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

TL;DR

This work analyzes complex dynamics in unidirectionally coupled non-isochronous Stuart-Landau oscillators by combining analytical amplitude- and phase-response theory with stability analysis and Lyapunov-based maps. Starting from a two-oscillator model, it derives phase-difference equations, characterizes amplitude death, phase locking via Arnold tongues, and resonance-induced isola bifurcations, revealing routes to quasiperiodic dynamics on torus attractors. Extending to a triadic network, the study shows how quasiperiodic forcing yields rich behaviors including QS, partial QS, and chaos, mapped across multidimensional parameter spaces. The results offer mechanistic insights for designing controllable dynamical architectures and have potential implications for neuromorphic and signal-processing applications where hierarchical coupling and shear (non-isochrony) play key roles.

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.

Figures (11)

• Figure 1: Illustration of amplitude death in the follower's dynamics. The plot depicts four scenarios with a phase portrait and a time series. The variable parameters are shown in the insets. Green lines correspond to the leader's dynamics and blue lines to the follower's dynamics. Parameters: $\alpha=(0.6,1.0,1.5)$, $r_2=1.0$, $\omega_2=3.0$.
• Figure 2: Arnold Tongues for $(2:3$, $1:1$, $1:2$, $1:3)$ phase locking of follower oscillator in $r_1-\omega_1$ plane. From left to right coupling coefficient $\epsilon$ increases and plotted for $\epsilon=(0.8, 1.5, 2.0)$. From top to bottom non-isochronous parameter $\alpha$ increases and is plotted for $\alpha=(0.6,1.0,1.5)$. For the plot $r_2=1.0$, and $\omega_2=3.0$.
• Figure 3: Follower amplitude response to leader frequency. The solid curve denotes the response. The dashed curve represents the fold conditions. Red $\odot$ denotes supercritical saddle-node bifurcation and blue $\odot$ represents subcritical saddle-node bifurcation. Parameters: $\alpha=1.0$, $\omega_2=3.0$, $\lambda_1=1.0$, $\lambda_2=2.0$.
• Figure 4: Follower amplitude response to leader amplitude. The solid curve denotes the response. The dashed curve represents the fold conditions. Red $\odot$ denotes supercritical saddle-node bifurcation and blue $\odot$ represents subcritical saddle-node bifurcation. Parameters: $\alpha=1.0$, $\omega_1=3.0$, $\omega_2=2.0$, $\lambda_2=2.0$.
• Figure 5: Illustration of stability in the amplitude response to the leader's frequency. The blue shaded regions represent phase locking. Red data points represent unstable (or saddle) responses, and blue data points represent stable responses. Markers $\odot$ represent numerically obtained values. Parameters: $\alpha=1.0$, $r_1=1.0$, $\omega_2=3.0$, $\lambda_2=2.0$.