Spectrum of a continuous closed symmetric chain with an arbitrary number of contours

A dynamical system with continuous time and continuous state space is studied. The system belongs to the class of Buslaev contour networks. Contour networks can be used to simulate of traffic on complex networks, as well as have other applications, in particular, to be used in modeling communication systems. Considered system contains a closed sequence of contours, each of which has two symmetrically located common points, called nodes, with adjacent contours. There is a segment on each contour. It is called a cluster and moves at a constant speed. This title is explained by the fact that in the discrete version of the transport model, such a segment corresponds to a group of particles located in adjacent cells and moving simultaneously, and each particle corresponds to a vehicle. Delays of clusters moving are caused by the impossibility of simultaneous passage of two clusters through a common node. The dynamics of the system is such that, from a certain moment in time, the states are belonging to a certain set (limit cycle) are periodically repeated. Every limit cycle corresponds to the value of the average cluster velocity. A value depends on the initial state in general case. System behavior on limit cycles is developed in dependence on initial conditions. Results are obtained on the nature of the behavior of the system under consideration at the limit cycle, on the value of the cycle period, on the behavior of the function of the state, called the delay potential. The possible values of the average velocity of the clusters are obtained for the prescribed values of the number of contours and the cluster length. Sufficient conditions for the existence of limit cycles for small cluster lengths with delays in motion are obtained.