The effect of characteristic times on collective modes of two quorum sensing coupled identical ring oscillators

There has long been interest in methods to achieve variability of collective modes in systems of coupled identical oscillators. Extensive multistability and a dynamically rich parameter space exist for two identical ring oscillators mean-field coupled by the quorum sensing mechanism when there is significant heterogeneity of the timescales of the ring elements’ variables. These findings have been shown in numerical and electrical circuit models based on the Repressilator, a ring oscillator using repressor genes as the inverter elements. Here we investigate the effect of reducing the degree of timescale heterogeneity. We find the 2D parameter map of dynamical regimes for the previously established heterogeneous timescale, using the coupling strength and the isolated oscillator strength for the parameters. The map shows a sea of anti-phase (AP) oscillation, with a large island of complex oscillations bounded by Neimark-Sacker torus bifurcation of the AP. The island is dominated by chaos produced by period doubling cascades of two prominent limit cycles (an unusual resonant cycle and a highly asymmetric cycle) and by torus destruction. Extensive regions of multistability exist, including between different oscillatory states, and, notably, coexistence of the steady state region and oscillatory states. Using numerical bifurcation analysis with timescale heterogeneity as a continuation parameter, we follow the evolution of the collective modes, reveal the appearance of new dynamical regimes, and determine changes in the multistability. We find that decreasing the heterogeneity causes: period-doubling cascades of the AP outside of the island, thereby creating new regimes of complexity; an increase in the dominance of chaos due to a reduction in the size of regions dominated by limit cycles; loss of stable oscillatory states inside steady state regions, converting those regions from multistable to sole dynamic steady state regions; and the appearance of in-phase oscillations and its sole dominance.