The strange attractor that pulls the strings of chaos from phase space
Edward Lorenz was looking at the phenomenon of rolling fluid convection.
The physical model of this is simple: place a gas in a solid rectangular box with a heat source on the bottom. The gas is heated from below and cooled from above.
To mathematically model this, he simplified a few fluid dynamics equations (called the Navier-Stokes equations) and ended up with a set of three nonlinear equations:
The variables x,y and z are visualised as coordinates in 3-dimensional space that can be plotted to a graph via the x,y and z axis.
It must be remembered that in these equations the variables are functions of time; The equations describe the rate of change of these three quantities with respect to time.
In this ‘rolling fluid convection’ scenario, x is proportional to the rate of convection, y to the horizontal temperature variation, and z to the vertical temperature variation.
The parameters P, R and B are values Lorenz used to vary the properties of the system. P represents the ratio of fluid viscosity to its thermal conductivity. R represents the difference in temperature between the top and bottom of the system and B is the ratio of height to width of the box used to hold the system.
The values Lorenz used are P = 10, R= 28, B = 8/3.
So how can these equations be plotted?
The parameter t (time) needs to be treated as though it proceeds in discrete “steps” ,instead of varying continuously. The smaller the steps, the more accurate the approximation.
For values of dt around 0.001 the approximation will not break down and the plotting will not be too slow. Then successive values of x, y, z can be recursively computed.
The previous state of (x,y,z) must be saved, in order to compute the next. As can be seen from the equations, each new value depends on all the old values,
On initial glance, these three equations appear simple enough to solve. But when the values are plotted we discover an extremely complicated dynamical system.
The system is chaotic. The values for the variables x,y and z (ie. the state of the system) taken at any one time are never repeated again. Infinite variety springs from these simple equations, that is, as long as time keeps up its relentless march.
When each of these unique variable states are plotted into phase space, instead of the random, chaotic pattern that might be expected, something else begins to emerge.
This seemingly ‘ordered’ form that presides over chaos from phase space has been dubbed a strange attractor. (see featured picture)
Finding this ‘higher dimensional order’ that informs the chaotic, has only become possible in the computer age. Manually plotting recursive points in phase space would not yield a pattern as we would not be able to plot enough points to recognise a pattern emerging.
Points with similar values may be arbitrarily near to each other or far apart.
Strange attractors are unique from other phase-space attractors in that one does not know exactly where on the attractor the system will be. Two points on the attractor that are near each other at one time will be arbitrarily far apart at later times. The only restriction is that the state of the system must remain on the attractor.
Strange attractors never close on themselves and therefor the motion of the system will never repeat.
The following three figures — made using Lorenz’s values for P, R and B show three time segments of the 3-D evolution of 2 trajectories (one in blue, the other in yellow) in the Lorenz attractor.
The blue and yellow trajectories start out, to all intents and purposes at the same point.
In fact they have the same y and z values and their x points only differ by a value of 0.00001
At t=1
Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one)
We can see that a short term prediction of the state of the yellow system, if blues system had been previously measured would hold good.
In the short term similar initial conditions yield similar results
t=2
but, after some time, the divergence becomes obvious.
t=3
This is the so called ‘butterfly effect’. It is impossible to predict a sufficiently advanced state of a chaotic system, even after being given the initial conditions to any arbitrary degree of accuracy.
(The blue and yellow lines started at the same y and z coordinates and the x coordinate only differred by a value of 0.00001)
In other words, the system has built into itself the property
of amplifying small perturbations until they become so significant that they affect the accuracy of the results.
Like our global weather and other chaotic systems it is only possible to predict with any accuracy its state in the short term. It is infinitely variable yet also inextricably bound by the attractor. It therefore retains a large scale consistency, despite its local variation.
With Lorenz’s experiment, his chosen paramater values had fallen on a sweet spot for chaos; For rich fractal variety ‘miraculously’ coming from such simple rules (equations)
Our world is full of such phase transitions; Where an ordered system changes to a chaotic system. We can see it in the flow of water through a pipe. The water flow is regular or laminar, until a certain threshold is reached by the increasing control parameter. The flow then becomes turbulant and chaotic.
In such ‘real world’ cases we seek to understand chaotic turbulence in order to remove it.
For Lorenz’s experiment the sweet spot was for a value of P=24.74
For values of P below this figure, the system is stable and evolves to one of two fixed point attractors.
It is predictable.
When P creeps above 24.74, the system becomes chaotic. The fixed points of the attractor become repulsors and the trajectory is repelled by them in a very complex way.
For other values of P, the system displays knotted periodic orbits
Interestingly (possibly), the art of knotting has a long and ancient history.
Before the accession of the Emperor Fo-hi (3300 BC) it is said that the Chinese were not acquainted with writing and used quipus, (talking knots) to keep records.
In the writings of Confucius(551–479 BC), it says that “the men of antiquity used knotted cords to convey their orders; those who succeeded them substituted signs or figures for these cords.”
Such knots are made with one single string!
