Collective dynamics and phase transitions of Stuart-Landau oscillators on a ring network: Interplay of asymmetric and symmetric couplings

Understanding limit-cycle oscillators' collective dynamics and phase transitions is crucial in science and engineering. We study a system of identical nonisochronous Stuart-Landau oscillators with asymmetric and symmetric couplings on a ring network. Due to the interplay of the two couplings, oscillators can have various modes of phase relationships: phase-lockings, and phase-drifts. Phase-locked states give rise to different symmetric collective states, whereas phase-drift states lead to asymmetric collective states. We derive the common frequency and amplitude solutions of oscillators in phase-locking states. By defining an order parameter for the system and computing quantities such as effective frequencies, amplitudes, and phase dispersions, we obtain a phase diagram of the system and identify distinct collective dynamical regimes: (i) Complete phase locking, resulting in full synchronization. (ii) Phase drifts among oscillators, leading to quasiperiodic traveling waves. (iii) Uniform phase distributions among oscillators, giving rise to various splay states. (iv) Oscillation death with different steady states. Continuous and discontinuous phase transitions operate across different collective dynamical regimes. The phase transition operating across complete synchronization and traveling wave states is continuous, while the phase transition operating across traveling wave and splay states is discontinuous and is associated with hysteresis. Furthermore, the oscillators impart an explosive oscillation death, even if all the oscillators have the same frequency, along the routes across the splay and complete synchronization states. Finally, we highlight the implications for biological and physical applications.