\documentclass[12pt]{article}

\usepackage{amsmath}
\usepackage{amssymb}

\usepackage{xcolor}

%\usepackage[textheight=8.5in, margin=1.6in]{geometry}
\usepackage[total={6in,8in}]{geometry}

\usepackage{tikz}
%\usepackage{pgfplots}
%\tikzstyle{ball} = [circle,shading=ball, ball color=black,
%    minimum size=1mm,inner sep=1.3pt]
%\usetikzlibrary{arrows.meta}
%\tikzset{>={Latex[length=2mm,width=1.5mm]}}

\usepackage{graphicx}

\usepackage{needspace}  % <-- to prevent bad pagebreak over environment titles
% Use:  \Needspace{3\baselineskip} before opening an environment

\usepackage{enumitem}
\setlist[enumerate]{leftmargin=*}

%\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\sn}{\mathfrak{S}}
\newcommand{\ve}{\varepsilon}
\newcommand{\ips}[2]{\langle{#1}, {#2}\rangle}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Span}{\mathrm{Span}}
\DeclareMathOperator{\rk}{\mathrm{rank}}
\DeclareMathOperator{\trace}{\mathrm{tr}}
\DeclareMathOperator{\im}{\mathrm{im}}
\DeclareMathOperator{\Sym}{\mathrm{Sym}}
\DeclareMathOperator{\Hom}{\mathrm{Hom}}
\DeclareMathOperator{\diag}{\mathrm{diag}}
\DeclareMathOperator{\glb}{\mathrm{glb}}
\DeclareMathOperator{\lub}{\mathrm{lub}}

\parskip = 0.05in
\parindent = 0.0in

%\pagestyle{empty}

\begin{document}
\centerline{Math 322 lecture for Monday, Week 10}
\bigskip

\centerline{\textsc{Equilibrium points for planar systems}}
\medskip

Consider a general planar system
\begin{align*}
  x'&= P(x,y)\\
  y'&= Q(x,y).
\end{align*}
By translating, we can assume that any equilibrium point we are interested in
sits at the origin.

Here are some types of equilibrium points.
\begin{enumerate}
  \item The origin is a {\bf center} if there exists~$\delta>0$ such that every
    trajectory with initial condition in~$B_{\delta}\setminus\left\{ (0,0)
    \right\}$ is a closed curve containing~$(0,0)$ in its interior.
  \item Let~$r(t,r_0,\theta_0)$ and~$\theta(t,r_0,\theta_0)$ denote the solution
    to our system in polar coordinates and with initial conditions~$r(0)=r_0$
    and~$\theta(0)=\theta_0$.  The origin is a {\bf stable focus} if there
    exists~$\delta>0$ such that~$0<r_0<\delta$ and~$\theta_0\in\R$
    imply~$r(t,r_0,\theta_0)\to(0,0)$ and~$|\theta(t,r_0,\theta_0)|\to\infty$
    as~$t\to\infty$.  It is an {\bf unstable focus} if the same holds
    as~$t\to-\infty$.
  \item The origin is a {\bf stable node} if there exists~$\delta>0$ such that
    for~$0<r_0<\delta$ and~$\theta_0\in\R$, we
    have~$r(t,r_0,\theta_0)\to(0,0)$ as~$t\to\infty$
    and~$\lim_{t\to\infty}\theta(t,r_0,\theta_0)$ exists.  In other words, the
    trajectories approach the origin with a well-defined tangent.  It's an
    {\bf unstable node} if the same holds with~$t\to-\infty$.  A node is called
    {\em proper} if every ray through the origin is tangent to some trajectory.
  \item The origin is a {\bf topological saddle} if it is locally homeomorphic
    to a saddle for a linear system.
  \item The origin is a {\bf center-focus} if there exists a sequence of closed
    solution curves~$\Gamma_n$ with~$\Gamma_{n+1}$ in the interior of~$\Gamma_n$
    such that~$\Gamma_k\to(0,0)$ as~$k\to\infty$ and such that every solution
    with initial condition between~$\Gamma_n$ and~$\Gamma_{n+1}$ spirals toward
    either~$\Gamma_n$ or~$\Gamma_{n+1}$ as~$t\to\pm\infty$.
\end{enumerate}

\textbf{Summary of results for hyperbolic equilibria.}

Let~$x_0=(0,0)$ be a hyperbolic equilibrium point and assume that~$P$ and~$Q$ are
continuously differentiable.  Then
\begin{enumerate}
  \item The point $x_0$ is a topological saddle if and only if the linearized system has a
    saddle at the origin.  (This follows from Hartman-Grobman.)
  \item If the linearized system has a center, then~$x_0$ is either a center, a
    focus, or a center-focus.  The case of a center-focus cannot occur if~$P$
    and~$Q$ are \emph{analytic} at~$x_0$, i.e., if they can be expressed as power series that
    converge in some disc about the origin.
  \item If~$x_0$ is a node then the linearized system it's a node or a focus for
    the nonlinear system.  Similarly, if it's a focus for the linearized system,
    then its a node or focus for the nonlinear system.  If~$f$ has continous
    second partials, then if~$x_0$ is a node for the linearized system, it is
    also a node for the nonlinear system, and similarly for foci. (See our text,
    Example~5, Section~2.10.)
\end{enumerate}
\medskip

{\bf Example.} Here is an example of a center-focus:
\begin{align*}
  x'&= -y+x\sqrt{x^2+y^2}\sin\left( \frac{1}{\sqrt{x^2+y^2}} \right)\\[10pt]
  y'&= x+y\sqrt{x^2+y^2}\sin\left( \frac{1}{\sqrt{x^2+y^2}} \right).
\end{align*}
for~$x^2+y^2\neq0$, and with~$f(0,0)=(0,0)$ where~$f$ is the right-hand side of
the above.  (In particular, it turns out that~$f$ is not analytic at the
origin.) Changing to polar coordinates gives the system
\begin{align*}
  r'&= r^2\sin\left( \frac{1}{r} \right)\\[4pt]
  \theta'&= 1
\end{align*}
for~$r>0$, and~$r'=0$ for~$r=0$.  So~$\theta=t+\theta_0$, and
if~$\sin(1/r)=0$, i.e., if~$r=\frac{1}{n\pi}$ for any~$n\in\Z_{>0}$, we have~$r'=0$.  So
the circles of radius~$\frac{1}{n\pi}$ are trajectories.  If
\[
  n\pi<\frac{1}{r}<(n+1)\pi,
\]
i.e., if
\[
  \frac{1}{(n+1)\pi}<r<\frac{1}{n\pi},
\]
then~$r'<0$ if~$n$ is odd and~$r'>0$ if~$n$ is even.  Which means the
trajectories will either spin inwards or outwards towards one of the circular
trajectories.  A partial picture appears below:
\medskip

\begin{center}
  \includegraphics[height=2in]{10mon-center-focus.png}
\end{center}
\medskip

{\bf Nonhyperbolic equilibria}.  Please set our text, Section~2.11 for a
description of possible behaviors for a nonhyperbolic equilibrium point for a
two-dimensional system.  In particular, please learn the meaning of the
following terms: sector, hyperbolic sector, parabolic sector, elliptic sector,
saddle-node.

Note the comment on p.~150: if the linearized system is nonzero, the only types
of equilibrium points that can occur beside those already mentioned for analytic
systems are saddle-nodes, critical points with elliptic domains, and cusps.  The
book gives examples of each of these:

{\bf saddle-node} (two hyperbolic sectors, one parabolic sector): 
\begin{align*}
  x'&= x^2\\
  y'&= y
\end{align*}

{\bf critical point with elliptic domain} (one elliptic sector, one hyperbolic sector,
  two parabolic sectors, four separatrices):
\begin{align*}
  x'&= y\\
  y'&=-x^3+4xy
\end{align*}

{\bf cusp} (two hyperbolic sectors, two separatrices):
\begin{align*}
  x'&= y\\
  y'&= x^2
\end{align*}
\end{document}


