On Huygens synchronization. Application to Van der Pol oscillators with distributed couplings

By Huygens synchronization it is understood synchronization of the oscillators through their connecting “environment”. In the last years modeling and analysis of the dynamics of this phenomenon (in various forms) has received a renewed attention. Among the recently discussed models one can find synchronization of two mechanical Van der Pol oscillators coupled through a distributed parameters elastic rod of finite length. We consider in this contribution an electronic analogue of the aforementioned mechanical model - two electronic Van der Pol oscillators coupled through a finite length LC line and give its description by a two point boundary value problem for a lossless telegraph equation. The problem is discussed by associating a system of functional differential equations obtained from the integration of the Riemann invariants along the characteristics. This “auxiliary” mathematical object allows statement and solving of various qualitative problems which afterwards are projected back on the problem for the telegraph equation.