Tag Archives: nonlinear dynamics

Turing Patterns: What do a leopard’s spots, vegetation in arid zones, and the formation of fingers have in common?

Please excuse my inconsistent posting of late, I have been deep down the rabbit hole of science. Last week, I attended the Society of Industrial and Applied Math (SIAM) dynamical systems conference. What fun!

I learned about Turing Patterns, named for mathematician Alan Turing. Complex patterns can arise from the balance between the diffusion of chemicals and the reaction of those chemicals. For this reason, Turing’s model is also called the Reaction-Diffusion model. In general, these kinds of patterns can arise when there’s some kind of competition.

This sounds abstract, but suspected examples in nature abound. Have you ever wondered how the leopard got his spots or what’s behind the patterns on seashells? We often don’t know the chemical mechanisms that produce the patterns, but we can mathematically reproduce them with generic models.

Image from wired.com discussion of Turing patterns.

Mary Silber and her grad student Karna Gowda presented research on Turing patterns in the vegetation of arid regions. When there isn’t enough precipitation to support uniform vegetation, what vegetation will you observe? If there’s too little water, their model yields a vegetation-free desert. Between “not enough” and “plenty” the model generates patterns, from spots to labyrinths to gaps. Their work expands at least two decades worth of study of Turing patterns in vegetation.


Figure by Karna Gowda, see the full article at SIAM news.

Silber and Gowda considered an area in the Horn of Africa (the bit that juts east below the Middle East). Here, stable patterns in the vegetation have been documented since the 1950s. They wanted to know how the patterns have changed with time. Have the wavelengths between vegetation bands changed? Are there signs of distress due to climate change? By comparing pictures taken by the RAF in the 1950s to recent satellite images, they found that the pattern were remarkably stable. The bands slowly travelled uphill, but they had the same wavelength and the same pattern. They only observed damage in areas with lots of new roads.


From google maps of the Horn of Africa! I screen-capped this from here.

Turing patterns have even been studied experimentally in zebrafish. Zebrafish stripes might appear stationary, but they will slowly change in response to perturbations. So scientists did just. Below is a figure from the paper. The left shows the pattern on the zebrafish, the right shows the predictions of the model.


Experimental perturbations to the patterns of zebrafish are well-predicted by the Turing model. Read more in this excellent Science paper.

The model has been used to explain the distribution of feather buds in chicks and hair follicles in mice. Turing’s equations have even been used to explain how fingers form.

If you want to learn more, the links above are a great start. And if you want to play with the patterns yourself, check out this super fun interactive. These waves aren’t stationary like the Turing patterns I described here, but they arise from similar mathematics. The interactive can make your computer work, fyi.


Reaction-diffusion pattern I generated with this online interactive. It’s super fun!


The Beautiful Lab

Across the country, thousands of labs study thousands of topics. In my lab, we study nonlinear dynamics in electrochemical oscillators. The dynamics of these oscillators can be used to make math models for other oscillators we might be very interested in, like heart cells, breathing, and neurons in the brain. Oscillators and their dynamics show up in many places. In a previous post on synchrony, I discuss some of these dynamics.

My experiments aren’t particularly much to look at. The beauty in mostly in the data. But here are a few of my better snaps over the years. There can also be science in the photographic technique. The bottom two photos were taken using reverse lens macro, a cheap way to do great zoom shots.

From top to bottom the photos below show: the electrochemical 3 electrode cell, the variable resistance resistors for each electrode, and a capacitor. The featured image is of some resistors.



Fun Science: Chaos

The common concept of chaos as the disordered unknown is a concept thousands of years old. Chaos as a branch of mathematics refers to an extremely ordered but complex behavior. In some lights, mathematical chaos might seem utterly random; different inspections reveal order. The first model of chaos emerged in the 1950s to model the movement of air in the atmosphere– the Lorenz model.

Defining Chaos in Mathematics

First, let’s discuss what chaos is in a mathematical sense (chaos on scholarpedia). In daily language, we would assume that something chaotic is not at all predictable. Chaos is deterministic, which means that if we know the position or value of a chaotic element exactly, then we know every value that it will ever have for all times. Many non chaotic systems are deterministic also; if we measure the position and velocity of a pendulum and its frictional loss, we can predict its path for all time.

If we are measuring some value, there is a limit to our possible accuracy. If you measure your weight, you are measuring your weight plus or minus some error associated with the instrument. We know some scales are more accurate, and some are less accurate. When we measure the position and velocity of the pendulum, there is some error, but the behavior at position x plus a little bit (or x+ε) is essentially identical to the behavior at position x. In a chaotic system, the paths following positions x+ε and x will be entirely different after some window of time. The paths will be similar for some window of time; the length of this window is determined by how chaotic the system is. This behavior is known as sensitive dependence upon initial conditions (or more famously, the butterfly effect).

The lower half of the image below shows the difference between two paths starting at x+ε and x in the Lorenz attractor. The paths are nearly the same, until they quickly become very different. You can look at the Lorenz attractor in more detail and play with it here. The link goes to a Java applet hosted by Caltech where you can change the parameters and see what happens in a time series and in a different representation called the state space.

The above image also shows the third of the three most common criteria for chaos: periodicity. This means that the system oscillates in some kind of way. The top half the picture above shows the time series of z variable (see link above for the full model if you are interested). If we plot the data in a different way, we get the image below. This image is called the Lorenz attractor. We see it loops around, but that it is bounded in space. That is, there are places we can say the path will not go. Intriguingly, the Lorenz attractor has a fractal dimension of 2.06. For more on fractals, you can read my posts on fractals and fractal dimensions.

Simple example of chaos: the double pendulum

Many pretty simple systems can exhibit chaos. The easiest example is the double pendulum, which is just a pendulum whose bottom is the top for a second pendulum.  The video below shows the complexity of behavior achieved by the double pendulum. Chaos has also been demonstrated in electrical circuits, biological systems, electrochemical reactions and planetary orbits. In another post, I will write about the amazingly universality of chaos, which shows how all of the diverse systems I listed are in fact very similar.

Fun Science: Art Resembling Science

Can you tell the two below pictures apart? Which of the following two images is a modern aboriginal painting, and which is a picture of an oscillatory chemical reaction?


The left image is of the Belousov-Zhabotinsky (BZ) chemical reaction. The right image is a painting from Western Australia using aboriginal techniques. They are strikingly similar. Could it just be a coincidence? I believe some models with bacteria growing competitively yield similar patterns; perhaps such patterns existed in nature. (EDIT: traveling wavefronts like above have been shown in slime molds. A search on “cAMP spiral waves” reveals many examples.)

Art: Aboriginal designs from Western Australia

The image on the right above came from a book about Aboriginal art called Balgo-4-04 that I found at the Kluge-Ruhe Aboriginal collection in Charlottesville, VA. Unfortunately, I did not think to write down the title of the exact work, or its info (hopefully I will go back soon and retrieve it).

The collection was put together by Warlayirti artists from Western Australia. I don’t know the year of this painting, but I think it is modern and based upon older sand painting techniques. Unfortunately I am not enough of an expert on this topic to provide any deep insights. If anyone else is, I’d love to learn more.

Science: Belousov-Zhabotinsky reaction

The image on the left is an image of the Belousov-Zhabotinsky (BZ) reaction (photo credit: Brandeis U Chemistry). For a more technical overview, check out the Scholarpedia page. Transition metals at different oxidation states lead to these colors; the particular metal can be varied to give different properties. Cerium and manganese, as well as many others, can be used in the reaction. The curling waves in the dish are traveling oscillations. The video below shows the patterns in time.

Another good youtube video showing how the BZ reaction is set up is here. The behaviors observed in the BZ reaction occur in other oscillatory systems. The spiral waves are 2D analogies to the 3D scroll waves that occur in the heart during ventricular fibrillation (VF). VF causes the heart to quiver and is deadly. In this link, wave-propagation in the heart is shown under several conditions (using a java plug-in). If you click “java applets” on the left, under the “introduction” header, you can choose VF, VT (ventricular tachycardia), and normal heart rhythm. You can then apply defibrillation to these rhythms and see what happens. The website is maintained by a scientist, Flavio Fenton, who researches nonlinear dynamics in hearts and biological systems.

For more discussion on oscillatory dynamics, check out my post on synchrony.

Fun Science: Synchrony

Have you ever wondered why fireflies flash at the same time ? Or how the heart contracts and relaxes? Why they had to shut down the Millennium Bridge in London for repairs? These are all questions related to synchrony. (See the following papers about these questions if you are interested in the mathematics: J Buck, 1988, Quar Rev Biol; Strogatz, 2000, Physica D; Strogatz et al, 2005, Nature; Michaels et al, 1987, Circulation Res.)

By synchrony, I am talking about the tendency of systems that periodically do something to align into patterns. This periodic action could be contracting (heart cells), chirping (crickets), firing (neurons), stepping (people walking), swinging (pendulums)… you get the idea. Synchrony can apply to the simplest or the most complicated interacting items, from transistors to crickets and neurons and people. The study requires only some kind of repetitive action.

How do all these systems synchronize? The elements communicate in some way. At a concert, thousands of people can clap together at the same beat because they hear each other. In the example below, 32 metronomes synchronize because the table is not fully rigid. Each metronome is slightly disturbed by the shaking in the table, and is slightly changed by it. As a dominant timing emerges, the metronomes synchronize. And each metronome retains its natural character– no oscillators stop or become greatly faster to achieve synchrony.

Synchrony isn’t always desirable. When the Millennium Bridge was opened, it was a new kind of bridge design. As thousands of people crossed it, it began to sway from side to side. Due to the slight swaying, people trying to maintain their balance began stepping with this rhythm, adding energy to this rhythm. This continued to the point where the bridge swung visibly. It was shut down, and dampening was added. The Millennium Bridge was similar to the famous Tacoma Narrows bridge, except that thousands of people had to act in unison to activate the resonant frequency. Synchrony is also believed to play a role in epilepsy. The theory is that a strong synchronous signal emerges, and this signal overwhelms the normal functions in the brain. So when we study synchrony, we wish to understand how it arises, and sometimes how to destroy it.

I study synchrony in electrochemical oscillators. Drop me a note or a comment if you have questions or thoughts.

Some more cool videos:

And a topic for future discussion, chaos

  • Chaotic double pendulum– amazingly, something as simple as a pendulum with two vertices exhibits some wild and chaotic behavior.