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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2001 • Question 4

Question

Math 6410 • ODE Fall 2001 • Question 4

Consider the system in the plane \[ \begin {aligned} u'&=-v+uw,\\ v'&=u+vw, \end {aligned} \tag {5} \] where \[ r^2=u^2+v^2,\qquad w=r^4-6r^2+8. \] Show that (5) has precisely two nontrivial periodic orbits and provide a “complete” phase plane analysis of the system.

Raw LaTeX

Consider the system in the plane
\[
\begin{aligned}
  u'&=-v+uw,\\
  v'&=u+vw,
\end{aligned}
\tag{5}
\]
where
\[
  r^2=u^2+v^2,\qquad w=r^4-6r^2+8.
\]
Show that (5) has precisely two nontrivial periodic orbits and provide a ``complete''
phase plane analysis of the system.

Topic Tags

Periodic Orbits, Limit Cycles, and Orbital Stability Phase Plane Analysis Poincare Maps and Periodic Solutions