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

TL;DR

This paper explores the complex behaviors of unidirectionally coupled non-isochronous Stuart-Landau oscillators, revealing how different parameters lead to various attractors including quasiperiodic oscillations, and providing insights for designing controllable dynamical systems.

Contribution

It provides a comprehensive nonlinear analysis of the dynamics of coupled non-isochronous Stuart-Landau oscillators, including stability and attractor classification, bridging theoretical predictions with potential experimental applications.

Findings

Weak forcing or coupling leads to quasiperiodic oscillations.

Parameter mapping classifies attractors into periodic, quasiperiodic, partially synchronized, and chaotic.

Steady-state stability analysis reveals instability of periodic attractors under certain conditions.

Abstract

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 toward 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 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.