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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2017 • Question 5

Question

Math 6410 • ODE Fall 2017 • Question 5

Consider the Brusselator system for an autocatalytic reaction. Show that it has a non-constant periodic solution. \[ \begin {cases} \dot {x}=2-13x+x^2y,\\ \dot {y}=12x-x^2y. \end {cases} \] [Hint: show the trapezoid with sides \(y=0\), \(x=\frac {2}{13}\), \(y=78\) and \(x+y=80\) is forward invariant.]

Raw LaTeX

Consider the Brusselator system for an autocatalytic reaction. Show that it has a
non-constant periodic solution.
\[
  \begin{cases}
    \dot{x}=2-13x+x^2y,\\
    \dot{y}=12x-x^2y.
  \end{cases}
\]
[Hint: show the trapezoid with sides \(y=0\), \(x=\frac{2}{13}\), \(y=78\) and
\(x+y=80\) is forward invariant.]

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