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Summary Notes for the Study of Chaos and Complexity

Beware of articles that mention one concept from complexity or chaos studies and project ideas from that one concept without looking at all the generalities that come out of both sciences. Different books use different terms for the study of complex systems. These terms emphasize different features: self-organizing systems, dynamical or emergent systems, complex adaptive systems, open dissipative structures,’ self-organized criticality, complex adaptive matter, or simply systems or networks.

Defnitions, Indicators, and Bibliography for the Study of Chaos and Complexity
Carolyn A. (Cary) Neeper, Ph.D.

Complex Systems: Networks of independent units interact using non-linear, open reactions far from equilibrium, thus ratcheting to higher levels of organization. Per Bak insisted that any theory of complexity would have to explain the ubiquity of fractals in nature and 1/f noise and the regularity of catastrophes. His theory for a mechanism of complexity: self-organized criticality (SOC).

In Chaos Theory Tamed, Garnett Williams, geologist, described complexity as a type of behavior that doesn't reach equilibrium (like a flat beach as compared to a sand pile). At least six independent units interact continually, acting on, responding, and adapting to fellow units using local rules that govern each unit to self-organize. Hierarchical complex systems adapt to become more complex over time, eventually exhibiting emergence--in which the parts can not explain the behavior or characteristics of the whole.

Examples: The brain has been called the most complex object in the universe. Other examples include weather systems, ecosystems, and the national economy. Complex systems may be called primarily spatial (DNA), temporal (commodity prices), both (the brain), or adaptive (ecosystems, the human body).

DETERMINISTIC CHAOS--Simple rules in nonlinear dynamical deterministic equations leading to outcomes, in part through feedback, that can’t be determined in the long run (that is, deterministic equations with stochastic, probabilistic, random-like behavior). Most nonlinear equations exhibit chaos within certain parameter ranges.
a) Extreme Sensitivity to Initial Conditions—results are predictable only in the short term. Physicist Arthur Peacocke has said that infinite precision in infinite time is needed to predict the future in a chaotic reaction. Small errors grow exponentially. This results in b) long-term unpredictability.
c) Strange Attractors are diagnostic for deterministic chaos; the data are bounded in non-repeating patterns that have fractal properties.
Examples: features of the weather, trajectory of pool balls on an oval table, turbulence, Pluto's position in orbit, EEG brain activity (petit mal seizures show limit cycles), electrical activity of the heart, population fluctuations, prices in the stock market, baby cries, asteroid movement, motion of atoms in an electromagnetic field, laser cavities.

Chaos cannot explain complexity; chaotic processes do not evolve, have no memory, and cannot produce fractal objects as do complex systems; they are seen within complex systems when certain parameters (in reactions that show chaotic behavior) reach critical values. The unpredictability of complex systems is not like that of chaos, which is deterministic and has few dimensions. Chaos does not produce 1/f noise; its data show white noise (all frequencies occur with equal probability). Datasets required for good analysis should include at least 5000 values; 10,000 being much better. Long-term behavior is very sensitive to changes in initial conditions.
Chaos analysis is weak in revealing details of a physical law or governing equation.
Why finding chaos in data is important:
1- Greater understanding. Identifying an orderly system in disguise.
2- Greater accuracy in short-term predictions.
3- Finding the time limits of reliable predictions.
4- Makes modeling easier.
5- Finding ways to control or stabilize chaos has a vast literature. Ideas are developing on how to avoid chaos in ecology. Chaos probably should be encouraged in physiology. Disease, mental depression, the brain, and the heart behave chaotically in healthy individuals and more regularly in unhealthy ones. Mixing processes and encoding electronic messages can exploit chaos.

Definitions To Clarify The Definitions Of Complexity and Chaos
Deterministic--something that follows a rule, like a mathematical equation and a given initial condition; its past and future are set by its current state and new input.
Dynamical system--one that moves or changes in time.
Equilibrium--particles or reactins at rest or in uniform motion.
Linear equations--f(a times x) = a times f(x) or f(x+y) = f(x) + f(y). Equations that are smooth, responding to changes in proportion to the stimulus. Some authors say linear systems don't exist in nature.
Non-linear equations are those in which the output is not directly proportional to the input. It doesn't plot as a straight line on ordinary graph paper. Changes occur in erratic, responses to input and are often much greater than the stimulus. Pulses can last a long time.
Open reaction--a reaction receiving energy and material from its surroundings. Entropy increases for all isolated (closed) systems
Parameter--an arbitrary constant in an equation.
Strange Attractor--a reproducible asymptotic trajectory in time on a phase space surface, wandering chaotically with an invariant probability distribution.
System--a group of interacting parts.


1) Synchronization--The coordination of activity of independent agents.
Living Examples: Applause, fireflies flashing, crickets signaling, menstrual cycles in women living in proximity, heartbeats of close associates, neural synchrony over many regions of the brain when conscious, riots(?).
Co-evolution: the fates of different species in the same environment are linked, as seen in parallel extinctions in different families (a function of criticality of the global ecology.)
Inanimate Examples: Molecules of water in Bénard cells, chemicals in Belousov-Zhabotinsky reaction.

2) Fractals--Patterns in nature that are self-similar at all scales, resulting from simple
Inanimate Examples: The coastline of Norway, fractal dimension 1.58.
The Koch Curve (Start with a triangle, divide each side in 3 equal parts, substitute the middle section with two lines of equal length, repeat.)
Snowflakes, molecules that either stick or don't stick in diffusion limited aggregation (DLA), the distribution of galaxies, music, the irregular shapes of rocks and mountains (simulated in computer programs), cloud borders, the branching of rivers, the crustal plates of the Earth.
The Mandelbrot Set: generated by the iteration of complex numbers z-squared + c. A pixel on the computer screen is colored black if after so many (say 500) iterations the result is less than 4. If the result is greater than 4 (rapidly on its way to infinity), the pixel is colored according to how many iterations it took to exceed 4.
Living Examples: the leaves of a fern, broccoli, cauliflower; the branching of rivers, blood vessels, bronchi, and the evolutionary tree of life.
Generalities: Per Bak--Fractals are snapshots of self-organized criticality--the mechanism for how complexity works.
Strange attractors, diagnostic of chaos, have fractal properties as a mathematical feature unrelated to the processes that produce physical properties in nature with fractal geometry--a confusion often found in popular literature.Chaos does not explain complexity.
Statistics: Data show no average, mean or variance; they require statistics of stable distributions.
A fractal is defined as an object having a fractal dimension greater than its topological dimension. A line has a fractal dimension of 1, a plane 2, a sphere 3. Jagged line 1.n. Crumpled paper 2.n.

3) 1/f Noise and the regularity of catastrophes--Ubiquitous power laws are observed in nature, including human activity.
Inanimate Examples: There are a small number of large earthquakes, a large number of small earthquakes, and a moderate number of moderate size earthquakes.
The size of earthquakes is not periodic; if there has been a long time between large earthquakes, there will be continue to be a long time between large earthquakes. If a short time, better see your Insurance agent.
Mandelbrot's study of cotton prices for some years, other commodity prices, the stock market.
Sand or rice pile avalanches, snow avalanches?
Flow rate of the Nile River, traffic jams.
Number of chemical reactions per molecule.
Pulsars rotational velocity x-rays from black holes, solar flare intensity, light from quasars.
Distribution of Internet nodes with certain numbers of links.
Living Examples: Extinction of species (as a percentage) occurring in a number of 4 million-year periods--i.e. the sizes of extinctions.
Species rank distribution, species abundance distribution, canopy gap sizes in the rain forest, tree sizes in a forest, the length of lifetime of families, the fraction of cities with so many inhabitants (Zipf's Law), the frequency of the use of words in English.
In ecology: the power law is seen where the driving force is slow and system response is rapid. Example: constant immigration is a driving force..
Generalities: frequency(f) is inversely proportional to power or size of event. This relationship breaks down at very small and very large values, depending on what is being measured. Data are represented by straight-line relationships on log log plots of frequency vs. size of event. The slope of the line indicates the fractal dimension of the event or object's measured characteristic.
The data are scale free. Example: the length of the coast of Norway increases as the measuring rod decreases in length.
The phenomena are due to scale-free avalanches in self-organized critical systems.
No laws in physics hint at the ubiquity of this observation.
Micromechanisms produce the behavior at all scales; large incidents require no special stimulus. Example: the size of push is not proportional to the resulting movement, as in linear physics.

4) Criticality--One more (or a small) input produces a sudden change.
Inanimate Examples: At a critical density of trees, fire set on one edge of an area will spread in a fractal pattern.
Eco-systems seem to operate close to critical points, making the results of one extinction (for example) unpredictable, touching all elements in the network. Constant immigration can be a force driving toward criticality.
The Logistics Map: Pn+1 = G(1-Pn). The critical point is G (growth rate) = 3.57.
Self-organized criticality is seen in faults and in volcanic activity.
Living Examples: In ant colonies tactile and chemical communication is maximum at critical population densities, resulting in stable patterns of raid activity.
In growing termite mounds pillars arise only after the termite population reaches a critical density.
Chance of infection that impacts the persistence of an infectious epidemic.
Per Bak successfully demonstrated that self-organized criticality (SOC) is the mechanism resulting in complex behavior, the phenomenon called complexity. He stated that any theory of complexity would have to explain fractals in nature and 1/f noise (with the regularity of catastrophes).
Complex natural systems self-organize to a critical point, often followed by characteristic routes to chaos (such as period doubling) as the input continues. At a critical point, organization into a new ordered state may occur (indicated by fractal behavior), and/or beyond the critical point chaos may occur, followed by more self-organization. Thus, order works with chaos and chance (at forks in the path (bifurcation points)) to drive systems to higher states of organization. Living systems seem to exist at the “edge of chaos,” near critical points.
When communication is maximal at a critical density of a population, it can avoid instability.

5) Self-organization--Pattern formation (or Bak's self-organized critical state) that occurs through interactions internal to the system, using only local information, without external intervention or reference to global patterns.
Inanimate Examples: Bénard cells (the spontaneous emergence of order in a network of water molecules), patterns in a thin layer of shaken sand, alpine and arctic raised stone circles due to frost heave, stable patterns in computer cellular automata, chemical reactions that oscillate, hurricanes, laminar flow, stars in the spiral arm of a galaxy, flocking behavior of computer boids.
Some complicated systems, including most 2-D cellular automata, do not show self-organized criticality, but can be tuned to do so by changing the rules. The rules for the Game of Life were a lucky find, resulting in stable or moving patterns after the program runs for a while: gliders, blinkers, etc.
Computer models of living examples (below) provide better understanding of the systems studied, realization of information that is missing in that understanding, and the possibility of making realistic predictions in the natural system. See MIT's Starlogo simulation program at http://www.media.mit.edu/starlogo or the web site http://beelab.cas.psu.edu.
Living Examples: flocking behavior of birds, fish schooling, seed development, slime mold migration, ornamentation on pollen grains and mushroom caps, lichen growth, symmetrical arrangement of flower parts, pigmentation patterns on animals and shells, ocular dominance stripes in the visual cortex of monkey brains, cell growth, differentiation in embryos.
Ant trails or raiding behavior (a function of chemical and tactile stimuli). Individual activity shows a low-level chaotic attractor.)
Bark beetles put out an attractive pheromone that results in larvae clustering while feeding on trees. This enables them to avoid being overwhelmed by the tree's defensive output of sticky resin.
Termite mounds: Pheromone concentrations are critical in eliciting responses. Pillars emerge only at a critical density of termites, and the rules change as the mound environment changes.
Demand for goods and services in economic markets, rush hour traffic, the cleanup after 9/11.
Generalities: Recent studies suggest that self-organization may have a different characteristic within contained organic systems than it does within groups of independent, free agents.
Stuart Kauffman points out that self-organization works with natural selection, resulting in evolution (and increasing complexity) at all levels of bio-systems. Rapid change in stable bio-patterns (punctuated equilibrium) usually occurs as a result of a driving force pushing the self-organizing system beyond a critical point. Environmental factors then play their part in driving natural selection (good adaptations in a species result in their reproduction).
Not all patterns in nature come from self-organization. Other possible mechanisms for pattern production may be 1) leadership of one agent, as a mother duck leading ducklings, 2) a recipe followed, as in the making of a spider web, 3) an envisioned blueprint, probably not in beaver dam building, and 4) use of templates, as in DNA and messenger RNA, the weaver bird using its own body to build a nest, and termites modeling the copularium around the growing queen and her mate.
Self-organization uses techniques of stigmergy, decentralized control, and dense heterarchies with positive feedback, amplification of fluctuations, and negative feedback Biological systems seem to be tuned by natural selection to avoid chaos, but they utilize tunable parameters and bifurcations to provide flexible responses to different circumstances under the same basic behavioral and physiological rules.

6) Emergence--The whole being more than the sum of the parts. The behavior of the whole cannot be ascertained by the behavior or properties of the parts, a result of self-organization.
Inanimate Examples: A checkers computer game that learns to win from experience with living and non-living opponents.
Superconductivity, superfluidity, crystals, lasers, ferromagnetism.
Living Examples:
Bak: Life is an emergent property of matter and energy.
Morowitz: a specie, a family of species.
Colony behavior in ants and termites.
The mind, your unique self, trust, love, beauty, health, soul, personality, appearance, pattern.
Bak: Moral evil--self-preservation and self-declaration gone beyond necessity and social viability.
David Korten: Our "suicide economy" driven by love of money--not love of life, community and beauty--in which the concentration of power is de-linked from obligations to individuals, governments, or societies.

7) Amplification of small events into huge effects over time.
Inanimate Examples: Erosion, weather patterns.
Kauffman and Bak's computer evolution model: They finally got self-organized criticality occurring when they chose for mutation, not random species, but those least fit for mutation (the working of extremal dynamics).
Living Examples: VHS market dominance over Beta, embryonic development, any example of contingency in history, Microsoft operating systems, Harry Potter book sales.
Generalities: The "real" butterfly effect (Bak): at critical densities random individual acts (small effects) will propagate through the whole colony. That critical density, however, is low enough to prevent activity from remaining the same too long. I.e. some disorder allows manipulation of information. Intermediate entropy allows for maximum information transfer.
A tornado is so well organized that a butterfly would have little influence on it.

I. Most natural systems, including human organizations, exhibit behavior characteristic of deterministic chaos and/or complexity. Studies in network theory suggest that such behavior is seen in interdependent webs (networks of independent agents) at many hierarchical levels of organization.
Therefore look to the whole organization or business, as well as to individual members and visitors. Both holistic and reductionist views are important.

II. Complex systems evolve through the action of open networks working within networks (chemicals, cells, tissues, organs), in which order (physical-chemical rules) works with chance and chaos in reactions that operate far from equilibrium. Michael Shermer uses the phrase “contingency working with necessity” to describe these interactions. Good communication is essential for stability.
Therefore be sure members, stakeholders, and visitors are always well informed.

III. At critical points (concentrations, densities) sudden changes occur (like phase changes) to produce states of chaos and/or to self-organize to ordered states with lower entropy that exhibit emergence (the whole being more than can be ascertained by the parts or their behavior).
In other words, relationships between individuals are critical; they must be nurtured.

IV. Outcomes in a complex system are predictable only in the immediate future; they are unpredictable in the long run. This is due to the extreme sensitivity to initial conditions in chaotic systems embedded in a complex system, to the historical role of contingency, and to the random elements inherent at bifurcation points (and at critical points), where a choice of path is made by the system. Once a system is heavily invested in a strong pattern, chaotic sensitivity may have little effect. A butterfly has little effect on a tornado.
Therefore, continually evaluate all feedback, both good and bad.

V. In complex systems small events can be amplified over time into large effects. Therefore, since most natural systems are complex--self-organizing on independent individual interactions--nothing you do is inconsequential. Our lives and our organizations matter.
Therefore, work together until everyone buys into a few clear guiding principles; then set individuals free to implement those principles.

IN SUMMARY--Applying Complexity Principles to human organizations:
1) Look to the whole organization as well as to individual members and visitors; both holistic and reductionist views are important.
2) Be sure members, stakeholders, and visitors are informed.
3) Relationships between individuals are critical; they must be nurtured.
4) Consider all feedback, both good and bad.
5) Work together until everyone buys into a few clear guiding principles; then set individuals free to implement those principles.

Bibliography and Personal Notes
for the Workshop
“Complexity and Chaos—Implications for Philosophy and Theology"
by Cary Neeper

For a more extensive bibliography on complexity
click here.
For recommended readings in science and religion
click here.

Bak, Per. How Nature Works: The Science of Self-Organized Criticality. New York: Springer-Verlag, 1996.
Note: A favorite. Enjoyable real life examples of complexity and a clarifying statement on the confusion of terms in the popular press.
Barbour, Ian G. When Science Meets Religion: Enemies, Strangers, or Partners? San Francisco: Harper San Francisco, 2000.
Note: An academic approach. Barbour does not emphasize the role of science as a source of religious inspiration and does not discuss the importance of distinguishing the sources of ones thinking and writing.
Barbour, Ian G. Religion and Science: Historical and Contemporary Issues. Harper San Francisco, 1997. Invaluable as a condensed update on recent science, its limits, and its complement in process theology.
Briggs, John and F. David Peat. Seven Lessons of Chaos: Spiritual Wisdom from the Science of Change. New York: Harper Collins, 1999.
Note: Briggs tends to link chaos with fractal geometry. In contrast, Per Bak states emphatically that “…simple chaotic systems cannot produce a spatial fractal structure like the coast of Norway…In short, chaos theory cannot explain complexity.”
Capra, Fritjof. The Web of Life: A New Scientific Understanding of Living Systems. New York: Doubleday, 1996.
Note: One of the most accessible explanations of basic concepts in chaos and complexity studies, though Capra has a few questionable notions about science.
Chaisson, Eric J. Cosmic Evolution: The Rise of Complexity in Nature. Cambridge, MA: Harvard University Press, 2001.
Note: An overview of self-organization working with natural selection at all levels of nature, including the Big Bang.
Gregersen, Niels Henrik. From Complexity to Life: On the Emergence of Life and Meaning. Oxford University Press, 2003.
Gribbin, John. Deep Simplicity: Bringing Order to Chaos and Complexity. New York: Random House, 2004.
Note: An excellent overview for the general reader, useful as a textbook, but beware of confusing the term complexity with a complicated racing bike.
Holland, John H. Emergence: From Chaos to Order. Reading, MA: Addison-Wesley, 1998.
Note: Another favorite. Includes amazing demonstrations of emergence in simple games and networks.
Holland, John H. and Heather Mimnaugh. Hidden Order: How Adaptation Builds Complexity. Reading, MA: Perseus Pr., 1996.
Note: A technical review of how modeling of complex systems should be done.
Kauffman, Stuart A. At Home in the Universe: The Search for the Laws of Self-Organization and Complexity. New York: Oxford University Press, 1995.
Note: An exciting review of Kauffman’s testable, mechanistic theory of evolution derived from complexity theory.
Kauffman, Stuart A. The Origins of Order: Self-Organization and Selection in Evolution. New York: Oxford University Press, 1993.
Note: The basics of how self-organization works with natural selection in biology.
Kiel, L. Douglas and Euel Elliott. Chaos Theory in the Social Sciences: Foundations and Applications. Ann Arbor: University of Michigan Press, 1996.
Note: An introduction to and review of direct applications.
Peitgen, H.O., and P.H. Richter. The Beauty of Fractals: Images of Complex Dynamical Systems. Berlin: Springer-Verlag, 1986.
Note: This is the classic that sparked interest in the Mandelbrot Set.
Polkinghorne, John. Quarks, Chaos and Christianity: Questions to Science and Religion. New York: Crossroad, 1994.
Note: As an Anglican and physicist, Polkinghorne integrates his theology with complexity theory then makes a leap to Christian doctrine.
Prigogine, Ilya. The End of Certainty: Time, Chaos, and the New Laws of Nature. New York: Simon and Schuster Inc., 1996.
Note: As a chemist, Prigogine makes his case that quantum mechanics needs to be reformulated with real probabilities, not probability amplitudes, to account for the real world of open, irreversible systems far from equilibrium.
Prigogine, Ilya and Isabelle Stengers. Order Out of Chaos. London: Heinemann, 1984.
Note: The early classic on dissipative and adaptive systems.
Raymo, Chet. Skeptics and True Believers: The Exhilarating Connection Between Science and Religion. New York: Walker and Company, 1998.
Note: An entertaining, easy read, a good book for a lay discussion group on the general topic “Science and Religion.”
Russell, Robert John, Nancey Murphy and Arthur R. Peacocke Editors. Chaos and Complexity: Scientific Perspectives on Divine Action, second edition. Berkeley, CA: The Center for Theology and the Natural Sciences and Vatican City State: Vatican Observatory Publications, 1997.
Note: Exploration of God acting in nature.
Solé, Ricard and Brian Goodwin. Signs of Life: How Complexity Pervades Biology. New York: Basic Books, 2000.
Note: A mathematician’s introduction to complexity and a good update on modern biology, including a summary of two theoretical mechanisms for the origin of life.
Stewart, Ian. Does God Play Dice? The New Mathematics of Chaos. Maiden, MA: Blackwell Publishers, Inc. 2002.
Note: Entertaining and thorough update on chaos and its historical roots.
Taylor, Barbara Brown. The Luminous Web: Essays on Science and Religion. Cambridge: Cowley Publications, 2000.
Note: An Episcopalian integrates complexity and chaos theory into her theology.
Thuan, Trinh Xuan. Chaos and Harmony: Perspectives on Scientific Revolutions of the Twentieth Century. Oxford: Oxford University Press, 2001.
Note: One of my favorite authors. A beautifully written overview of modern science.
Thuan, Trinh Xuan. The Secret Melody. New York: Oxford University Press, 1995.
Note: Thuan’s even-handed review of cosmological theories.
Waldrop, M. Mitchell. Complexity: The Emerging Science At the Edge of Order and Chaos. New York: Simon and Schuster, 1992. Note: The amusing story of how it all began at the Santa Fe Institute--physicists talking to economists, for example.
Ward, Peter D. and Donald Brownlee. Rare Earth: Why Complex Life is Uncommon in the Universe. New York: Springer Verlag Copernicus, 2000.
Note: Though the new sciences suggest that life arose independently throughout the universe, most of that life is probably very small. Big animals may require special conditions to evolve, like big planetary neighbors that clean up dangerous asteroids.
Wheatley, Margaret J. Leadership and the New Science: Discovering Order in a Chaotic World. San Francisco: Berrett-Koehler, 1999.
Note: If you have time for only one book, read this one. Clear explanations of the new sciences are followed by practical advice on how to use the lessons learned.