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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

2005 • Question 3

Question

Math 6410 • ODE 2005 • Question 3

Problem. Let \(A\) be an \(N\times N\) matrix, none of whose eigenvalues have zero real part. Show that if \(b:(-\infty ,\infty )\to \mathbb {R}^N\) is a continuous function which is periodic of period 1, \((b(t+1)=b(t),\ t\in \mathbb {R})\), then the equation \[ x'=Ax+b(t), \] has a unique periodic solution of period 1.

Raw LaTeX

\textbf{Problem.}
Let \(A\) be an \(N\times N\) matrix, none of whose eigenvalues have zero real part. Show
that if \(b:(-\infty,\infty)\to\mathbb{R}^N\) is a continuous function which is periodic of
period 1, \((b(t+1)=b(t),\ t\in\mathbb{R})\), then the equation
\[
  x'=Ax+b(t),
\]
has a unique periodic solution of period 1.

Topic Tags

Gronwall Inequality and Continuous Dependence Linear Systems and Matrix Exponentials Poincare Maps and Periodic Solutions Spectral Theory of Operators