# “Hopfed Turingles”

## Pattern Formation in a simple reaction-diffusion system

With this explorable you can discover a variety of spatio-temporal patterns that can be generated with a very famous and simple autocatalytic reaction diffusion system known as the **Gray-Scott model**. In the model two substances \(U\) and \(V\) interact and diffuse in a two-dimensional container. Although only two types of simple reactions occur, the system generates a wealth of different stable and dynamic spatio-temporal patterns depending on system parameters.

# “Stranger Things”

## Strange attractors

This explorable illustrates the structure and beauty of * strange attractors* of two-dimensional discrete maps. These maps generate sequences of pairs of number \((x_n,y_n)\) where the index \(n=0,1,2,...\) denotes the step of the iteration process that starts at the point \((x_0,y_0)\). The map is defined by two functions \(f(x,y)\) and \(g(x,y)\) that determine the point \((x_{n+1},y_{n+1})\) given \((x_{n},y_{n})\):

# “Ride my Kuramotocycle!”

## The Kuramoto model

This explorable illustrates the ** Kuramoto model** for phase coupled oscillators. This model is used to describe synchronization phenomena in natural systems, e.g. the flash synchronization of fire flies or wall-mounted clocks.

# “Into the Dark”

## Collective intelligence

This explorable illustrates how a school of fish can collectively find an optimal location, e.g. a dark, unexposed region in their environment simply by light-dependent speed control. The explorable is based on the model discussed in Flock'n Roll, which you may want to explore first.

# “Lotka Martini”

## The Lotka-Volterra model

This explorable illustrates the dynamics of a ** predator-prey model** on a hexagonal lattice. In the model a prey species reproduces
spontaneously but is also food to the predator species. The predator requires the prey for reproduction. The system is an example of an

**system, in which two dynamical entities interact in such a way that the activator (in this case the prey) activates the inhibitor (the predator) that in turn down-regulates the activator in a feedback loop. Activator-inhibitor systems often exhibit oscillatory behavior, like the famous**

*activator-inhibitor***, a paradigmatic model for predator prey dynamics.**

*Lotka-Volterra System*# “Cycledelic”

## The spatial rock-paper-scissors game

This explorable of a pattern forming system is derived from a model that was designed to understand co-existance of cyclicly interacting species in a spatially extended model ecosystem. Despite its simplicity, it can generate a rich set of complex spatio-temporal patterns depending on the choice of parameters and initial conditions.

# “Kick it like Chirikov”

## The kicked rotator (standard map)

In this explorable you can investigate the dynamics of a famous two-dimensional, time discrete map, known as the standard or Chirikovâ€“Taylor map, one of the most famous simple systems that exhibits determinstic chaos.

# “Epidemonic”

## The SIRS epidemic model

This explorable illustrates the dynamics of the SIRS epidemic model, a generic model that captures disease dynamics in a populations or related contagion phenomena.

# “Spin Wheels”

## Phase-coupled oscillators on a lattice

This explorable illustrates some interesting and beautiful properties of oscillators that are spatially arranged on a lattice and interact with their neighbors.