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Synchronization of relativistic particles in the hyperbolic Kuramoto model
Louis M Ritchie1, M A Lohe1, Anthony G Williams1
1Centre for Complex Systems and Structure of Matter, Department of Physics, University of Adelaide, Adelaide 5005, Australia.
Chaos (Woodbury, N.Y.)
|June 3, 2018
Summary
This study introduces a noncompact Kuramoto model using hyperbolic functions, demonstrating synchronization in relativistic particle systems. Synchronization occurs universally for positive couplings, with potential singularities under specific conditions.
Area of Science:
- Theoretical Physics
- Nonlinear Dynamics
- Statistical Mechanics
Background:
- The Kuramoto model describes synchronization in coupled oscillators.
- Standard Kuramoto model relies on compact SO(2) symmetry.
- Exploring noncompact symmetry extensions is crucial for broader applications.
Purpose of the Study:
- To formulate and analyze a noncompact version of the Kuramoto model.
- To investigate synchronization phenomena in systems with noncompact symmetry.
- To establish a physical interpretation in terms of relativistic particle dynamics.
Main Methods:
- Replacing SO(2) with the noncompact group SO(1,1).
- Expressing system equations using hyperbolic angles.
- Numerical simulations and exact solutions for N=2.
- Analysis using order and disorder parameters.
Main Results:
- Synchronization occurs for any positive couplings, parameters, and exponents.
- No critical values for coupling constants are needed for synchronization.
- Singularities can arise with negative couplings or restricted initial values.
- Physical interpretation as synchronized relativistic particles.
Conclusions:
- The noncompact Kuramoto model exhibits robust synchronization.
- The model provides insights into collective behavior in relativistic systems.
- The framework is extendable to higher dimensions with SO(p,q) covariance.
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