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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2014 • Question 5

Question

Math 6410 • ODE Fall 2014 • Question 5

Consider Selkov’s model of glycolysis, the process whereby living cells break down sugar, where \(x\) and \(y\) are concentrations of ADP and F6P. Show that it has a nonconstant periodic solution. \[ \begin {aligned} \dot {x} &= -x+\frac {y}{10}+x^2y,\\ \dot {y} &= \frac {1}{2}-\frac {y}{10}-x^2y \end {aligned} \] [Hint: show the polygon with vertices \((0,0)\), \((0,5)\), \((\frac {1}{2},5)\) and \((\frac {11}{2},0)\) is forward invariant.]

Raw LaTeX

Consider Selkov's model of glycolysis, the process whereby living cells break down
sugar, where \(x\) and \(y\) are concentrations of ADP and F6P. Show that it has a
nonconstant periodic solution.
\[
\begin{aligned}
  \dot{x} &= -x+\frac{y}{10}+x^2y,\\
  \dot{y} &= \frac{1}{2}-\frac{y}{10}-x^2y
\end{aligned}
\]
[Hint: show the polygon with vertices \((0,0)\), \((0,5)\), \((\frac{1}{2},5)\) and
\((\frac{11}{2},0)\) 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