Clustering and Bellerophon state in Kuramoto model with second-order coupling

Area of Science:

• Complex systems
• Nonlinear dynamics
• Statistical physics

Background:

• The Kuramoto model is a fundamental tool for studying synchronization in coupled oscillator systems.
• Understanding synchronization transitions is crucial for various fields, including neuroscience and power grids.
• Investigating frequency distributions, like the bimodal Lorentzian, reveals complex emergent behaviors.

Purpose of the Study:

• To analyze clustering and synchronization transitions in the Kuramoto model with second-order coupling.
• To explore the impact of a bimodal Lorentzian frequency distribution on synchronization dynamics.
• To characterize emergent states, such as Bellerophon states, within this system.

Main Methods:

• Linear stability analysis to determine critical coupling strengths.
• Ott-Antonsen ansatz for theoretical treatment of the system.
• Numerical simulations to verify theoretical predictions and explore parameter space.

Main Results:

• The critical coupling strength for synchronization was theoretically derived and numerically confirmed.
• Multiple synchronization pathways were identified, including first- and second-order transitions and multiple bifurcations.
• Bellerophon states were observed and their dynamical features were fully characterized under specific parameter conditions.

Conclusions:

• The Kuramoto model with a bimodal Lorentzian frequency distribution exhibits rich synchronization phenomena.
• System parameters significantly influence the type of synchronization transition and emergent states.
• The findings provide insights into the complex dynamics of coupled oscillator systems with non-uniform frequency distributions.