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\centerline{{\sc Problem set 5}}
\centerline{Math 207A, Fall 2018}
\centerline{Due: Fri., Nov. 2}


\bigskip\bigskip\noindent
{\bf 1.} A simple model for the potential energy of two uncharged molecules a distance $r$ apart, with strong repulsion at small distances and weak attraction at large distances, is the Lennard-Jones potential
\[
V(r) = 4\epsilon\left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right],
\]
where $\epsilon, \sigma > 0$ are positive constants. Sketch the $(r,m\dot{r})$-phase plane for the motion of a particle of mass $m$ and position $r(t)>0$ in this potential. Sketch graphs of $r(t)$ versus $t$ for various values of the energy of the particle.

\bigskip\noindent
{\bf 2.} The KPP or Fisher equation
\[
u_t = u_{xx} + u(1-u)
\]
is a PDE that describes the diffusion of a spatially distributed population with logistic growth. Traveling wave solutions $u=u(x-ct)$
satisfy the ODE
\[
u'' + cu' + u(1-u) = 0.
\]
Sketch the phase plane of this ODE for various values of the wave speed $c \ge 0$. For what values of $c$ are there nonnegative, bounded traveling waves?
Sketch the graph of $u(\xi)$ versus $\xi$ for these values of $c$. What do these traveling waves describe?


\bigskip\noindent
{\bf 3.} Consider a linear system $x_t = A(t) x$ where the continuous matrix-valued function $A(t) = A(t+1)$ is $1$-periodic, and $\Phi(t,t_0)$ is the fundamental matrix. Let $M = \Phi(1,0)$ be the monodromy matrix and $L=\log M$ its logarithm. Show that there exists a $1$-periodic matrix $\Psi(t) = \Psi(t+1)$
such that
\[
\Phi(t,0) = \Psi(t) e^{t L}.
\]

\smallskip\noindent
\textsc{Hint.} You can assume that every nonsingular matrix $M$ has a (possibly complex-valued) matrix logarithm $L = \log M$ such that $M = e^L$.

\newpage\noindent
{\bf 4.}  Consider the nonlinear system
\[
x_t = - x + y + 3y^2,\qquad y_t = y.
\]
(a) Sketch the phase plane, and show that its flow map $\varphi_t : \Rl^2 \to \Rl^2$ is given by
\[
\varphi_t(x,y) = \left(x e^{-t} + y \sinh t + y^2(e^{2t}-e^{-t}), y e^t\right).
\]
What are the stable and unstable manifolds of $(0,0)$?

\smallskip\noindent
(b) Linearize the system at the equilibrium $(x,y) = (0,0)$. Classify the equilibrium, sketch the phase plane of the linearized system, and show that its flow map $e^{tA} : \Rl^2\to\Rl^2$ is given by
\[
e^{tA} = \left(\begin{array}{cc} e^{-t} & \sinh t \\ 0 & e^t\end{array} \right).
\]
What are the stable and unstable subspaces?

\smallskip\noindent
(c) Show that the flow of the nonlinear system is mapped to the flow of the linearized system by $h : \Rl^2\to \Rl^2$
where
\[
h(x,y) = \left(x - y^2, y\right).
\]



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