# Proving the existence and stability of a periodic solution

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.

• Yes, it can be done in a fairly elementary way. The Poincare map takes $[-1,\frac{4rc}{2r+\mu}]$ into its interior, and the existence of its fixed point is a consequence of the Intermediate Value Property. The resulting periodic solution is globally asymptotically stable, because the difference of any two solutions satisfies a linear homogeneous ODE which can be solved in a closed form. May 3 '19 at 10:55