Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.
One of the big applications of phase portraits, a method for visualizing changes in a dynamic system, was done by Edward Lorenz, who wondered in 1961 if math could be used to predict the weather. He developed 12 equations involving several variables including temperature, pressure, wind speed, and so on. He fortunately had computers to help him with the calculations and…he found his models didn’t do a good job of accurately getting down the weather. Short term, everything was fine but the further out one went then the worse the model became. This isn’t surprising because of the many factors going into the system. Lorenz decided to simplify his models by focusing on the convection and current of cold/hot air. This motion is circular in nature as the warm air rises and the cool air sinks. 3 total differential equations were developed to examine this, and Lorenz was very confident his new work would resolve the long-term lack of predictability (Parker 85-7, Bradley, Stewart 121).
Instead, each new run of his simulation gave him a different result! Close conditions could lead to radically different results. And yes, it turns out that the simulation would upon each iteration round the prior answer from 6 significant digits to 3, leading to some error but not enough to account for the results seen. And when the results were plotted in phase space, the portrait became a set of butterfly wings. The middle was a bunch of saddles allowing for a transition from one loop to another. The chaos was present. Lorenz released his results in the Journal of Atmospheric Science entitled “Deterministic Nonperiodic Flow” in 1963, explaining how long-term forecasting was never going to be a possibility. Instead, the first strange attractor, the Lorenz attractor, was discovered. For others, this led to the popular “Butterfly effect” that is so often quoted (Parker 88-90, Chang, Bradley).
A similar study into nature was conducted by Andrei Kolmogorov in the 1930s. He was interested in turbulence because he felt it was nestling eddy currents forming within each other. Lev Landau wanted to know how those eddies form, and so in the mid-1940s started to explore how the Hopf bifurcation came about. This was the moment when random motions in the fluid suddenly became periodic and started cyclic motion. As a fluid flows over an object in the path of the flow, no eddies form if the speed of the fluid is slow. Now, increase the speed just enough and you will have eddies form and the faster you go the further away and longer the eddies become. These do translate into phase space rather well. The slow flow is a fixed point attractor, the faster one a limit cycle and the fastest results in a torus. All of this assumes we reached that Hopf bifurcation and so entered a period motion – of a sort. If indeed period, then the frequency is stablished and regular eddies will form. If quasiperiodic, we have a secondary frequency and a new bifurcation arises. Eddies stack up (Parker 91-4).
To David Ruelle, this was a crazy result and too complicated for any practical use. He felt the initial conditions of the system should be enough to determine what is happening to the system. If an infinite amount of frequencies were possible, then Lorenz’ theory should be terribly wrong. Ruelle set out to figure out what was going on and worked with Floris Takens on the math. Turns out, only three independent motions are required for turbulence, plus a strange attractor (95-6).
But don’t think that astronomy was left out. Michael Henon was studying globular star clusters which are full of old, red stars in close proximity to one another and therefore undergo chaotic motion. In 1960, Henon finishes his Ph.D. work on them and presents his results. After taking many simplifications and assumptions into account, Henon found that the cluster will eventually undergo a core collapse as time progresses, and stars start to fly away as energy is lost. This system is therefore dissipative and continues on. In 1962, Henon joined up with Carl Heiles to further investigate and developed equations for the orbits then developed 2D cross sections to investigate. Many different curves were present but none allowed a star to return to its original position and the initial conditions did impact the trajectory taken. Years later, he recognizes that he had a strange attractor on his hands and finds that his phase portrait has a dimension between 1 and 2, demonstrating “space was being stretched and folded” as the cluster progressed in its life (98-101).
How about in particle physics, a region of seemingly compounding complexity? In 1970 Michael Feigenbaum decided to pursuit the chaos he suspected in it: the perturbation theory. Particles hitting each other and thus causing further changes was best attacked with this method but it took lots of calculations and then to find some pattern in it all…yes, you see the issues. Logarithms, exponentials, powers, many different fits were tried but to no avail. Then in 1975 Feigenbaum hears of bifurcation results and decides to see if some doubling effect was happening. After trying many different fits, He found something: when you compare the difference in distances between the bifurcations and find the successive ratios converge to 4.669! Further refinements narrowed down more decimal places, but the result is clear: bifurcation, a chaotic characteristic, is present in particle collision mechanics (120-4).
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Evidence for the Chaos
Of course all of these results are interesting, but what are some practical, hands-on tests that we can perform to see the validity of phase portraits and strange attractors in chaos theory? One such way was done in the Swinney-Gollub Experiment, which builds on the work of Ruelle and Takens. In 1977, Harry Swinney and Jerry Gollub used a device invented by M.M. Couette to see if the expected chaotic behavior would crop up. This device consists of 2 cylinders of different diameters with liquid between them. The inner cylinder rotates and the changes in the fluid cause flowing, with the total height of 1 foot, an outer diameter of 2 inches, and a total separation between cylinders of 1/8 of an inch. Aluminum powder was added to the mix and lasers recorded the speed via the Doppler Effect and as the cylinder spun the changes in frequency could be determined. As that velocity increased, waves of different frequencies began to stack up, with only a Fourier analysis able to discern the finer details. Upon completing that for the data collected, many interesting patterns emerged with several spikes of different heights indicating quasiperiodic motion. However, certain velocities would also result on long series of spikes of the same height, indicating chaos. The first transition ended up being quasiperiodic but the second was chaotic (Parker 105-9, Gollub).
Ruelle read up on the experiment and notices it predicts much of his work but notices that the experiment only focused on specific regions of the flow. What was happening for the entire batch of contents? If strange attractors were happening here and there, were they everywhere in the flow? Around 1980, James Crutchfield, J.D. Farmer, Norman Packard, and Robert Shaw resolve the data issue by simulating a different flow: a dripping tap. We have all encountered the rhythmic beat of a leaky faucet, but when the drip becomes the smallest flow we possible get then water can stack up in different ways and therefore regularity isn’t happening anymore. By placing a microphone at the bottom, we can record the impact and get a visualization as intensity changes. What we end up with is a graph with spikes, and after a Fourier analysis was done it was indeed a strange attractor much like Henon’s! (Parker 110-1)
Predicting the Chaos?
As strange as it may sound, scientists have possible found a kink into the chaos machine, and it is…machines. Scientists from the University of Maryland have found a breakthrough with machine learning, when they developed an algorithm that enabled the machine to study chaotic systems and make better predictions based off it, in this case the Kuramoto-Sivashinksky equation (which deals with flames and plasmas). The algorithm took 5 constant data points and using the past behavior data as a basis for comparison, the machine would update its predictions as it compared its projected to the actual results. The machine was able to predict to 8 factors of Lyapunov time, or the length it takes before the paths similar systems can take begin to separate exponentially. Chaos still wins, but the ability to predict is powerful and can lead to better forecasting models (Wolchover).
Bradley, Larry. “The Butterfly Effect.” Stsci.edu.
Cheng, Kenneth. “Edward N. Lorenz, a Meteorologist and a Father of Chaos Theory, Dies at 90.” Nytime.com. New York Times, 17 Apr. 2008. Web. 18 Jun. 2018.
Gollub, J.P. and Harry L. Swinney. “Onset of Turbulence in a Rotating Fluid.” Physical Review Letters 6 Oct. 1975. Print.
Parker, Barry. Chaos in the Cosmos. Plenum Press, New York. 1996. Print. 85-96, 98-101.
Stewart, Ian. Calculating the Cosmos. Basic Books, New York 2016. Print. 121.
Wolchover, Natalie. “Machine Learning’s ‘Amazing’ Ability to Predict Chaos.” Quantamagazine.com. Quanta, 18 Apr. 2018. Web. 24 Sept. 2018.
© 2018 Leonard Kelley