Complex dynamics and route to quasiperiodic synchronization in non-isochronous directed Stuart–Landau triads

The coupled Stuart–Landau equation serves as a fundamental model for exploring synchronization and emergent behavior in complex dynamical systems. However, understanding its dynamics from a comprehensive nonlinear perspective remains challenging due to the multifaceted influence of coupling topology, interaction strength, and oscillator frequency detuning. Despite extensive theoretical investigations over the decades, numerous aspects remain unexplored, particularly those that bridge theoretical predictions with experimental observations—an essential step towards deepening our understanding of real-world dynamical phenomena. This work investigates the complex dynamics of unidirectionally coupled non-isochronous Stuart–Landau oscillators. Calculations of steady-states and their stability analysis further reveal that periodic attractors corresponding to weak forcing or coupling regimes are dynamically unstable, which pushes the system towards quasiperiodic oscillation on the torus attractor. The connection between amplitudes in the synchronized state and the isola bifurcation is established. The mapping of parameter values with the kind of attractor of the oscillatory system is presented and classified into periodic, quasiperiodic, partially synchronized, and chaotic regions. The results of this study can be leveraged to design complex yet controllable dynamical architectures.