Nonlinear Anti-(Parity-Time) symmetric dimer

TL;DR

This paper introduces a nonlinear anti-PT symmetric dimer, analyzing its solution families, bifurcations, and stability, with experimental relevance in electric circuits and detailed numerical validation of its dynamic behavior.

Contribution

It presents the first detailed analysis of a nonlinear anti-PT symmetric dimer, including solution classification, bifurcation analysis, and stability properties, supported by experimental context and numerical simulations.

Findings

Four solution families identified, including symmetric and asymmetric branches.

Critical thresholds for existence and bifurcations analytically derived and numerically confirmed.

Only the upper symmetric branch is spectrally and dynamically stable.

Abstract

In the present work we propose a nonlinear anti-$\mathcal{PT}$-symmetric dimer, that at the linear level has been experimentally created in the realm of electric circuit resonators. We find four families of solutions, the so-called upper and lower branches, both in a symmetric and in an asymmetric (symmetry-broken) form. We unveil analytically and confirm numerically the critical thresholds for the existence of such branches and explore the bifurcations (such as saddle-node ones) that delimit their existence, as well as transcritical ones that lead to their potential exchange of stability. We find that out of the four relevant branches, only one, the upper symmetric branch, corresponds to a spectrally and dynamically robust solution. We subsequently leverage detailed direct numerical computations in order to explore the dynamics of the different states, corroborating our spectral analysis results.