Emergence and evolution of unusual inhomogeneous limit cycles displacing hyperchaos in three quorum-sensing coupled identical ring oscillators

Physica D: Nonlinear Phenomena. 2023. Vol. 455. Article 133902.

Stankevich N., Volkov E.

We demonstrate that strongly asymmetric limit cycles can be observed in the system of three identical ring oscillators (3-gene networks known as Repressilators) globally coupled by signal molecule diffusion added to the model in a way like the known bacterial ‘‘quorum-sensing’’ mechanism. These cycles are stable over a wide interval of the coupling strengths where they expel the dominant hyperchaotic regime existing in three Repressilators in very large areas of parameters. The bifurcations of the inhomogeneous limit cycle, with a high-amplitude orbit for one oscillator and two lowamplitude identical orbits for the other two, are traced. Bifurcation analysis reveals an unusual cascade of bifurcations ended in the appearance of a new limit cycle with splitted (slightly nonidentical) lowamplitude orbits. Both cycles lose stability giving birth inhomogeneous chaos in the small parameter interval. Hyperchaos dominates in the parameter plane around the ‘‘island’’ with inhomogeneous limit cycles, and this accounts for very long hyper chaotic transients when the system is returning to stable asymmetric cycles after their perturbations. In turn, it is the cycles that contribute the asymmetric and often rather long pieces in hyperchaotic trajectories. The presented cycles differ from the known asymmetric attractors: inhomogeneous limit cycles born from ‘‘oscillation death’’ and the cycles observing in ‘‘smallest chimeras’’.

Keywords: hyperchaos, гиперхаос, broken symmetry, coupled repressilators, quorum-sensing, связанные репрессиляторы, чувство плотности, потеря симметрии

Publication based on the results of: Развитие аналитических и численных методов исследования многомерных динамических систем: этап I (2023)