Coherent regimes of globally coupled dynamical systems

Area of Science:

• Dynamical systems theory
• Statistical physics
• Nonlinear dynamics

Background:

• Describing large populations of coupled dynamical systems is computationally challenging.
• Existing methods often struggle with parameter diversity and global coupling.
• Understanding collective phenomena like synchronization and pattern formation is crucial.

Purpose of the Study:

• To develop a simplified theoretical framework for analyzing the mean-field dynamics of diverse, globally coupled systems.
• To enable quantitative study of transitions between collective regimes.
• To investigate phenomena like oscillator death and full locking.

Main Methods:

• The proposed method reduces complex population dynamics to a few macroscopic degrees of freedom.
• It is applicable to populations of any size and functional form within the coherence region.
• The method requires linear variation or a narrow distribution of the dispersed parameter.

Main Results:

• The mean-field dynamics can be effectively described by a reduced set of macroscopic variables.
• Transitions between collective regimes are analyzed as bifurcations of these macroscopic variables.
• The approach quantitatively captures phenomena such as oscillator death and the route to full locking in chaotic oscillators with time-scale mismatch.

Conclusions:

• The developed method provides a powerful approximation for studying complex population dynamics.
• It offers new insights into the transitions between different collective behaviors.
• This framework is valuable for analyzing synchronization and pattern formation in diverse coupled systems.