Hierarchical clustering in mean-field coupled Stuart-Landau oscillators

Area of Science:

• Nonlinear dynamics
• Complex systems science
• Theoretical physics

Background:

• Clustered solutions in oscillator networks are crucial for understanding transitions from synchrony to complex spatiotemporal patterns.
• Existing knowledge on cluster formation and differentiation in coupled oscillator systems remains limited.
• These clusters serve as a vital link between fully synchronized and incoherent states in dynamical systems.

Purpose of the Study:

• To investigate the emergence of 3-cluster solutions from 2-cluster solutions in globally coupled Stuart-Landau oscillators.
• To analyze the organization and parameter space landscape of different 3-cluster solutions.
• To identify the underlying mechanism governing the transition between different cluster states.

Main Methods:

• Analysis of globally coupled Stuart-Landau oscillators.
• Focus on bifurcations from 2-cluster to 3-cluster solutions.
• Characterization of cluster arrangements and parameter space organization.

Main Results:

• Identified a codimension-two point, termed the "Type-II cluster singularity," which dictates the arrangement of clusters.
• Demonstrated the emergence of 3-cluster solutions from 2-cluster solutions.
• Revealed a hierarchical structure within multi-cluster solutions.

Conclusions:

• The Type-II cluster singularity is a key organizing principle for multi-cluster states in oscillator networks.
• Understanding cluster transitions provides insights into the emergence of complex dynamics from simpler states.
• The hierarchical structure of multi-cluster solutions offers a framework for further investigation into complex network dynamics.