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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2021 • Question 4

Question

Math 6410 • ODE Fall 2021 • Question 4

  1. For a general dynamical system, give the definitions of an invariant set, attracting set and \(\omega \)-limit sets.

  2. For the system

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

    identify all critical points and their stability, and use the Poincare-Bendixson theorem to show that this system has a periodic orbit in the annular region \(1<x^2+y^2<2\).

Raw LaTeX

\begin{enumerate}

\item
For a general dynamical system, give the definitions of an invariant set, attracting set
and \(\omega\)-limit sets.


\item
For the system

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

identify all critical points and their stability, and use the Poincare-Bendixson theorem
to show that this system has a periodic orbit in the annular region
\(1<x^2+y^2<2\).

\end{enumerate}

Topic Tags

Invariant Sets, Attracting Sets, and Omega-Limit Sets Periodic Orbits, Limit Cycles, and Orbital Stability Poincare-Bendixson Theorem Stability of Linear Systems