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Self-sustained oscillations in a ring array of coupled lossless transmission lines. (English) Zbl 0840.34080

The authors consider a ring array of mutually coupled lossless transmission lines. They assume the transmission lines to be resistively coupled and the capacity and inductive coupling among the system to be neglectable. Also they assume that each linked transmission line is identical and terminates at each end by a lumped linear or nonlinear circuit element. By employing telegrapher’s equation at each line together with a coupling term in the initial-boundary condition, they derive a difference-differential system of neutral type, which is equivalent to the original partial differential equations governing the coupled lines. They believe this is the first time that a diffusion system of neutral functional differential equations is derived.

The authors investigate the global bifurcation of the neutral equations. The global Hopf bifurcation analysis and the existence of self-sustained phase-locked and synchronous periodic solutions of large amplitudes is proved. They draw some conclusions and discuss briefly some of the implications of the lossless transmission line problem.


MSC:
34K18Bifurcation theory of functional differential equations
34C23Bifurcation (ODE)
34K40Neutral functional-differential equations
34C25Periodic solutions of ODE