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Oscillation quenching in third order Phase Locked Loop coupled by mean field diffusive coupling

We explore analytically the oscillation quenching phenomena (amplitude death and oscillation death) in a coupled third order phase locked loop (PLL) both in periodic and chaotic mode. The phase locked loops are coupled through mean field diffusive coupling. The lower and upper limits of the quenched state are identified in the parameter space of the coupled PLL using Routh-Hurwitz technique. We further observe that the ability of convergence to the quenched state of coupled PLLs depends on the design parameters. For identical system both the system converges to homogeneous steady state whereas for non-identical parameter values they converge to inhomogeneous steady state. It is also observed that for identical systems the quenched state is wider than non-identical case. When the systems parameters are so chosen that each isolated loops are chaotic in nature, in that case we observe the quenched state is relatively narrow. All these phenomena are also demonstrated through numerical simulations.