How to find a codimension-one heterocliniccycle between two periodic orbits

Global bifurcations involving saddle periodic orbits have recentlybeen recognized as being involved in various new types of organizingcenters for complicated dynamics. The main emphasis has been onheteroclinic connections between saddle equilibria and saddleperiodic orbits, called EtoP orbits for short, which can be found invector fields in R³. Thanks to the development of dedicatednumerical techniques, EtoP orbits have been found in a number ofthree-dimensional model vector fields arising in applications.    We are concerned here with the case of heteroclinic connectionsbetween two saddle periodic orbits, called PtoP orbits for short. Ahomoclinic orbit from a periodic orbit to itself is an example of aPtoP connection, but is generically structurally stable in a phasespace of any dimension. The issue that we address here is that, untilnow, no example of a concrete vector field with a non-structurallystable PtoP connection was known. We present an example of a PtoPheteroclinic cycle of codimension one between two different saddleperiodic orbits in a four-dimensional vector field model ofintracellular calcium dynamics. We first show that this model is agood candidate system for the existence of such a PtoP cycle and thendemonstrate how a PtoP cycle can be detected and continued in systemparameters using a numerical setup that is based on Lin's method.