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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2000 • Question 5

Question

Math 6410 • ODE Fall 2000 • Question 5

Consider the system \[ \begin {aligned} \dot {x}_1&=x_2,\\ \dot {x}_1&=-x_1+x_2(1-x_1^2-2x_2^2). \end {aligned} \] Prove, by means of the Poincaré-Bendixson theorem, that there exists at least one periodic orbit in the annulus \(1/2<x_1^2+x_2^2<1\).

Raw LaTeX

Consider the system
\[
\begin{aligned}
  \dot{x}_1&=x_2,\\
  \dot{x}_1&=-x_1+x_2(1-x_1^2-2x_2^2).
\end{aligned}
\]
Prove, by means of the Poincar\'e-Bendixson theorem, that there exists at least one
periodic orbit in the annulus \(1/2<x_1^2+x_2^2<1\).

Topic Tags

Periodic Orbits, Limit Cycles, and Orbital Stability Poincare Maps and Periodic Solutions Poincare-Bendixson Theorem Sobolev Spaces and Energy Estimates