Mathematics Qualifying Exam Question Bank

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Question

Math 6410 • ODE

Fall 2001 • Question 3

Question

Math 6410 • ODE Fall 2001 • Question 3

Let \[ f:\mathbb {R}^N\to \mathbb {R}^N \] be a \(C^1\) function and consider the system of differential equations \[ u'=f(u). \tag {4} \] Let there exist, for a given \(v\in \mathbb {R}^N\), a compact subset \(K\) of \(\mathbb {R}^N\), such that \[ u(t,v)\in K,\qquad t\in I_v, \] where \(I_v\) is the maximal interval of existence of the solution \(u(t,v)\) of (4) which satisfies \[ u(0,v)=v. \] Sketch proofs of the following:

\[ I_v=(-\infty ,\infty ). \]
The \(\omega \) limit set \(\Gamma (v)\) (define this term) is a nonempty, compact, connected, and invariant set.

What more can you say about the set \(\Gamma (v)\) in case of dimension \(N=2\)? Are there similar statements in higher dimensions?

Let
\[
  f:\R^N\to\R^N
\]
be a \(C^1\) function and consider the system of differential equations
\[
  u'=f(u).
  \tag{4}
\]
Let there exist, for a given \(v\in\R^N\), a compact subset \(K\) of \(\R^N\), such that
\[
  u(t,v)\in K,\qquad t\in I_v,
\]
where \(I_v\) is the maximal interval of existence of the solution \(u(t,v)\) of (4) which
satisfies
\[
  u(0,v)=v.
\]
Sketch proofs of the following:
\begin{enumerate}
\item
\[
  I_v=(-\infty,\infty).
\]
\item The \(\omega\) limit set \(\Gamma(v)\) (define this term) is a nonempty, compact,
connected, and invariant set.
\end{enumerate}
What more can you say about the set \(\Gamma(v)\) in case of dimension \(N=2\)? Are there
similar statements in higher dimensions?

Topic Tags

Center Manifold Theory and Invariant Manifolds Existence and Uniqueness Theorems Gronwall Inequality and Continuous Dependence Invariant Sets, Attracting Sets, and Omega-Limit Sets