A Strange Attraction – A Brief Study Of Strange Attractors And Their Implications… In Tribute To Edward Lorenz
June 2, 2009
The Lorenz attractor, named after Edward Lorenz, is a fractal structure corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its ‘lemniscate’ shape (a term used in algebraic geometry to refer to an object that has a likeness to the figure eight’s form).
The oscillator was originally used by Lorenz as a simplified model of convection flow in the earths atmosphere. The map (see both figure 1 and 2 below) shows how the state of a dynamical system (three variables varying in time, all plotted in a 3-D phase-space) evolves in a complex, non-repeating pattern.
Figure 1. appeared in the Nature journal 31 August 2000, pp 949 as part of an article titled The Lorenz Attractor Exists, written by Ian Stewart. It was created as part of an OpenGL interactive viewer and rendered on a farm of Dec Alphas using ProRay.
In 1961, Lorenz had managed to create a skeleton of a weather system from a handful of differential equations. He kept a continuous simulation running on an extremely primitive analog computer that would output a day’s progress in the simulation every minute as a line of text on a roll of paper. Evidently, the whole system was very successful at producing “weather-like” output – nothing ever happened the same way twice, but there was an underlying order that delighted Lorenz and his associates.
…Line by line, the winds and temperatures in Lorenz’s printouts seemed to behave in a recognizable earthly way. They matched his cherished intuition about the weather, his sense that it repeated itself, displaying familiar patterns over time, pressure rising and falling, the airstream swinging north and south. (GLEICK, J. Chaos: Making a New Science.)
What Edward Lorenz had discovered was a chaotic system. Even though a computer had control of the simulation, and certainly possessed the capability to generate random numbers at will, there was nothing random about any portion of the way the simulation was supposed to work. It merely followed the laws of calculus as set down by Sir Isaac Newton himself and outputted a day’s worth of virtual weather at the end of each minute. Lorenz’s initial brush with chaos is described best by James Gleick’s own words, from Chaos:
One day in the winter of 1961, wanting to examine one sequence at greater length, Lorenz took a shortcut. Instead of starting the whole run over, he started midway through. To give the machine its initial conditions, he typed the numbers straight from the earlier printout. Then he walked down the hall to get away from the noise and drink a cup of coffee. When he returned an hour later, he saw something unexpected, something that planted a seed for a new science.This new run should have exactly duplicated the old. Lorenz had copied the numbers into the machine himself. The program had not changed. Yet as he stared at the new printout, Lorenz saw his weather diverging so rapidly from the pattern of the last run that, within just a few months, all resemblance had disappeared. He looked at one set of numbers, then back at the other. He might as well have chosen two random weathers out of a hat. His first thought was that another vacuum tube had gone bad.
Suddenly he realized the truth. There had been no malfunction. The problem lay in the numbers he had typed. In the computer’s memory, six decimal places were stored: .506127. On the printout to save space, just three appeared: .506. Lorenz had entered the shorter, rounded-off numbers, assuming that the difference-one part in a thousand-was inconsequential.
It was a reasonable assumption. If a weather satellite can read ocean-surface temperature to within one part in a thousand, its operators consider themselves lucky. Lorenz’s Royal McBee was implementing the classical program. It used a purely deterministic system of equations. Given a particular starting point, the weather would unfold exactly the same way each time. Given a slightly different starting point, the weather should unfold in a slightly different way. A small numerical error was like a small puff of wind – surely the small puffs faded or canceled each other out before they could change important, large-scale features of the weather. Yet in Lorenz’s particular system of equations, small errors proved catastrophic.
And there is the show-stopper: small errors prove catastrophic. Lorenz entitled a 1979 paper, “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” and the title stuck. Today, sensitive dependence on initial conditions is referred to as “The Butterfly Effect.”
For the purposes of experimentation, Lorenz created a new system with three nonlinear differential equations:
dx / dt = a (y – x)
dy / dt = x (b – z) – y
dz / dt = xy – c z
It was a reduced model of convection, similar to the swirls of cream in a hot cup of coffee, only much, much, much simpler. And yet the complexity and never ending richness that it generates within a phase-space, demonstrating the sovereign and noble interplay behind the basins of attraction and the unpredictability behind their subtle networks of force are startling!
Ever since my first glance at one of these graphical extrapolations between the relationship of 3 variables within a dynamical system, as plotted within a 3-D phase-space, I have had a vision of multidimensional phase-spaces harboring similar ‘key’ shapes and patterns… Multidimensional phase-spaces where variables intertwine with one another, with each one affecting the other eventually (even if only in a minuscule manner) along great chains of interconnected cycles of influence that span across vast and varying scales of attention… Patterns slowly tugging on each other, some leading and defining, others lagging and holding back… But all working into waves of turbulent flow that undulate across all frequencies, and ripple and fold back over each other in ever more complex arrangements… Arrangements that can be likened to what we call quasicrystals, as seen in Animation 1 below.
And, as if to pay homage to this stupefying and almost unknowable complex lattice of experiential flow, I have had a deep sense of longing to port the basic essence (as Lorenz originally first described it) of this boundless and very real ideal into the sonic realm of rhythmic form, wondering whether listening to its chaotic nature might in some way provide further insight into the aural aesthetics of never repeating, endless cyclical flow. Some of my efforts towards this aim can be heard here…
Lorenz’s system, although simple in the eyes of a physicist or mathematician, is actually an insolvable problem except by numerical means. To aid in my own understanding of the system, I developed a simple program to numerically solve Lorenz’s system of equations. Why? Well… Mainly because solving the equations by hand would require many, many BIC pens and generate reams and reams of paper that would stack up into the edges of the atmosphere… And then, having amassed all these Chaotic numerical data entries, I would then have re-assimilate it all by plotting the data into a phase space graph in super fine detail, so as to avoid the over lapping of my data stream… I think you’ve got the idea as to how lengthy and tedious a process this would be. So the program allows the user to specify a set of initial conditions and outputs data in a format suitable for importing into a spreadsheet or graphing package or porting into software synthesis patches created within the Max/MSP multimedia program.
Be it said, while playing around with this program I have reproduced the Lorenz attractor time and again, sometimes leaving the variables the same thereby watching exactly the same patterns unfold in front of my eyes, and then at other times varying the initial conditions and so witnessing strong deviations in the trail’s flow through the 3-D phase-space. But all the time it manages to maintain its butterfly shape, like two wise old owl’s eyes peering back at me knowing the subtle influence attraction eventually induces over all things, despite the utter lack of correlation between phase-space plots. I can almost understand why people must have thought, at first glance, there are only what appear to be random fluctuations coming from what we expect to be a completely deterministic set of equations. Why? Because that type of behavior was completely unheard of before. It simply has no place in our romantic ideals of solvable mathematical problems and solutions. Why would anyone believe that there would/could be equations that were literally unsolvable? Years after people first noticed this unpredictable type of behavior, it was all too often discarded by mathematicians simply as errors in calculation by machines.
But thankfully (so thankfully), Lorenz was the first to recognize this erratic behavior as something other than error. He had the sense to understand that science had been trying too hard to understand the details, pinning dismissive ‘reasons’ (that were akin to romanticized human ideals about universal perfection, much like mankind did when at one stage in history we thought that the earth resided at the center of the universe as we dismissed the obvious facts) to ‘seemingly’ trivial and/or meaningless problems/issues… Thus we simply over looked probably the most important clue that nature and math has to offer us about our universe and its dynamical unfolding in space-time… And, once again, all because the world wasn’t quite fitting into our romantic notion that the universe can only ever offer us deducible and solvable problems. Basically Lorenz realized that everyone had been trying to view the world through a microscope, seeing only a limited and highly detailed view of things, a view that had little or no relation to the greater, bigger picture. But when Lorenz put that microscope aside and looked with his own eyes, what he saw was an undeniable order, born out of the randomness… The ideal of yin and yang, the duality and contradiction of knowing the unknowable, an idea that Roger Pernrose, M. C. Escher, Kurt Gödel and Douglas R. Hofstadter have entertained at length.
“The law of nature is change (chaos), while the dream of man is order.” (Henry Adams; http://www.thepomoblog.com/papers/pomo44.htm)
That’s the beauty of the underlying order in chaos. A system that is completely free willed when viewed from a certain perspective also just happens to fall within a completely deterministic, predictable pattern… And later Adams perceptibly pushed his idea that one step further by simply stating, “Chaos often breeds life… While order simply breeds habit.” And it is here, in my humble opinion, that we start to find a suitable definition for life; a definition that can provide a basis for the dynamics involved in the beginnings of creation. It is my belief that we did not arise from a single god or divine creator… Rather we were born from Chaos’ gentle nurturing grasp, a hold that has slowly bubbled over into existence over millions/billions of years on this little blue green planet orbiting around this sun (and very possibly other rocks around suns). An almost infinite number of forces connected into one long chain of events, just as the Buddhist theory of Interdependent Origination suggests, brimming over with causes and effects…
Just as weather systems arose from the forces of convection that originated from the sun heating the earth’s surfaces at differing rates, thereby causing air to rise in some areas and cause pulls of air from around these updrafts that in turn cause falls in others areas, slowly building and intermingling into more complex swirls and systems over time, which overlapped one another… Combine these factors along with other variables, such as the specific heat capacity of the chemicals on the ground… And in the atmosphere, along with their weight, density and pressure (all which affect fluidity)… Well you can begin to see the complexity of these dynamical systems.
And life began in a very similar manner too. But just before life could get a firm grip on the world in which they had been deposited via the forces of accretion, elemental atoms had to first find their own balance in between themselves by spreading/sharing energetic charges/differences with one another, and in so doing finding preferred arrangements of stability that stretched way beyond coincidence. Then the resulting molecules formed might simply either sit in earthy deposits as mineral ores… OR perhaps, if lucky (?) enough to be dissolved in solution, they might be able to slowly work their way into more stable arrangements, as fatty acids do, for example:
Whatever the process, these basins of attraction slowly worked through the world of atoms and then through the world of molecules, causing harmony by ordering themselves into predefined and less energetic arrangements. And if in those gravitational pockets of matter (pockets that we call planets) a healthy and diverse balance of elements exists, along with good ambient temperatures where water may fluctuate cyclically between the solid, liquid and gaseous states of its being, then the basic structures of self repeating reactions that emulate life as we know it might be given chances to form and propagate…
Eventually basic single celled organisms might have randomly drifted through the oceans of Earth, seeking out molecular arrangements that might allow them to continue to grow. Attractions to food… Slowly developing, as proffered by the Chaos inherent in these strange attractions, so they moved along the evolutionary chain into more complex arrangements, until one fine day (depending on your point of view, of course) humans arrived. And as we all know so well from personal experience, the attractors/desires that our bodies adhere to i.e. hunger (a need for, or an attraction to food), sexual urges (the need to find a mate in order to procreate so as to continue the lineage of ourselves/species), etc… allowed us to work with the environment and times that we found ourselves in, thereby beautifully adapting just as an algorithm of chaos natural would around the basins of attraction presented, till behaviors developed and honed the feedback system within our brains into beautifully and flexible program of adaption that could bow to the ever changing environmental conditions. But know them all? Well… The shear plethora of attractors that played out in our story of creation are far too numerous and mind-boggling to know, stretching through many scales of being, from the quantum world of forms so small to the vast galactic planes of multidimensional structure. Perhaps we may only ever know a hand-full of immediately obvious ones…
But either way, through random fluctuations around the multidimensional and multitudinous basins of attraction that exist within the Universe in which we find ourselves, around who’s attractions we are nimbly able to skirt, we weave our own unique and majestic tapestry of complexity into predictable sheets of experiential folds that ‘flap’ beautifully, if somewhat unpredictably, in the winds of chance…
I mean… Can you imagine just how stuck we’d all now be if the universe was prone to repeating itself exactly? Even if only on the smallest of levels? I doubt the diverse flow of life that we all take for granted and know so well today would have ever been able to escape from out of the centers of attractions who simply repeated exactly the same patterns over and over again. The chaotic orbits around the forces of these repetitious basins that everything is attracted to/graced with would simply and snuggly reside within their comfortable centers, hybernating in a dormant slumber, and be blissfully unaware of their true potential. Perhaps this is the ultimate fate of attraction? And one that singularities i.e. black holes in space time, know so well… An inescapable gravitation… Like an addiction… A calling or even a longing to repeat without variance.
I can’t begin to express how fortunate I feel to have this chance to be able to skate around the edges of attractions, too numerous to mention/know them all, and so to be a part of this Chaotic and unrepeatable universal dance… Everyday this Knowing comes to me, penetrating deeply my experience of Being… The fortune of winning the lottery of existence in this eternal and infinite dance is in itself divine providence. I am just a small part of this universe afforded the improbable experience of being able to know itself. And that is exactly what I intend to do… And do so with an attractor we all know so well… An attractor we call Love.
Because when you can begin to see the hugely improbable, chaotic journey that we have all (and I mean all of us, from the atomic beginnings of life through to all the organisms inbetween that eventually gave rise to us here and now) had to make in order to get here… You begin to free yourself from preconceptions of fear and the false ideals of mine and yours that this consumerist, elitist society ingrains into us… And once free, you can then begin to see and realize that if it wasn’t for Love… We simply wouldn’t be here right now.