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An imaginary resistor (Z) based electronic dimer is used to describe the gyroscopic and resistive coupled two-level systems in quaternionic space. We successfully used the quaternionic coefficients to characterize the different classes of non-Hermitian systems: the pseudo-Hermitian and anti-Hermitian, both having exceptional points (EPs) separating the exact and the broken phases. Remarkably, the EPs conical dynamic is fully described to follow a hyperbolic/parabolic shape in the case of parity-time symmetry (PTS)/anti-PTS, respectively. Interestingly, our results demonstrated that the gyroscopic coupling mechanism allows identical non-dissipative but nonoscillatory systems with negative capacitor to exhibit PTS-like behavior.

We investigate on the right and left-handed (RH/LH) balanced gain and loss non-Hermitian electrical transmission lines (ETL) modeled using an imaginary resistor. The hamiltonian of each system is successfully derived in the framework of the tight-binding theory. We discuss the underlying symmetries and calculate the breaking thresholds of the Parity time (PT) and Anti PT (APT) symmetry phase transitions. Moreover, the modes dynamic characterization reveal the existence of critical points beyond which operation requires thresholdless transition and the eigen modes are either real or complex. We also present a self-consistent theory to investigate on their scattering properties. In particular, we demonstrate that the Anderson-like localized modes mostly emerge in the broken phase where eigen modes are complex with the localization length proportional to the cell numbers. Importantly, the mode localization is most likely to occur in structures consisting of a large cells number for RH ETL while LH ETL will confined modes even for infinitesimal small cell numbers. These results unveil embryonic applications in cryptography, switching and system control.

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