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The simplest ring oscillator is made from three strongly nonlinear elementsrepressing each other unidirectionally resulting in the emergence of a limitcycle. A popular implementation of this scheme uses repressive genes inbacteria creating the synthetic genetic oscillator known as the Repressilator.Here, we consider the main collective modes produced when two identicalRepressilators are mean-field coupled via the quorum sensing (QS) mechanismwhich is realized via production of diffusive signal molecules. Using the rateof the repressor's production and the value of coupling strength as thebifurcation parameters, we performed analysis of dynamical regimes startingfrom the two Andronov-Hopf bifurcations of unstable homogeneous steady state,which generate in-phase and anti-phase limit cycles. Pitchfork bifurcation ofthe unstable in-phase cycle leads to creation of inhomogeneous limit cycleswith very different amplitudes in contrast to well-known asymmetrical limitcycles arising from oscillation death. Neimark-Sacker bifurcation of theanti-phase cycle determines the border of an island in two-parameter spacecontaining almost all the interesting regimes including the set of resonantlimit cycles, the area with stable inhomogenous cycle, and very large areaswith chaotic regimes resulting from torus destruction, period doubling ofresonant cycles and inhomogenous cycles. We discuss the structure of chaosskeleton to show the role of inhomogeneous cycles in its formation. Manyregions of multistability and transitions between regimes are presented. Theseresults provide new insights into the coupling-dependent mechanisms ofmultistability and collective regime symmetry breaking in populations ofidentical multidimensional oscillators.

Emergence of multistability and strongly asymmetric collective modes in two quorum sensing coupled identical ring oscillators