Proving the existence and stability of a periodic solution

Question

I want to prove the existence of a periodic solution for this differential

$$ \frac{dN}{dt}=(1+cos(\alpha t))r(c-N)-\mu N. $$

Plotting the solution of this equation in Matlab produces the graph below

I was thinking of using a Poincare return map to prove the existence? and then using Floquet Theory for the stability? Would this work?

Any help would be much appreciated.

Thanks.