Delays. Nonlinearity. Synchronization

A rather common model of the synchronization is represented by several local oscillators coupled to some (possibly distributed) transmission environment. This model goes back to Huygens. If the transmission environment is represented by a one dimensional distributed parameter structure (vibrating string in the mechanical case or an electric transmission line for electrical systems) then some functional differential equations may be associated by integration along the characteristics. Consequently, the synchronization of the two local oscillators can be analyzed as a problem of forced oscillations for functional differential equations. In this paper two LC local nonlinear oscillators are viewed as connected to a lossless LC transmission line of infinite length. This is the electrical analogue of the Huygens like case where two nonlinear pendula are coupled to a vibrating string.