I will discuss the outcome  of  periodic perturbations of attracting cyclic dynamics.  
The system to be perturbed may be either a periodic orbit, a heteroclinic cycle or a flow-invariant torus. 
We look for frequency locked solutions that return after an integer multiple of the period of the perturbation.
The analysis consists in reducing to discrete-time dynamics on a cylinder and the golden number has a surprise participation.

This is joint work with Alexandre Rodrigues (ISEG, U. Lisboa).