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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1996 • Question 11

Question

Math 6410 • ODE Fall 1996 • Question 11

Give a brief sketch of the proof of the following theorem:

Theorem: Let \[ E=\{x:\phi '(x)=0\} \] and let \(M\) be the largest invariant set of (2.14) in \(E\). Then every bounded (for \(t\ge 0\)) semiorbit of (2.14) tends to \(M\) as \(t\to \infty \).

Raw LaTeX

Give a brief sketch of the proof of the following theorem:

\textbf{Theorem:} Let
\[
E=\{x:\phi'(x)=0\}
\]
and let \(M\) be the largest invariant set of (2.14) in \(E\). Then every bounded (for
\(t\ge0\)) semiorbit of (2.14) tends to \(M\) as \(t\to\infty\).

Topic Tags

Center Manifold Theory and Invariant Manifolds Invariant Sets, Attracting Sets, and Omega-Limit Sets