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

TL;DR

This paper investigates the emergence and stability of asymmetric limit cycles in three coupled ring oscillators with quorum sensing, revealing their role in displacing hyperchaos and contributing to complex transient behaviors.

Contribution

It introduces the discovery of stable inhomogeneous limit cycles in quorum-sensing coupled oscillators and analyzes their bifurcations and role in chaotic dynamics.

Findings

Stable asymmetric limit cycles exist over wide parameter ranges.

These cycles displace hyperchaotic regimes and influence transient dynamics.

Bifurcation cascades lead to new limit cycles with split low-amplitude orbits.

Abstract

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 low-amplitude 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) low-amplitude orbits. Both cycles lose stability giving birth inhomogeneous chaos in the small parameter interval. Hyperchaos dominates in the parameter plane around the "island" with inhomogenous 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 "oscillating death" and the cycles observing in "smallest chimeras".