Various oscillation patterns in phase models with locally attractive and globally repulsive couplings

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Theoretical Physics

Background:

• Coupled oscillator systems are fundamental in understanding emergent phenomena.
• Existing models typically incorporate either local or global coupling, limiting observed patterns.
• The interplay between different coupling types remains an underexplored area.

Purpose of the Study:

• To investigate a novel one-dimensional phase model combining local attraction and global repulsion.
• To identify and characterize novel spatiotemporal patterns arising from this hybrid coupling.
• To analyze the dynamics of relative phase oscillations, particularly those with unusual stability properties.

Main Methods:

• Development and analysis of a one-dimensional phase model with mixed coupling.
• Numerical simulations to observe spatiotemporal patterns and synchronization states.
• Theoretical analysis of phase space dynamics, including stability of periodic orbits.

Main Results:

• The model exhibits unique spatiotemporal patterns not seen in purely local or global coupling systems.
• Observed states include in-phase synchronization, traveling waves, and three types of relative phase oscillations.
• A notable finding is a locally unstable but globally attractive oscillation, linked to saddle two-cluster states.

Conclusions:

• Hybrid coupling (local attraction and global repulsion) generates complex and novel system behaviors.
• The identified oscillation type highlights the intricate dynamics possible in coupled systems.
• Further research into the mechanism of these oscillations and their relation to heteroclinic orbits is warranted.