Bifurcations of periodic orbits in the system of coupled extended Stuart-Landau oscillators.

We investigate the bifurcations of periodic orbits and resulting collective dynamics in a system of two coupled identical bistable extended Stuart-Landau (Bautin) oscillators under frequency detuning. Using a symmetry-preserving phase-difference reduction under strict mathematical closure, we analytically and numerically trace the birth, evolution, and exact local stability thresholds of both symmetric and asymmetric periodic orbits. Special attention is directed to the emergence of asymmetric periodic orbits that break the inherent permutation symmetry of the system. We map these bifurcation transitions onto the coupling-detuning parameter plane, uncovering its intricate non-uniform landscape governed by a series of higher-codimension degeneracy points. In particular, we show that the emergence of both stable and saddle asymmetric invariant tori significantly expands the rich multistable complexity of the coupled extended Stuart-Landau oscillators. Building upon the resolved bifurcation structure of the periodic orbits, we single out the regions of symmetric and asymmetric synchronous (phase-locked) regimes and analyze their characteristics. Finally, we study oscillation quenching and identify the regions of a globally dominant amplitude death regime. These findings offer a unified framework for controlling complex collective behaviors in bistable oscillator networks.