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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2021 • Question 3

Question

Math 6410 • ODE Fall 2021 • Question 3

Suppose \(A(t)\) is a periodic function, \(A(t+1)=A(t)\) and \(0<A(t)<1\). Prove that the differential equation \(u'=u(A(t)-u)\) has a unique, positive, stable, periodic solution. Prove that the trivial solution \(u(t)=0\) is unstable. Hint: Use the Poincare map.

Raw LaTeX

Suppose \(A(t)\) is a periodic function, \(A(t+1)=A(t)\) and \(0<A(t)<1\).
Prove that the differential equation \(u'=u(A(t)-u)\) has a unique, positive,
stable, periodic solution. Prove that the trivial solution \(u(t)=0\) is
unstable. Hint: Use the Poincare map.

Topic Tags

Periodic Orbits, Limit Cycles, and Orbital Stability Poincare Maps and Periodic Solutions Poincare-Bendixson Theorem