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Chaos Theory |:-:| Deep Chaos & Strange Attractors

Part 3: Strange Attractor in Chaos Theory

To define an attractor is not simple. Tsonis gives the definition of attractors as "a limit set that collects trajectories".(1)   A strange attractor is simply the pattern of the pathway, in visual form, produced by graphing the behavior of a system. Since many, if not most, nonlinear systems are unpredictable and yet patterned, it is called strange and since it tends to produce a fractal geometric shape, it is said to be attracted to that shape. A system confines a particular entity and its related objects or processes to an imaginary or real frame as the subject of study, this is its "state space" or phase space. The behavior in this state space tends to contract in certain areas, this contraction is called "the attractor". The attractor is actually "a set of points such that all trajectories nearby converge to it". Now tell me what an attractor is. You can't and neither can I, even with Tsonis's definition. Scientists, mathematicians, and computer specialists can show you pictures of how they operate, but they cannot tell you what they are. Maybe, that is why Daniel Stein, compares Chaos/Complexity to a "theological concept", because lots of people talk about it but no one knows what it really is.(2)  (Found in the Preface to the first volume of lectures given at the 1988 Complex Systems Summer School for the Santa Fe Institute in New Mexico)

Several researchers have defined and studied strange attractors. The first was Lorenz in "Deterministic nonperiodic flow" in 1963 (3), and Later Ruelle in "Sensitive dependence on initial condition and turbulent behavior of dynamical systems" in 1979 (4) When computer simulation came along, the first fractal shape identified took the form of a butterfly; it arose from graphing the changes in weather systems modeled by Lorenz. Lorenz's attractor shows just how and why weather prognostication is so involved and notoriously wrong because of the butterfly effect. This amusing name reflects the possibility that a "butterfly in the Amazon might, in principle, ultimately alter the weather in Kansas."(5)   For an in depth story of how this butterfly effect developed into the science of Chaos and Complexity, see James Gleick's CHAOS and Mitchell Waldrop's COMPLEXITY(6).Kauffman explains that the tiny differences in initial conditions make "vast differences in the subsequent behavior of the system"(7)  as Lorenz illustrated in his weather prognosticator.

Chaos is centered on the concept of the strange attractor. Watch the flow of water from you faucet as you turn the water on to give faster and faster outpour; you will see activity from smooth delivery to gushing states. These various kinds of flow represent different patterns to which the flow is attracted. The feedback process is the feedback displayed in most natural systems in nature. There are four basic kinds of feedback or cycles a system can display: these are the attractors.(8)Tsonis gives the definition of attractors as "a limit set that collects trajectories"(9) The four kinds of attractors(10) are:

1. Point attractor, such as a pendulum swinging back and forth and eventually stopping at a point. The Attractor may come as a point, in which case, it gives a steady state where no change is made.
2. Periodic attractor, just add a mainspring to the pendulum to compensate for friction and the pendulum now has a limited cycle in its phase space. The periodic attractor portrays processes that repeat themselves.
3. Torus attractor, picture walking on a large doughnut, going over, under and around its outside surface area, circling, but never repeating exactly the same path you went before.(11)  The torus attractor depicts processes that stay in a confined area but wander from place to place in that area. (These first three attractors are not associated with Chaos theory because they are fixed attractors.(12)
4. Strange attractor, this attractor deals with the three-body problem of stability. The strange attractor shows processes that are stable, confined and yet never do the same thing twice.
Three non-linear equation solutions exhibit a fractal structure in computer simulations of the strange attractor.(13)  In other words, each solution curve tended to the same area, the attractor area, and cycled around randomly without any particular set number of times, never crossing itself, staying in the same phase space, and displaying self-similarity at any scale.(14)   The operative term here is self-similarity. Each event, each process, each period, each end-state in phase-space is never precisely identical to another; it is similar but not identical.The attractor acts on the system as a whole and collects the trajectories of perturbation in the environment. (These trajectories of perturbation are the positive and negative events going on in and around the system.) Though these systems are unstable, they have patterned order and boundary.


  French mathematician Georges Julia studied these chaotic orbits in complex analytical systems back in the 1920s, but Benoit Mandelbrot, in the early 1970s, gave some rules for computation. His work on noise interference problems revealed distinct ratios between order and disorder on any scale he used. The seemingly chaotic behavior of noise displayed a fractal structure.(15) Mandelbrot recognized a self-similar pattern that the fractals formed. He then cross-linked this new geometrical idea with hundreds of examples, from cotton prices to the regularity of the flooding of the Nile River.

Mitchell Feigenbaum found the constants or ratios that are responsible for the phase transition state when order turns to chaos. These Feigenbaum numbers helped to predict the onset of turbulence (chaos) in systems: applications in the real world began. Optics, economics, electronics, chemistry, biology, and psychology quickly used this new analytic tool. Fractal geometry is now being used to graphically show change and evolution in technology, sociology, economics, psychotherapy, medicine, psychology, astronomy, evolutionary theory, and the metaphorical application is spreading to art, humanities, philosophy and theology.(16)


Kauffman explains that the tiny differences in initial conditions make "vast differences in the subsequent behavior of the system."(17)  Unstable or aperiodic systems are unable to resist small disturbances and will display complex behavior, making prediction impossible and measurements will appear random. Human history is an excellent example of aperiodic behavior. Civilization may appear to rise and fall, but things never happen in the same way. Small events or single personalities may change the world around them.(18)

The symbolic use of chaos to delineate the interactions of a system and its environment can be more enlightening with chaos theory as the tool, especially when explicating an historical personage or situation. Here the human being or set of human beings creates a pattern in the time-space, this pattern is the basin of attraction within which the attractor or multiple attractors form. The Newtonian paradigm of linear mechanics does not reveal all the ramifications that effect the event or person. Newtonian expectations proposed smooth transformations that can be plotted by linear actions or reactions; but chaos/complexity will allow the researcher to see the symbolic interaction of the person or event with their environment.

1. Tsonis, Anastasios A. Chaos: From Theory to Applications. New York: Plenum Press. 1992. P. 67. return

2. Kellert, Stephen. In the Wake of Chaos: Unpredictable Order in Synamical Systems. Chicago, Ill.: The University of Chicago Press. 1993. return

3. Stein, Daniel L. ed. Lectures in the Sciences of Complexity. Vol. 1. Redwood City, California: Addison-Wesley Publishing Co. 1989. P. XIII.

4. Lorenz, Edward N. "Deterministic nonperiodic flow". Journal of Atoms.Sci. 20:130. 1963.  return

5. Ruelle, David. "Sensitive dependence on initial condition and turbulent behavior of dynamical systems". Ann. N. Y. Acad. Sci. 316:408. 1979. See also: Grassberger, P. and Procaccia, I. "Measuring the strangeness of strange attractors". Physica 9D:189. 1983. And, Meyer-Kress, G. ed. Dimensions and Entropies in Chaotic Systems: Quantification of Complex Behavior. , Berlin: Springer-Verlag. 1986. return

6. Ibid.  return

7. Waldrop, M. Mitchell. Complexity: The Emerging Science at the Edge of Order and Chaos. N. Y, N. Y.: Simon & Schuster: 1992.  return

8. Kauffman, Stuart A. The Origins of Order: Self-Organization and Selection in Evolution. New York: Oxford University Press. 1993. P. 178. return

9. Tsonis, Anastasios A. Chaos: From Theory to Applications. (New York: Plenum Press. 1992). This book teaches and applies the theory of nonlinear dynamical systems to problems of weather prediction, noise reduction, and neural networks.  return

10. Attractor refers to sets that "attract" orbits and hence determine typical long-term behavior. It is also possible to have sets in phase space on which the dynamics can be exceedingly complicated, but which are not attracting. In such cases orbits placed exactly on the set stay there forever, but typical neighboring orbits eventually leave the neighborhood of the set, never to return. One indication of the possibility of complex behavior on such nonattracting (unstable) sets is periodic orbits whose number increases exponentially with their period, as well as the presence of the uncountable number of nonperiodic orbits. Nonattracting unstable chaotic sets can have important observable macroscopic consequences. Three such consequences are the phenomena of chaotic transients, fractal basin boundaries, and chaotic scattering.
11. For more complete descriptions see: Ian Stewart "Portraits of Chaos" in New Scientist. Nov. 1989. P. 45. and James P. Crutchfield, J. D. Farmer, N. H. Packard, and R. S. Shaw in "Chaos" from Scientific American. Dec. 1986. P. 50.  return

12. In most cases, you can predict what will happen no matter what the initial conditions were, but in extreme conditions, they too become chaotic. Cramer gives some illustrations of what happens at high energy or angular momentum when the two-body system does exhibit chaos or reordering. Cramer, Freidrich. Chaos and Order: The Complex Structure of Living Systems. New York: VCH Publishers. 1993. Pp.121-2.  return
13. The use of computer simulations is one reason this field of research has been realized only in the past 20 years. The computations are vast and computers make it easier, also they can show detailed pictographs of the form and growth of fractals. The image often used to describe a fractal structure is the Russian nesting dolls--each one, inside another, growing progressively smaller, but always identical. Fractals reveal self-similarity no matter how deeply you look into the forms. return

14. To be as concise as possible, the strange attractor exhibits two seemingly contradictory effects converging into a new system. You might say that one comes to a crossroads and takes both paths at once and winds up circling forever both of the areas and never crossing the same spot twice. This is actually a word picture of what the Lorenz attractor, or fractal, looks like. Though the system looks random at first, it will retain its shape and space, thus displaying order. This gives researchers a way to investigate the way a system changes its behavior in response to a change in the parameters describing the system and its environment.   return
15. Fractal got its name from MandelbrotÕs sonÕs Latin book, fractus, meaning to break. return

16. Scott, George. (ed.) Time, Rhythm and Chaos: In the New Dialogue with Nature. Ames: Iowa State University Press. 1991. Complexity theory in the research area of self-organization gives an idea of the widespread nature of this new analytical tool of Chaos.   return
17. Kauffman, Stuart A. The Origins of Order: Self-Organization and Selection in Evolution. New York: Oxford University Press. 1993. P. 178.  return

18. Kellert. Pp. 3-5.   return




Deep chaos is the fractal dimension where patterns of self-similarity reveal themselves in descending scales of order. Uri Merry likens them to "a set of wooden Russian dolls, each containing a smaller replica of itself within."(1)  This complexity can occur in natural and man-made systems, as well as in social structure; therefore because it is so ubiquitous to nature, it has no agreed-upon definition. Çambel describes complex systems from 15 categories, but to be succinct, complex systems have size, purpose, and are dynamic.(2)    Cramer(3) gives his definition in the form of a logarithm taken from information theory which in essence means that "the more complex a system, the more information it is capable of carrying.(4)   In deep chaos, there is a displacement of being, the chthonic realm of turmoil; it is the dimension between states. It is here in the deep chaotic state that the system becomes complex, and hence the term Complexity enters in. Kauffman and Christopher Langdon speak of the edge of chaos as the place where systems are at their optimum performance potential.(5)

When the constraints on a system are sufficiently strong, (many positive and negative perturbations), the system can adjust to its environment in several different ways. There may be several solutions possible from the whole basin of attraction, and chance alone cannot decide which of these solutions will be realized. It is the attractor that will help determine the solution. The fact that one solution among many does occur gives the system a historical dimension, a sort of memory of a past event that took place at a critical moment and which will affect its further evolution. This is the phase transition of the system, the place where the system is isotropic; it has no preferred direction to go in, it is an either/or decision, the past old ways or the future new ways. You might visualize the phase transition as the coin tossed into the air; while it is in the air there is only probability, no actual choice has been made until it lands. There is no observance of transilience (leaping from one state to another) in the system, but there is a phase transition that takes place at the edge of chaos before an actual self-organization into another state.

There might be some similarity of phase transition to the sense of NOW. The sense of NOW worries all philosophers and theologians and even scientists. What is NOW? Karl Popper reckons NOW as a single frame in a filmstrip--the future and past are all known within the whole of the strip.(6)   Einstein worried about NOW as a physics question--NOW was special for humans, but did not have a meaning in physics.(7)   The NOW in chaos theory is the phase transition state where all choices are open. Paul Tillich(8)  spoke eloquently about living oneÕs life in the Eternal Now. Perhaps that is precisely what we do, each moment is a phase transition to the next, our choices moment by moment determine the life we live.


1. Merry, Uri. Coping with Uncertainty: Insights from the New Sciences of Chaos, Self-Organization and Complexity. Westport, CT.: Praeger Publishers. 1995. P. 40.

2. Çambel, A. B. Applied Chaos Theory: A Paradigm for Complexity. Academic Press, Inc. San Diego, CA 1993. P. 2-4.

3. Cramer, Friedrich. Chaos and Order: The Complex Structure of Living Systems. Trans. David I. Loewus. New York: VCH Publishers. 1993. Pp. 210-218. This section contains CramerÕs view of complexity. He gives its practical aspect as an indeterminate whole, its teleological aspect as the whole emerging, and a subcritical stage in which intentional randomization of the universe as a deterministic system allows for free will because it involves complexity. Therefore the universe is deterministic and indeterministic at the same time. He also refutes that his theory is mysticism or scientism. His explanation is esoteric, but this might be accounted for in translation. The Forward by Ilya Prigogine says Cramer deserves international acclaim. Hey, what do I know?
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4. Cramer gets his information for the logarithm from N. Pippenger. "Complexity Theory." Scientific American. June. 1978. Pp. 90-100.

5. Kauffman. Stuart A. The Origins of Order: Self-Organization and Selection in Evolution. New York: Oxford University Press. 1993. Pp. 181-218.
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6. Popper, Karl. The Open Universe. Totowa, N. J.: Rowman & Littlefield. 1956.

7. Prigogine, I. and Stengers, I. Order out of Chaos . New York: Bantam Books. 1984. P. 214 .

8. Tillich, Paul. The Eternal Now.
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