Periodic solutions of travelling-wave type in circular gene networks

We consider circular chains of unidirectionally coupled ordinary differential equations which are mathematical models of artificial gene networks. We study the problems of the existence and stability of special periodic solutions, the so-called travelling waves, in these chains. We establish that the number of such periodic solutions grows unboundedly as the number of links in the chain grows. However, at most one of these travelling waves can be stable. We give an explicit algorithm for choosing the stable cycle.