User:Khevsational/sandbox

Introduction
Swarmalators are generalization of phase oscillators that swarm around in space as they synchronize in time. They were introduced to model the many systems which both sync and swarm, such as vinegar eels, magnetic domain walls, and Japanese tree frogs.

Model
The swarmalator model in generic form is



\begin{array}{rcl} \dot{x}_i & = & \frac{1}{N} \sum_j I_{att}(x_j-x_i) F(\theta_j - \theta_i) - I_{rep}(x_j - x_i) \\ \dot{\theta}_i & = & \frac{1}{N} \sum_j G(\theta_j - \theta_i) H(x_j - x_i) \end{array} $$

While in general position could be in 2D or 3D $$x_i \in \mathbb{R}^d\ $$, the original swarmalator model introduced is a 2D model $$x \in \mathbb{R}^2$$ and the choices for $$ I_{att}(x) $$ etc were



\begin{array}{rcl} \dot{x}_i & = & \frac{1}{N} \sum_j (x_j - x_i) \left(1 + J \cos(\theta_j - \theta_i)\right) - \frac{x_j - x_i}{|x_j - x_i|^2} \\ \dot{\theta}_i & = & \frac{K}{N} \sum_j \frac{\sin (\theta_j - \theta_i)}{|x_j - x_i|} \end{array} $$

There are two parameters $$ J $$ and $$ K $$ are parameters: $$ J $$  controls the strength of phase-space attraction/repulsion, while $$ K $$  describes the phase coupling strength.

Phenomena
The model above produces 6 collective states depicted in the Figure below



The rainbow order parameters



\begin{array}{rcl} W_{\pm} & = & S_{\pm} e^{i \phi_{\pm}} = &\frac{1}{N} \sum_j e^{i(\phi_j - \theta_i)} \end{array} $$

where $$ \phi_i := arctan(y_i / x_i) $$ is the spatial angle can distinguish between the models states as shown in the Figure below.

Puzzles
There are several unresolved puzzles and open questions related to swarmalators:


 * Melting point $$ K_m $$: What is the value $$ K_m $$ at which the static async state melts into the active phase wave state?


 * Splitting point $$ K_s $$: What is the splitting point $$ K_s $$ at which the active phase wave splits into the splintered phase wave?


 * Rainbow order parameters: Can you derive an expression for the the supercritical branch of the $$ S_{\pm}(K) $$ in the active phase wave and splintered phase wave states?


 * $$ N_c $$: What determines the number of clusters formed in the splintered phase wave?

1D swarmalator model
A simpler swarmalator model where the spatial motion is confined to a 1D ring $$ x_i \in \mathbb{S}^1 $$ has also been proposed



\begin{array}{rcl} \dot{x}_i & = \nu_i + \frac{J}{N} \sum_j \sin(x_j - x_i) \cos(\theta_j - \theta_i) \\ \dot{\theta}_i & = \omega_i + \frac{K}{N} \sum_j \sin(\theta_j - \theta_i) \cos(x_j - x_i) \end{array} $$

where $$ \nu_i, \omega_i $$ are the (random) natural frequencies of the i-th swarmalator and are drawn from certain distributions $$ g(.) $$. This 1D model corresponds to the angular component of the 2D swarmalator model. The restriction to this simpler topology allows for a greater analysis. For instance, the model with identical natural frequencies, distributed frequencies, and distributed couplings have all been analyzed.