Bifurcations and Singularities for Coupled Oscillators with Inertia and Frustration

Area of Science:

• Complex systems
• Nonlinear dynamics
• Theoretical physics

Background:

• The Kuramoto model is a standard framework for studying synchronization phenomena in coupled oscillator systems.
• Understanding the impact of system parameters on synchronization transition types is crucial for diverse applications.

Purpose of the Study:

• To investigate how nonzero inertia affects the nature of synchronization transitions in Kuramoto-like models.
• To analyze the role of singularities in controlling system dynamics near bifurcations.

Main Methods:

• Utilized an unstable manifold expansion, inspired by Crawford's work.
• Analyzed the behavior of singularities in the vicinity of bifurcations.
• Performed numerical simulations to validate theoretical predictions.

Main Results:

• Demonstrated that any small, nonzero inertia can change synchronization transitions from continuous to discontinuous, or vice versa.
• Identified that singularities near bifurcations are key determinants of qualitative system behavior, not mere artifacts.
• Numerical tests confirmed the theoretical findings regarding inertia's impact and singularity control.

Conclusions:

• Nonzero inertia is a critical factor that fundamentally alters synchronization transition types in Kuramoto-like models.
• The presence and behavior of singularities near bifurcations play a crucial role in governing the system's dynamics.
• This study provides a deeper understanding of synchronization dynamics in coupled oscillator systems with inertia.