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Cell[CellGroupData[{
Cell["Qualitative Behavior of Non-Linear Systems:  Limit Cycles.", 
  "Subsection",
  FontFamily->"Helvetica",
  FontSize->24],

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Cell[TextData[{
  "We again consider a non-linear autonomous 2 x 2 system of ODE's.\n       \
x' = f(x,y)\n       y' = g(x,y)\nwhere for simplicity we assume that all \
partial derivatives of f and g with respect to x and y exist and are \
continuous.  The Poincare-Bendixson Theorem asserts that any solution (x(t), \
y(t)) of such a system exhibits one of three behaviors as t goes to infinity. \
 The first possibility is divergence--i.e. (x,y) goes to infinity as t \
increases.  The second possibility is convergence to\na critical point--i.e. \
\n\n       ",
  Cell[BoxData[
      \(TextForm
      \`\(lim\+\(t\[LongRightArrow]\[Infinity]\)\) 
          \((x \((t)\), y \((t)\))\)\  = \ \((x\_0, y\_0)\)\)]],
  ",\n       \nwhere (",
  Cell[BoxData[
      \(TextForm\`\(\(x\_0, y\_0)\)\ \)\)]],
  "is a constant solution of the system.  The last possibility,\nand the one \
we will illustrate here, is that of convergence to a so-called\nlimit cycle.  \
That is, it can happen for some systems and some intial conditions that ( \
x(t), y(t) ) is asymptotically periodic.  A simple example of such behavior \
is exhibited by solutions of the linear system \n\n             x'(t) = y(t)\n\
             y'(t) = -x(t).\n             \nThe general solution is \n\n      \
       ( x(t), y(t) ) = ( r cos (t+",
  Cell[BoxData[
      \(TextForm\`\(\(\(t\_0)\), \ r\ sin \((t + t\_0)\)\ )\), \)\)]],
  "\n             \nwhere ",
  Cell[BoxData[
      \(TextForm
      \`\(t\_0\) and\ r\ are\ constants . \ \ One\ can\ easily\ see\ that\ 
        when\ r\ is\)]],
  " not zero that\nall solution curves are circles, and (x(t), y(t) ) moves \
around one of these\ncircles at a constant rate.  In this case, non-zero \
solutions are not just asymptotically periodic; they are simply periodic... \
period.\n\nA more complicated example of a limit cycle is provided by \
changing the previous example slightly so that it becomes non-linear.  That \
is, we take the right side of the system"
}], "Text",
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Cell["\<\
for some small but non-zero number a.  Note that (0,0) is still a \
critical point of this system.  We   look at a direction field
\t\t\t\
\>", "Text",
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