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Department of Mathematics

Question

Math 6410 • ODE

Spring 2014 • Question 6

Question

Math 6410 • ODE Spring 2014 • Question 6

Consider the system \[ \begin {aligned} \dot {x} &= \mu +x^2+y^2,\\ \dot {y} &= -y+x^2 \end {aligned} \]

  1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
  2. Using an approximate center manifold, determine the nature of the bifurcation at the origin at \(\mu =0\).

Raw LaTeX

Consider the system
\[
\begin{aligned}
  \dot{x} &= \mu+x^2+y^2,\\
  \dot{y} &= -y+x^2
\end{aligned}
\]
\begin{enumerate}
\item[(a)] State the Center Manifold Theorem for rest points. Briefly explain its
importance in bifurcation theory.
\item[(b)] Using an approximate center manifold, determine the nature of the bifurcation
at the origin at \(\mu=0\).
\end{enumerate}

Topic Tags

Bifurcation Theory Center Manifold Theory and Invariant Manifolds Phase Plane Analysis