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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2004 • Question 6

Question

Math 6410 • ODE Fall 2004 • Question 6

Stable and Unstable Manifolds. Find the stable and unstable manifolds in the neighborhood of the origin for the system \[ \begin {aligned} x' &= -x,\\ y' &= y+x^2,\\ z' &= z+y^2. \end {aligned} \] How are the stable and unstable manifolds for the linearization of this system related to those of the nonlinear system?

Raw LaTeX

\textbf{Stable and Unstable Manifolds.}
Find the stable and unstable manifolds in the neighborhood of the origin for the system
\[
\begin{aligned}
  x' &= -x,\\
  y' &= y+x^2,\\
  z' &= z+y^2.
\end{aligned}
\]
How are the stable and unstable manifolds for the linearization of this system related
to those of the nonlinear system?

Topic Tags

Center Manifold Theory and Invariant Manifolds Contraction Mapping and Nonlinear Equations Linear Systems and Matrix Exponentials