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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2024 • Question 4

Question

Math 6410 • ODE Fall 2024 • Question 4

  1. For the system \(\dot {u}=f(u)\) where \(f\) is a continuously differentiable function on \(\mathbb {R}^d\), give the definition of an invariant set, attracting set and \(\omega \)-limit set.

  2. Determine the \(\omega \)-limit sets for this system and prove their existence. Sketch the phase portrait.

    \[ \dot {x}=x-y-x^3 \]
    \[ \dot {y}=-x+y-y^3. \]

Raw LaTeX

\begin{enumerate}

\item
[3] For the system \(\dot{u}=f(u)\) where \(f\) is a continuously differentiable function on \(\mathbb{R}^d\),
give the definition of an invariant set, attracting set and \(\omega\)-limit set.


\item
[22] Determine the \(\omega\)-limit sets for this system and prove their existence. Sketch the
phase portrait.

\[
\dot{x}=x-y-x^3
\]

\[
\dot{y}=-x+y-y^3.
\]


\end{enumerate}

Topic Tags

Center Manifold Theory and Invariant Manifolds Existence and Uniqueness Theorems Global Existence and Finite-Time Blow-Up Invariant Sets, Attracting Sets, and Omega-Limit Sets