# Reference for periodic solution of linear differential system

I am looking for a reference (article, book) about the following results :

Let $$A(t)$$ be a $$(n,n)~ T-$$periodic matrix, i.e. $$A(t+T) = A(t) ~\forall t$$

Similarly, let $$B$$ be a $$T-$$periodic map from $$\mathbb{R}$$ to $$\mathbb{R}^n$$ i.e. $$\exists T >0 , \forall 1 \leqslant i \leqslant n, B_i(t+T) = B_i(t)$$

Then, the first order linear differential system $$\dot{Y}(t) = A(t)Y(t) + B(t)$$ admits a $$T-$$periodic solution.

I understand that this is close to Floquet theory in which $$B$$ is chosen to be $$0_{\mathbb{R}^n}$$

I keep finding results on slightly different/more general problems. (ODE systems with impulse for instance, see here, or nonlinear ode systems, see also here, which might work ?)... Can someone point me toward literature encompassing this result ? Thank you.

## 1 Answer

See the chapter on periodic coefficients in Linear Ordinary Differential Equations by Coddington and Levinson. There is a section on first order inhomogeneous systems.

• Thank you. Theorem 4.6-4.7 was exactly what I needed. Oct 14 '19 at 14:43
• @Nufeu No problem at all. Good luck. Oct 14 '19 at 14:51