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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2004 • Question 3

Question

Math 6410 • ODE Fall 2004 • Question 3

Periodic Orbits. A model of an autocatalytic chemical reaction is given by the Brusselator system \[ \begin {aligned} \frac {dx}{dt} &= 1-4x+x^2y,\\ \frac {dy}{dt} &= 3x-x^2y. \end {aligned} \tag {1} \] Show that the trapezoidal region with vertices \((\frac {1}{4},0)\), \((13,0)\), \((1,12)\), \((\frac {1}{4},12)\), is an invariant set for (1). Show that the system has a nonconstant periodic trajectory in \(\mathbb {R}^2\).

Raw LaTeX

\textbf{Periodic Orbits.}
A model of an autocatalytic chemical reaction is given by the Brusselator system
\[
\begin{aligned}
  \frac{dx}{dt} &= 1-4x+x^2y,\\
  \frac{dy}{dt} &= 3x-x^2y.
\end{aligned}
\tag{1}
\]
Show that the trapezoidal region with vertices
\((\frac{1}{4},0)\), \((13,0)\), \((1,12)\), \((\frac{1}{4},12)\), is an invariant set for (1).
Show that the system has a nonconstant periodic trajectory in \(\R^2\).

Topic Tags

Center Manifold Theory and Invariant Manifolds Invariant Sets, Attracting Sets, and Omega-Limit Sets Periodic Orbits, Limit Cycles, and Orbital Stability Poincare Maps and Periodic Solutions