The Swarmalator

The Swarmalator taken from O'Keeffe and Strogatz (arXiv). N is the number of particles, K is the strength of the phase coupling (neighbours want to have same colour for K > 0, as different as possible for K < 0) and J is the attraction between phases (J > 0 like attracts like, J < 0 opposite phases attract). dt is the timestep for Euler integration (warning can go unstable for large dt!). Restart by clicking the N slider below which moves all the particles to a random position and gives then a random phase/colour.

Phases:

• Static Sync. e.g. (J,K) = (0.1,1)
• Static Async. e.g. (J,K) = (0.1,-1)
• Static Phase Wave. e.g. (J,K) = (1,0)
• Splintered Phase Wave. e.g. (J,K) = (2,-0.3) this can take some time to fully develop.
• Active Phase Wave. e.g. (J,K) = (1,-0.8)

Markov Model

A

Moiré patterns

Co-ordinate Change

The parabola $Y = X^2$. Click and drag the origin to translate and click and drag the axes to rotate.

Cubic Integration

Click on the x-axis to set the left limit, then click and drag to evaluate the integral. Click again to reset.

Verhulst Diagram

Click somewhere on the x-axis to start the iteration.

Regula Falsi

Click on the graph at two points to construct the linear approximation to the root. The two points must have opposite sign or you'll have to start again!

Newton–Raphson method

Perform one step of the Newton-Rhapson metthod by clicking on the function.

Complex Operations

Click a button to see the effect of that operation. Move the red dots around to change the inputs.

Cubic Differentiation

Use the sliders to adjust the function, click the function to see the tangent lines.

Quadratic Differentiation

Use the sliders to adjust the function, click the function to see the tangent lines.