Скачать или смотреть Viscosity and norm of phase gradient in a Kuramoto model on the sphere

In this simulation of a Kuramoto model on the sphere with noise, the color hue indicates the vorticity, while the radial or z-coordinate depend on the norm of the gradient. The color-coding has been chosen to emphasize the annihilation of singularities, which are points around which the phase makes a whole turn. Most of these annihilations occur at the beginning of the simulation.
The variables in this equation are angles, or phases, describing the state of a so-called phase oscillator. This oscillator could be a pendulum, a firefly, or any other system with a periodic motion. The dynamics of each oscillator is driven by two terms: a local term, which is just a rotation at constant angular speed, and a coupling to the four nearest neighbors. The coupling is proportional to the negative of the sine of the phase difference, which tends to align the phases, a phenomenon called synchronization.
The initial phases of the oscillators are random. The coupling soon leads neighboring oscillators to have similar phases. However, there is a topological feature, which is that the system develops singularities, which are points around which the phase makes a full turn. This singularities can move over time, and in some cases annihilate each other.
The simulation grid used here is obtained by projecting on the sphere a regular square grid on the faces of a cube.
The video has two parts, showing the same simulation with two different visualizations:
3D view: 0:00
2D view: 1:28
The color hue depends on the vorticity of the field, which is almost zero except near the singularities. The radial coordinate (in 3D) and the z-coordinate (in 2D) depend on the norm of the gradient of the phase. In the 3D part, the point of view of the observer turns around the sphere on an orbit in a plane containing the center of the sphere. The 2D part uses equirectangular coordinates.

Render time: Part 1 - 1 hour 20 minutes
Part 2 - 1 hour 14 minutes
Color scheme: Plasma by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap

Music: Cylinder Four by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/...
Source: http://chriszabriskie.com/cylinders/
Artist: http://chriszabriskie.com/

See also https://images.math.cnrs.fr/Des-ondes... for more explanations (in French) on a few previous simulations of wave equations.

#Kuramoto #synchronisation #synchronization

The simulation solves a partial differential equation by discretization.
C code: https://github.com/nilsberglund-orlea...
https://www.idpoisson.fr/berglund/sof...