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I just found this very abreviated but really well written intro to chaos theory, this should help anyone that is having trouble understanding this post.
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Chaos and Complexity
One of the themes straddling both biological and physical sciences is the quest for a mathematical model of phenomena of emergence (spontaneous creation of order), and in particular adaptation, and a physical justification of their dynamics (which seems to violate physical laws).
The physicist Sadi Carnot, one of the founding fathers of Thermodynamics, realized that the statistical behavior of a complex system can be predicted if its parts were all identical and their interactions weak. At the beginning of the century, another French physicist, Henri Poincare`, realizing that the behavior of a complex system can become unpredictable if it consists of few parts that interact strongly, invented "chaos" theory. A system is said to exhibit the property of chaos if a slight change in the initial conditions results in large-scale differences in the result. Later, Bernard Derrida will show that a system goes through a transition from order to chaos if the strength of the interactions among its parts is gradually increased. But then very "disordered" systems spontaneously "crystallize" into a higher degree of order.
First of all, the subject is "complexity", because a system must be complex enough for any property to "emerge" out of it. Complexity can be formally defined as nonlinearity.
The world is mostly nonlinear. The science of nonlinear dynamics was originally christened "chaos theory" because from nonlinear equations unpredictable solutions emerge.
A very useful abstraction to describe the evolution of a system in time is that of a "phase space". Our ordinary space has only three dimensions (width, height, depth) but in theory we can think of spaces with any number of dimensions. A useful abstraction is that of a space with six dimensions, three of which are the usual spatial dimentions. The other three are the components of velocity along those spatial dimensions. In ordinary 3-dimensional space, a "point" can only represent the position of a system. In 6-dimensional phase space, a point represents both the position and the motion of the system. The evolution of a system is represented by some sort of shape in phase space.
The shapes that chaotic systems produce in phase space are called "strange attractors" because the system will tend towards the kinds of state described by the points in the phase space that lie within them.
The program then becomes that of applying the theory of nonlinear dynamic systems to Biology.
Inevitably, this implies that the processes that govern human development are the same that act on the simplest organisms (and even some nonliving systems). They are processes of emergent order and complexity, of how structure arises from the interaction of many independent units. The same processes recurr at every level, from morphology to behavior.
Darwin's vision of natural selection as a creator of order is probably not sufficient to explain all the spontaneous order exhibited by both living and dead matter. At every level of science (including the brain and life) the spontaneous emergence of order, or self-organization of complex systems, is a common theme.
Koestler and Salthe have shown how complexity entails hierarchical organization. Von Bertalanffi's general systems theory, Haken's synergetics, and Prigogine's non-equilibrium Thermodynamics belong to the class of mathematical disciplines that are trying to extend Physics to dynamic systems.
These theories have in common the fact that they deal with self-organization (how collections of parts can produce structures) and attempt at providing a unifying view of the universe at different levels of organization (from living organisms to physical systems to societies).
Holarchies
The Hungarian writer and philosopher Arthur Koestler first brought together a wealth of biological, physical, anthropological and philosophical notions to construct a unified theory of open hierarchical systems.
Langauge has to do with a hierarchical process of spelling out implicit ideas in explicit terms by means of rules and feedbacks. Organisms and societies also exhibit the same hierarchical structure. In these hierarchies, each intermediary entity ("holon") functions as a self-contained whole relative to its subordinates and as one of the dependent parts of its superordinates. Each holon tends to persist and assert its pattern of activity.
Wherever there is life, it must be hierarchically organized. Life exhibits an integrative property (that manifests itself as symbiosis) that enables the gradual construction of complex hierarchies out of simple holons. In nature there are no separated, indivisible, self-contained units. An "individual" is an oxymoron. An organism is a hierarchy of self-regulating holons (a "holarchy") that work in coordination with their environment. Holons at the higher levels of the hierarchy enjoy progressively more degrees of freedom and holons at the lower levels of the hierarchy have progressively less degrees of freedom. Moving up the hierarchy, we encounter more and more complex, flexible and creative patterns of activity. Moving down the hierarchy behavior becomes more and more mechanized.
A hierarchical process is also involved in perception and memorization: it gradually reduces the percept to its fundamental elements. A dual hierarchical processis involved in recalling: it gradually reconstructs the percept.
Hierarchical processes of the same nature can be found in the development of the embryo, in the evolution of species and in consciousness itself (which should be analyzed not in the context of the mind/body dichotomy but in the context of a multi-levelled hierarchy and of degrees of consciousness).
They all share common themes: a tendency towards integration (a force that is inherent in the concept of hierarchic order, even if it seems to challenge the second law of Thermodynamics as it increases order), an openess at the top of the hierarchy (towards higher and higher levels of complexity) and the possibility of infinite regression.
Hierarchies from Complexity
Stanley Salthe, by combining the metaphysics of Justus Buchler and Michael Conrad's "statistical state model" of the evolutionary process, has developed what amounts to a theory of everything: an ontology of the world, a formal theory of hierarchies and a model of the evolution of the world.
The world is viewed as a determinate machine of unlimited complexity. Within complexity, discontinuities arise. The basic structure of this world must allow for complexity that is spontaneously stable and that can be broken down in things divided by boundaries. The most natural way for the world to satisfy this requirement is to employ a hierarchical structure, which is also implied by Buchler's principle of ordinality: Nature (i.e., our representation of the world) is a hierarchy of entities existing at different levels of organization. Hierarchical structure turns out to be a consequence of complexity.
Entities are defined by four criteria: boundaries, scale, integration, continuity. An entity has size, is limited by boundaries, and consists of an integrated system which varies continuously in time.
Entities at different levels interact through mutual constraints, each constraint carrying information for the level it operates upon. A process can be described by a triad of contiguous levels: the one it occurs at, its context (what the philosopher Mario Bunge calls "environment") and its causes (Bunge's "structure"). In general, a lower level provides initiating conditions for a process and an upper level provides boundary conditions. Representing a dynamic system hierarchically requires a triadic structure.
Aggregation occurs upon differentiation. Differentiation interpolates levels between the original two and the new entities aggregate in such a way that affects the structure of the upper levels: every time a new level emerges, the entire hierarchy must reorganize itself.
Salthe also recalls a view of complexity due to the physicist Howard Hunt Pattee: complexity as the result of interactions between physical and symbolic systems. A physical system is dependent on the rates at which processes occur, whereas a symbolic system is not. Symbolic systems frequently serve as constraints applied to the operation of physical systems, and frequently appear as products of the activity of physical systems (e.g., the genome in a cell). A physical system can be said to be "complex" when a part of it functions as a symbolic system (as a representation, and therefore as an observer) for another part of it.
These abstract principles can then be applied to organic evolution. Over time, Nature generates entities of gradually more limited scope and more precise form and behavior. This process populates the hierarchy of intermediate levels of organization as the hierarchy spontaneously reorganizes itself. The same model applies to all open systems, whether organisms or ecosystems or planets.
By applying principles of complex systems to biological and social phenomena, Salthe attempts to reformulate Biology on development rather than on evolution. His approach is non-Darwinian to the extent that development, and not evolution, is the fundamental process in self-organization. Evolution is merely the result of a margin of error. His theory rests on a bold fusion of hierarchy theory, Information Theory and Semiotics.
Salthe is looking for a grand theory of nature, which turns out to be essentially a theory of change, which turns out to be essentially a theory of emergence.
Dissipative Systems
By far, though, the most influential school of thought has been the one related to Ilya Prigogine's non-equilibrium Thermodynamics, which redefined the way scientists approach natural phenomena and brought self-organizing processes to the forefront of the study of complex systems. His theory found a stunning number and variety of fields of application, from Chemistry to Sociology. In his framework, the most difficult problems of Biology, from morphogenesis to evolution, found a natural model.
Classical Physics describes the world as a static and reversible system that undergoes no evolution, whose information is constant in time. Classical Physics is the science of being. Thermodynamics, instead, describes an evolving world in which irreversible processes occurs. Thermodynamics is the science of becoming.
The second law of Thermodynamics, in particular, describes the world as evolving from order to disorder, while biological evolution is about the complex emerging from the simple (i.e. order arising from disorder). While apparently contradictory, these two views show that irreversible processes are an essential part of the universe.
Furthermore, conditions far from equilibrium foster phenomena such as life that classical Physics does not cover at all.
Irreversible processes and non-equilibrium states turn out to be fundamental features of the real world.
Prigogine distinguishes between "conservative" systems (which are governed by the three conservation laws for energy, translational momentum and angular momentum, and which give rise to reversible processes) and "dissipative" systems (subject to fluxes of energy and/or matter). The latter give rise to irreversible processes.
The theme of science is order. Order can come either from equilibrium systems or from non-equilibrium systems that are sustained by a constant source (or, dually, by a persistent dissipation) of matter/energy. In the latter systems, order is generated by the flux of matter/energy. All living organisms (as well as systems such as the biosphere) are non-equilibrium systems.
Prigogine proved that, under special circumstances, the distance from equilibrium and the nonlinearity of a system drive the system to ordered configurations, i.e. create order. The science of being and the science of becoming describe dual aspects of Nature.
What is needed is a combination of factors that are exactly the ones found in living matter: a system made of a large collection of independent units which are interacting with each other, a flow of energy through the system that drives the system away from equilibrium, and nonlinearity. Nonlinearity expresses the fact that a perturbation of the system may reverberate and have disproportionate effects.
Non-equilibrium and nonlinearity favor the spontaneous development of self-organizing systems, which maintain their internal organization, regardless of the general increase in entropy, by expelling matter and energy in the environment.
When such a system is driven away from equilibrium, local fluctuations appear. This means that in places the system gets very unstable. Localized tendencies to deviate from equilibrium are amplified. When a threshold of instability is reached, one of these runaway fluctuations is so amplified that it takes over as a macroscopic pattern. Order appears from disorder through what are initially small fluctuations within the system. Most fluctuations die along the way, but some survive the instability and carry the system beyond the threshold: those fluctuations "create" new form for the system. Fluctuations become sources of innovation and diversification.
The potentialities of nonlinearity are dormant at equilibrium but are revelead by non-equilibrium: multiple solutions appear and therefore diversification of behavior becomes possible.
Technically speaking, nonlinear systems driven away from equilibrium can generate instabilities that lead to bifurcations (and symmetry breaking beyond bifurcation). When the system reaches the bifurcation point, it is impossible to determine which path it will take next. Chance rules. Once the path is chosen, determinism resumes.
The multiplicity of solutions in nonlinear systems can even be interpreted as a process of gradual "emancipation" from the environment.
Most of Nature is made of such "dissipative" systems, of systems subject to fluxes of energy and/or matter. Dissipative systems conserve their identity thanks to the interaction with the external world. In dissipative structures, non-equilibrium becomes a source of order.
These considerations apply very much to living organisms, which are prime examples of dissipative structures in non-equilibrium. Prigogine's theory explains how life can exist and evolution work towards higher and higher forms of life. A "minimum entropy principle" characterizes living organisms: stable near-equilibrium dissipative systems minimize their rate of entropy production.
From non-equilibrium Thermodynamics a wealth of concepts has originated: invariant manifolds, attractors, fractals, stability, bifurcation analysis, normal forms, chaos, Lyapunov exponents, entropies. Catastrophe and chaos theories turn out to be merely special cases of nonlinear non-equilibrium systems.
In concluding, self-organization is the spontaneous emergence of ordered structure and behavior in open systems that are in a state far from equilibrium described mathematically by nonlinear equations.
Catastrophe Theory
Rene' Thom's catastrophe theory, originally formulated in 1967 and popularized ten years later by the work of the British mathematician Erich Zeeman, became a widely used tool for classifying the solutions of nonlinear systems in the neighborhood of stability breakdown.
In the beginning, Thom, a French mathematician, was interested in structural stability in topology (stability of topological form) and was convinced of the possibility of finding general laws of form evolution regardless of the underlying substance of form, as already stated at the beginning of the century by D'Arcy Thompson.
Thom's goal was to explain the "succession of form". Our universe presents us with forms (that we can perceive and name). A form is defined, first and foremost, by its stability: a form lasts in space and time. Forms change. The history of the universe, insofar as we are concerned, is a ceaseless creation, destruction and transformation of form. Life itself is, ultimately, creation, growth and decaying of form.
Every physical form is represented by a mathematical quantity called "attractor" in a space of internal variables. If the attractor satisfies the mathematical property of being "structurally stable", then the physical form is the stable form of an object. Changes in form, or morphogenesis, are due to the capture of the attractors of the old form by the attractors of the new form. All morphogenesis is due to the conflict between attractors. What catastrophe theory does is to "geometrize" the concept of "conflict".
The universe of objects can be divided into domains of different attractors. Such domains are separated by shock waves. Shock wave surfaces are singularities called "catastrophes". A catastrophe is a state beyond which the system is detroyed in an irreversible manner. Technically speaking, the "ensembles de catastrophes" are hypersurfaces that divide the parameter space in regions of completely different dynamics.
The bottom line is that dynamics and form become dual properties of nonlinear systems.
This is a purely geometric theory of morphogenesis, His laws are independent of the substance, structure and internal forces of the system.
Thom proves that in a 4-dimensional space there exist only 7 types of elementary catastrophes. Elementary catastrophes include: "fold", destruction of an attractor which is captured by a lesser potential; "cusp", bufurcation of an attractor into two attractors; etc. From these singularities, more and more complex catastrophes unfold, until the final catastrophe. Elementary catastrophes are "local accidents". The form of an object is due to the accumulation of many of these "accidents".