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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2007 • Question 5

Question

Math 6410 • ODE Fall 2007 • Question 5

Let \(A\) be a \(N\times N\) matrix none of whose eigenvalues have zero real part and consider the system \begin{equation} x'=Ax+h(t), \tag {0.7} \end{equation} where \(h:\mathbb {R}\to \mathbb {R}^N\) is a continuous function which satisfies \[ h(t+T)=h(t),\quad \forall t\in \mathbb {R}. \] Show that for all such \(h\) (0.7) has a unique solution \(x\) such that \[ x(t+T)=x(t),\quad \forall t\in \mathbb {R}. \] What stability properties does this periodic orbit have?

Raw LaTeX

Let \(A\) be a \(N\times N\) matrix none of whose eigenvalues have
zero real part and consider the system
\begin{equation}
  x'=Ax+h(t),
  \tag{0.7}
\end{equation}
where \(h:\R\to\R^N\) is a continuous function which satisfies
\[
  h(t+T)=h(t),\quad \forall t\in\R.
\]
Show that for all such \(h\) (0.7) has a unique solution \(x\) such that
\[
  x(t+T)=x(t),\quad \forall t\in\R.
\]
What stability properties does this periodic orbit have?

Topic Tags

Existence and Uniqueness Theorems Linear Systems and Matrix Exponentials Periodic Orbits, Limit Cycles, and Orbital Stability Stability of Linear Systems