THE FEIGENBAUM CONSTANT

HELPING SCIENTISTS CONTROL CHAOS

Guest post by EMILY MAVERICK

MSU SciComm blog contest winner


You’re driving across the country on a late night in July. You need a place to stay the night so you take the next exit ramp and find yourself on a sleepy Main St. in Middle-of-Nowhere, Kansas. You see a red light ahead and stop, because even though there’s no one else on the road, you follow the rules. Twenty seconds later the light turns green and you take a left toward the nearest motel for the night.


The next morning you awake to the hustle and bustle of people going about their business. Well, as much hustle and bustle as one would expect in Middle-of-Nowhere, Kansas. Just as you pull out of the motel parking lot, POP! Your front tire is toast! You slowly roll toward Main St. in search of a mechanic, stopping at the same red light from last night. There are a few more cars at the intersection this morning, but everyone follows the rules and twenty seconds later you turn onto Main St. and make it to the body shop.

As you wait for your car to be repaired you start noticing some strange things. A truck decorated like an ear of corn goes by followed closely by a trailer full of farm animals. You ask the mechanic what’s going on and he tells you it’s the first day of the county fair. “Every popcorn lover within two state lines will be flooding in over the next few hours!” You’ve got a good view of that intersection on Main St. and as cars trickle into town it starts getting pretty entertaining. Some drivers are looking at their phones trying to follow their GPS to the fairgrounds. Some are jumping the light when they see an opening because they need funnel cake now. As more and more cars pile into town the intersection gets even crazier! Now some drivers are following Siri blindly and some have gotten out of their cars to yell at other drivers. An amateur stilt-walker is taking forever to cross the street, completely blocking one arm of the intersection for a while. The sheer number of cars matriculating through town has led to utter chaos!


Although I made up this town and its county fair, what happened at the intersection illustrates a remarkably common natural phenomenon called the bifurcation route to chaos. To see what I mean, let’s build a graph of the time drivers waited at the intersection as the number of cars in town changed. Panel A, below, shows your progress late at night on Thursday, when you were the only one on the road and you waited twenty seconds to turn. Panel B shows what happened in the morning when there were a few more cars but everyone still only waited twenty seconds because the drivers followed the very simple rules of the traffic light cycle.

However, once cars started coming into town for the fair, new outcomes revealed themselves in the traffic patterns. The drivers that really wanted funnel cake spent less time waiting because they anticipated the light turning green and jumped the light a little bit. On the other hand, drivers who waited for their GPS to reroute them hesitated to do anything, and so they spent a little longer than twenty seconds at the light. These new outcomes look like prongs in panel C. The lower prong represents slightly shorter waiting times and upper prong slightly longer ones. As more and more cars matriculate through the intersection the graph splits over and over again as new outcomes occur—like panel D.


One of these prongs represents angry drivers who got out of their cars to yell, one represents the drivers that had to wait for the stilt walker, and another prong represents the drivers who ignore the rules as they follow Siri. Each split in the graph is what mathematicians would call a “bifurcation” and all throughout the natural world bifurcations can hint that a system is approaching chaos.


Mitchell Feigenbaum was a mathematician and physicist who studied mathematical equations that reproduced these bifurcation routes to chaos. He found many different equations that resulted in bifurcating graphs that eventually devolved into chaos, but what shocked him was what all the models had in common.


If we slice up our bifurcation diagram at each of the bifurcations, you will notice that the slices get thinner and thinner, like the panel on the right. In our case, this means that the more cars that come to town the more sensitive the system becomes to additional vehicles and the faster it becomes chaotic. Feigenbaum discovered that all his bifurcating models looked like this too. Specifically, he concluded that the width of one slice divided by the width of the next slice converged on a constant value: ~4.669. Since the discovery of this so-called Feigenbaum constant in 1975, no one has figured out what this ratio really means. All we know is that it can be found all around us – in fluid dynamics, in neuroscience, and in population biology – and that makes it universal.


Although we may not know exactly what the value means, we know the Feigenbaum constant exists and therefore we can harness its power. Last year, for example, Omer Tzuk at Tel Aviv University and colleagues suggested that the Feigenbaum constant can be used to predict which ecosystems are most susceptible to drought. These scientists created a mathematical model of an ecosystem and observed how plant growth was affected when they changed the levels of rainfall in the model.


Specifically, they measured the mass of plant vegetation that grew as they reduced rainfall each season. In seasons with plenty of rain the growing plant mass reached a single predictable value every time they ran the simulation. But when they reduced the amount of rain that fell, their graph bifurcated. Now the mass of plant matter that grew had two possible values during each season. They observed more and more bifurcations at even lower precipitation levels until there was no predictable value of vegetation each season.


In the real world, an ecosystem in this chaotic state would not be able to support a diversity of organisms and would collapse. The authors think that acquiring data on plant biomass in real ecosystems could potentially help scientists predict how sensitive those ecosystems are to drought. For example, if plant biomass fluctuates between two different values during the season, scientists might conclude that the ecosystem is in the second slice of the bifurcation diagram. This information could act as a warning sign that the ecosystem may be more sensitive to a drought in the upcoming season – just like our traffic intersection became more sensitive to additional cars after the first bifurcation. This is the Feigenbaum constant in action.


Given the universality of the Feigenbaum constant, it’s exciting to think about how many other real-world problems could be solved using bifurcation analysis. Could we save a species on the cusp of extinction? Could we anticipate natural disasters before they happen? Or even predict when an epilepsy patient will have their next seizure? The answers to these questions and more may be held in the meaning of the Feigenbaum constant. When we unlock the mystery, we will be one step closer to controlling chaos.


EMILY MAVERICK is a PhD candidate at Colorado State University studying the proteins that control electricity in our nervous systems. When not doing experiments or writing about cool science, you'll find her in the mountains backpacking or rock climbing. You can find more of her musings on Twitter: @EmilyEMaverick or on her personal blog: www.scitations.blogspot.com


IMAGE CREDIT: Wikimedia commons


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