
\]
and show that $E(t)\ne \Phi(t;0)$. Why doesn't the result of Problem 3 apply here?


\bigskip\noindent
{\bf 5.} Consider the Lorenz equations
\begin{align*}
&x_t = \sigma(y-x),
\\
&y_t= r x - y - xz,
\\
&z_t = xy - \beta z
\end{align*}
where $r, \beta, \sigma > 0$ are positive parameters. Determine the linearized stability of the  
equilibrium solution $(x,y,z) = (0,0,0)$. When is this equilibrium hyperbolic?
\end{document}             % End of document.
