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# Recent Explorables

# Ride my Kuramotocycle!

April 14, 2018

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.
The model is defined as a system of \(N\) oscillators. Each oscillator has a phase variable \(\theta_n(t)\) (illustrated by the angular position on a circle below), and an angular frequency \(\omega_n\) that captures how fast the oscillator moves around the circle.

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# Maggots in the Wiggle Room

April 8, 2018

This explorable illustrates an evolutionary process in an "ecosystem" of interacting species (cartoon maggots, in this case). Individuals move around in their enviroment, replicate and eat each other. Optionally, mutations can generate new species. The system is similar to the Explorable A Patchwork Darwinge, only a bit more animalistc and dynamically slightly different. However, for this one here, you need a bit more patience in order to observe interesting effects.

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# A Patchwork Darwinge

February 27, 2018

This explorable illustrates how the combination of variation and selection in a model biological system can increase the average fitness of a population of mutants of a species over time. Fitness of each mutant quantifies how well it can reproduce compared to other mutants. Variation introduces new mutants. Sometimes a mutant's fitness is lower than its parent's, sometimes higher. When lower, the mutant typically goes extinct, if higher the mutant can outperform others and proliferate in the population.

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# Into the Dark

February 12, 2018

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 is how it works: The swarm here consists of 100 individuals. Each individual moves around at a constant speed and changes direction according to three rules:

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# Barista's Secret

February 6, 2018

This explorable illustrates a process known as percolation. Percolation is a topic very important for understanding processes in physics, biology, geology, hydrology, horstology, epidemiology, and other fields. Percolation theory is the mathematical tool designed for understanding these processes.
Percolation is best understood in terms of the following physical situation: Let's say you have a porous medium, e.g. ground coffee in your percolator or porous soil of some depth. You pour a liquid on the medium and would like to know whether the liquid can make it through the medium and if so how this depends on the porosity of the medium (i.

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# Lotka Martini

January 28, 2018

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.

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# Double Trouble

January 12, 2018

This explorable illustrates the beautiful dynamical features of the double pendulum, a famous idealized nonlinear mechanical system that exhibits deterministic chaos. The double pendulum is essentially two simple pendula joined by a bearing. It's a classic complex system in which a simple setup generates rich and seemingly unpredictable behavior. The only force that is acting on it is gravity. There's no friction.
Initially, the pendulum is raised to some position and has therefore some potential energy with respect to the central pivot.

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# Critical HexSIRSize

January 5, 2018

This explorable illustrates the behavior of a contagion process near its critical point. Contagion processes, for example transmissable infectious diseases, typically exhibit a critical point, a threshold below which the disease will die out, and above which the disease is sustained in a population. Interesting dynamical things happen when the system is near its critical point.
First press the Play button. While the system is doing its thing, keep on reading.

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# Orli's Flock'n Roll

December 30, 2017

This explorable can be sluggish when viewed in Safari. It works much better in Chrome/Firefox
Important Note This explorable is an updated version of the previous Flock'n Roll Explorable. This new version is much better. It has Orli's Magic Switch. The implementation of the magic switch was suggested by Orli (*), a 6yr old girl and future collective behavior scientist who --rightfully-- pointed out that the original Flock'n Roll Explorable lacked creativity in the chosen color scheme.

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# Hokus Fractus!

December 29, 2017

This explorable illustrates one of the simplest ways to generate fractals by an iteration process in which elements of a structure are replaced by a smaller version of the whole structure. Similar to the Weeds & Trees Explorable, these structures can be viewed as Lindenmayer systems. A great variety of examples of such fractals exist.
Here you can explore some of the most famous ones (e.g. the Koch Snowflake and the Sierpinski Triangle) and a few that aren't so well known (e.

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# Particularly Stuck

October 9, 2017

This explorable illustrates a process known as diffusion-limited aggregation (DLA). It's a kinetic process driven by randomly diffusing particles that gives rise to fractal structures, reminiscent of things we see in natural systems. The process has been investigated in a number of scientific studies, e.g. the seminal paper by Witten & Sander.
The Model: The system is initialized by say 300 diffusing particles that move about randomly in the plane.

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# Cycledelic

October 3, 2017

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.
The Model: The foundation of the model is a set of 3 species A, B and C that are distributed in space and locally interact in a cyclic way: When species A (red) encounters species B (green), A "eats" B and replicates.

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# Kick it like Chirikov

October 3, 2017

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.
The Model: The system is defined by the iterative equations
[\begin{align} p _{n+1} &= p _n+K\sin(x_n) \mod 2\pi\newline x _{n+1} &= x _n +p _{n+1} \mod 2\pi \end{align}]
for the two variables $x$ and $p$.

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# Epidemonic

October 3, 2017

This explorable illustrates the dynamics of the SIRS epidemic model, a generic model that captures disease dynamics in a populations or related contagion phenomena.
The Model: Susceptible individuals (S) can be infected by coming in contact with other infected (I) individuals. Once infected they can transmit the disease until they recover (R) and become immune. After some time immunity wanes and individuals become susceptible again.

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# Keith Haring's Mexican Hat

October 3, 2017

This explorable illustrates one of the most basic pattern forming mechanism: Local excitation and long range inhibition. This mechanism or similar ones are responsible for patterns observed in neural tissue, animal fur and spatial heterogeneity in social systems.
This is how it works:
The state of the two dimensional system is defined by a state variable $u(x,t)$ at each position $x$ at time $t$. The state variable can have values in the range $[-1,1]$.

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# Spin Wheels

September 10, 2017

This explorable illustrates some interesting and beautiful properties of oscillators that are spatially arranged on a lattice and interact with their neighbors.
This is how it works: Each pixel in the panel on the right represents an oscillator. The state of each oscillator $n$ is defined by its phase $\theta_n(t)$, the angle of a circle. Because of this, we represent the state of each oscillator by a color in the cyclic rainbow spectrum.

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# Weeds & Trees

August 7, 2017

This explorable illustrates how fractal patterns observed in natural systems, particularly structural properties of some plants, can approximately be modeled by simple iterative models. Sometimes these models are refered to as Lindenmayer systems.
The pattern that you see is generated by a basic structure to start with and applying a simple replacement rule.
Initially we have a root branch (stem) of base length $L_0$. To this root we attach 3 leaf branches of lengths $l_1$, $l_2$, $l_3$ and at angles $\theta_1$, $\theta_2$, and $\theta_3$, respectively.

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# Flock'n Roll

August 2, 2017

This explorable can be sluggish when viewed in Safari. It works much better in Chrome/Firefox
This explorable illustrates of an intuitive dynamic model for collective motion (swarming) in animal groups. The model can be used to describe collective behavior observed in schooling fish or flocking birds, for example. The details of the model are described in a 2002 paper by Iain Couzin and colleagues.
Here's a short summary of how it works:

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