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Transition to synchronization in a Kuramoto model with the first- and second-order interaction terms
Keren Li1, Shen Ma1, Haihong Li1
1School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China.
Summary
This study explores a Kuramoto model with complex interactions, revealing multiple stable states. The most stable state exhibits synchronized oscillators with a single peak phase distribution.
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 emergent collective behaviors, such as synchronization, is crucial in various scientific fields.
- Investigating higher-order interaction terms can reveal more complex dynamics than simpler models.
Purpose of the Study:
- To analyze a Kuramoto model incorporating both first-order and second-order interaction terms.
- To identify and characterize the different stable states (attractors) within this extended model.
- To map the parameter space and understand the conditions leading to various synchronization patterns.
Main Methods:
- Simulation of the Kuramoto model with modified interaction terms.
- Analysis of attractor coexistence and their properties.
- Investigation of phase distributions (unimodal, bimodal) of oscillators.
- Construction of transition diagrams using forward and backward continuation.
- Phase diagram construction in the parameter space.
Main Results:
- The model exhibits the coexistence of multiple attractors, indicating diverse possible system states.
- Attractors are distinguishable by the phase distributions of the coupled oscillators.
- The synchronous state with a unimodal phase distribution is found to be the most stable.
- Cluster synchrony with an evenly distributed bimodal phase distribution is identified as the least stable state.
- A comprehensive phase diagram illustrating the model's behavior across different parameters is presented.
Conclusions:
- The inclusion of higher-order interactions significantly enriches the dynamics of the Kuramoto model.
- The stability of different synchronization states is directly linked to the phase distribution of oscillators.
- The findings provide insights into the complex synchronization phenomena and their parameter dependencies.
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