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\centerline{{\sc Problem set 2}}
\centerline{Math 207A, Fall 2018}
\centerline{Due: Fri., Oct. 12}


\bigskip\noindent
{\bf 1.} Suppose that $x(t) = \cos t$ is a solution of the autonomous, scalar ODE $x_t = f(x)$ for some smooth function $f : \Rl \to \Rl$. Show that $x(t) = -\sin t$ is also a solution.

\bigskip\noindent
{\bf 2.} Prove that a continuously differentiable function $f : \Rl^n\to \Rl^n$ is locally Lipschitz continuous. If, in addition, there exists
a constant $M \ge 0$ such that $|\partial f/\partial x| \le M$ for all $x\in \Rl^n$, prove that $f$ is globally Lipscitz continuous.

\smallskip\noindent \textsc{Hint.} Note that
\[
f(x) - f(y) = \int_0^1 \frac{d}{dt} f\left(t x+(1-t)y\right)\, dt.
\]

\bigskip\noindent
{\bf 3.} Compute the Picard iterates for the following scalar initial value problems, and discuss their convergence:
\[
(a)\ x_t = x,\quad x(0) = 1;\qquad (b)\ x_t = 2t - 2\sqrt{\max(0,x)},\quad x(0) = 0.
\]


\bigskip\noindent
{\bf 4.} Consider the following initial value problem for $x: \Rl \to \Rl$
\[
x_t + x = \cos t,\qquad x(0) = x_0.
\]
(a) How would you classify this ODE? What do general theorems say about the local/global existence and uniqueness of solutions?

\smallskip\noindent
(b) Define a Poincar\'e map $P : \Rl\to \Rl$ by $P(x_0) = x(2\pi)$, where $x(t)$ is the solution in (a). Compute $P$ and find its fixed point.
Show that the fixed point of $P$ corresponds to a $2\pi$-periodic solution of the original ODE. Discuss the stability of this solution.

\bigskip\noindent
{\bf 5.} Consider the following $2\times 2$-system for $(x(t),y(t))$:
\[
x_t = x - y - x^3,\qquad y_t = x + y - y^3.
\]
(a) What do general theorems say about the local/global existence and uniqueness of solutions of the initial value problem
with $x(0) = x_0$, $y(0) = y_0$?

\smallskip\noindent
(b) Let $V(x,y) = x^2 + y^2$. Compute
\[
\frac{d}{dt} V\left(x(t),y(t)\right)
\]
and use the result to show that the solution of the initial value problem exists globally forwards in time for all $t \ge 0$.

\smallskip\noindent
(c) Let $0<a<1$ and $b >2$. If $(x_0,y_0) \ne (0,0)$, show that the solution satisfies
\[
a\le x^2(t) + y^2(t) \le b
\]
for all sufficiently large $t > 0$. Do you have any guesses for the long time behavior of the solution?

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