Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2012 • Question 6

Question

Math 6410 • ODE Spring 2012 • Question 6

  1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
  2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= -xy,\\ \dot {y} &= -y+x^2-2y^2. \end {aligned} \]

Raw LaTeX

\begin{enumerate}
\item[(a)] State the Center Manifold Theorem for rest points. Briefly explain its
importance in bifurcation theory.
\item[(b)] Construct an approximation to the center manifold at the origin and use it
to determine the local behavior of solutions.
\[
\begin{aligned}
  \dot{x} &= -xy,\\
  \dot{y} &= -y+x^2-2y^2.
\end{aligned}
\]
\end{enumerate}

Topic Tags

Bifurcation Theory Center Manifold Theory and Invariant Manifolds Phase Plane Analysis Poincare Maps and Periodic Solutions