Aggregation of conservative systems into the chain with an attractive cycle

We consider a set of an arbitrary number of conservative systems of arbitrary orders. It is assumed that each of the systems admits a non-degenerate family of single-frequency periodic motions and the entire set of systems considered as one conservative system possesses the same property. It is assumed that periodic oscillations in each system have individual phase shifts. The problem of aggregation of a coupled system with an attractive cycle is solved. We pick a leading conservative system, which becomes closed-loop system through the van der Pol dissipation type feedback. The leading system possesses an attractive cycle. The next (slave) conservative system is coupled to the leading one by means of a one-way coupling control. Van der Pol type dissipation is applied, so that the coupled system also has an attractive cycle. Further, each subsequent system joins the previous system: a chain of weakly coupled conservative systems with an attractive cycle is aggregated. In the chain, the previous system, as the master, acts on the subsequent one, as the slave, through one-way coupling control. The selected coupling controls have a universal character and can be applied to any conservative systems that satisfy the assumptions introduced. A numerical example of the formation of a chain of three identical mathematical pendulums, which are aggregated by the proposed method into a chain of weakly coupled systems with an attractive cycle, is given.