Math 6410 • ODE

For a system of equations \(\frac {dx}{dt}=A(t)x\), where \(x\in \mathbb {R}^n\) and \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\),

1. What is Abel’s formula? (State without proof).
2. Suppose \(A(t)\) is periodic, \(A(t+T)=A(t)\), \(T>0\). Define the monodromy matrix \(M\).
3. Show that the eigenvalues \(\mu _i\), \(i=1,\cdots ,n\), of the monodromy matrix satisfy \[ \prod _{i=1}^n \mu _i = \exp \left (\int _0^T \operatorname {trace}(A(t))\,dt\right ). \]
4. Suppose \(\left |\prod _{i=1}^n \mu _i\right |>1\). Prove that the zero solution of \(\frac {dx}{dt}=A(t)x\) is unstable. If \(\left |\prod _{i=1}^n \mu _i\right |<1\), does that guarantee that the zero solution of \(\frac {dx}{dt}=A(t)x\) is stable? Why or why not?

5. Suppose \[ A(t)= \begin {pmatrix} \lambda (t) & \omega (t)\\ -\omega (t) & \lambda (t) \end {pmatrix}, \]

   where both \(\lambda (t)\) and \(\omega (t)\) are periodic functions of \(t\) with period \(T=1\). Find a condition on \(\lambda (t)\) that determines stability or instability of the zero solution of \(\frac {dx}{dt}=A(t)x\).

   Hint: Examine \(\frac {d}{dt}(x_1^2+x_2^2)\).