Theories of Chaos and Complexity

• General Systems Theory Developed in the 1940-1960 period, limited by the lack of computers to do simulations
• Chaos theory is the study of deterministic, rule-based systems that nevertheless appear to be acting completely unpredictably, even at random
• This is most striking in the physical sciences, where measurement is precise and theory would lead us to expect things to be predictable.  Meteorology is a common example.
• The What is Chaos lecture provides a good outline
• contrasts chaos theory to deterministic science
• refers to Newton's Three Laws of Motion as an example of deterministic science
• chaos theory deals with situations which are highly sensitive to initial conditions
• no measurements can be perfectly precise, which means that systems which are highly dependent on initial conditions are unstable
• Chaos is not just due to errors in measurement,  but can be shown in pure mathematics, generating "strange attractors" out of fairly simple formulas.
• Complexity is not the same as chaos, it refers to systems where a large number of variables interact with each other.  Complex systems may or may not lead to chaotic behavior.
• We are interested in studying complexity  - in general - because we may find similar patterns emerging in diverse systems.  Otherwise, we can study each complex system independently.  Thus, we believe that systems as different as the brain, families, the climate, auto traffic, earthquakes and financial markets may have things in common.

• this equation models the growth of a population in a limited environment.  The larger it grows, the less the opportunity for further growth
• At first, the population growth is smooth, then it becomes chaotic before stopping
• The results are best graphed with a computer program.
• But we may understand the process better if we plot it out in a spreadsheet program things  such as the Excel program I have written here
• If this math is too much for you, the important thing to remember is that computing certain formulas over and over, inputting the results from each calculation into the next calculation, sometimes leads to very complex results, such as those show in the graphic of a plot of the logistic equation
• scientists believe this models how many complex phenomena evolve in the real world

• fixed point - constant, not changing
• limit cycle -  changing within a fixed pattern, periodic, e.g., motion of the planets.  These can be studied with mathematics, e.g., Newton's laws, calculus...
• Strange Attractors are patterns which have irregular, nonlinear shapes and which emerge out of repeated iterations of a system.  They must be studied with computer simulation.  They often make beautiful graphics.
• There is evidence that strange attractors exist in the brain, e.g., in the olfactory lobes of rabbits

• evolution requires a large number of competing entities, which can survive or not survive depending on circumstances in their environment
• these entities must also have the ability to mutate or combine to form new entities

• entities which maintain their structure despite environmental pressure to change
• if part of the structure is destroyed, the other parts act to replace or compensate for it
• the ability to create new components which serve to strengthen and maintain the existing system

• The archetypes can be viewed as the attractors of the mind as a complex system
• Jung did not know about chaos theory, but he observed patterns which recurred in mind after mind.  These include:
• The Self - a "zero" archetype for Jung, where we begin
• The Shadow -  the "other" or "Dark Side" - evil
• Anima (female) and Animus (male), both of which are found in both men and women, and combine to form the Soul (Thirdness in Peircian terms)
• The Syzygy - Fourthness, an autopoietic system, represented for Jung by Mandalas.

• The Network Archetype - a variant of Peirce's Thirdness archetype.
• The Hierarchy Archetype - also a variant of Thirdness
• Synergy or emergence or Fourthness
• The Dual Network Archetype - this achieves Fourthness by combining the Thirdnesses of Network and Hierarchy.
• The Self - not in Jung's sense, but in the sense of an image of the whole living within the whole.  This is a fractal archetype in that the part is a recreation of the whole.  And one may find that within the self there are sub-selves, each of which recreate the whole in a different way
• Evolution and Autopoiesis - the synergy of these two archetypes working together creates an archetype of dynamics