Quasi-stable structures in circular gene networks

A new mathematical model is proposed for a circular gene network representing a system of unidirectionally coupled ordinary differential equations. The existence and stability of special periodic motions (traveling waves) for this system is studied. It is shown that, with a suitable choice of parameters and an increasing number $m$ of equations in the system, the number of coexisting traveling waves increases indefinitely, but all of them (except for a single stable periodic solution for odd $m$) are quasistable. The quasi-stability of a cycle means that some of its multipliers are asymptotically close to the unit circle, while the other multipliers (except for a simple unit one) are less than unity in absolute value.