This explorable illustrates how remarkable spatio-temporal patterns can emerge when two dynamical phenomena, synchronization and collective motion, are combined. In the model, a bunch of oscillators move around in space and interact. Each oscillator has an internal oscillatory phase. An oscillator's movement and change of internal phase both depend on the positions and internal phases of all other oscillators. Because of this entanglement of spatial forces and phase coupling the oscillators are called swarmalators.

This explorables is a combination of two models, one that explains the emergence of dynamic patterns and collective behavior in schools of fish or flocks of birds, the second, the Schelling model, captures the geographic segregation of populations of different kinds of individuals. When these two models are combined, a great variety of beautiful dynamic swarming patterns can be generated. These patterns show traces of the generic swarming states like "milling" and segregation effects within these dynamic states.

This explorable illustrates the wealth of dynamical states exhibited in a network of Janus Oscillators. Janus oscillators are a generalization of ordinary phase coupled oscillators (see Complexity Explorable "Ride my Kuramotocycle!"). Like the Roman god Janus, a Janus oscillator has two "faces", each corresponding to a distinct internal phase variable. This slight generalization produces a surprisingly large array of complex phenomena in a simple ring arrangement. Press play and keep on reading.

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.

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 \((x_{n},y_{n})\):

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.

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.

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 activator-inhibitor 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 Lotka-Volterra System, a paradigmatic model for predator prey dynamics.

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.

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

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

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