what's In the Box?!!??!

Definition

It operates in a non-linear way

That it is iterative (the output of one cycle  becomes the input of the next

Small variations in initial conditions lead to large  differences in outcomes

The evolution of a nonlinear, dynamical, complex system can be marked by a series of phases, each of which constrains the behavior of the system to be in consonance with a reigning attractor(s). Such phases and their attractors can be likened to the stages of human development: infancy, childhood, adolescence, and so on. Each stage has its own characteristic set of behaviors, developmental tasks, cognitive patterns, emotional issues, and attitudes (although, of course, there is some variation among different people). Though a child may sometimes behave like an adult (and vice versa), the long term behavior is what falls under the sway of the attractor. Technically, in a dynamical system, an attractor is a pattern in phase or state space called a phase portrait to which values of variables settle into after transients die out. More generally, an attractor can be considered a circumscribed or constrained range in a system which seemingly underlies and "attracts" how a system is functioning within particular environmental (internal and external) conditions. The dynamics of the system as well as current conditions determine the system’s attractors. When attractors change, the behavior in the system changes because it is operating under a different set of governing principles. The change of attractors is called bifurcation, and is brought about from far-from-equilibrium conditions which can be considered as a change in parameter values toward a critical threshold.

Types of Attractors:

Far-from-equilibrium:

The term used by the Prigogine School for those conditions leading to self- organization and the emergence of dissipative structures. Far-from-equilibrium conditions move the system away from its equilibrium state, activating the nonlinearity inherent in the system. Far-from-equilibrium conditions are another way of talking about the changes in the values of parameters leading-up to a bifurcation and the emergence of new attractor(s) in a dynamical system. Furthermore, to some extent, far-from-equilibrium conditions are similar to "edge of chaos" in cellular automata and random boolean networks.

Adaptation:

In the theory of Darwinian Evolution, adaptation is the ongoing process by which an organism becomes "fit" to a changing environment. Adaptation occurs when modifications of an organism prove helpful to the continuation of the species in a changed environment. These modifications result from both random mutations and recombination of genetic material (e.g., by means of sexual reproduction). In general, through the mechanism of natural selection, those modifications that aid in the survival of species survival are maintained. However, insights from the study of complex, adaptive systems are suggesting that natural selection operates on systems which already contain a great deal of order simply as a result of self- organizing processes following the internal dynamics of a system (Kauffman’s "order for free"). A fundamental characteristic of complex, adaptive systems is their capacity to adapt by changing the rules of interaction among their component agents. In that way, adaptation consists of "learning" new rules through accumulating new experiences.

From the Greek "anacoluthon" (inconsistency in logic), a general term for system processes or methods facilitating self-organization and emergence. In these processes traditional procedures are followed while at the same time they are transgressed, thereby allowing the emergence of something radically new. An example of an anacoluthian process is the crossing-over of chromosomes from both parents in sexual reproduction. An example in a business or institution when people from diverse organizational functions are brought together in a project team, hopefully resulting in the emergence of an innovative organizational structure.

A "graphical" way to measure and explore the adaptive (fitness) value of different configurations of some elements in a system. Each configuration and its neighbor configurations (i.e., slight modifications of it) are graphed as lower or higher peaks on a landscape-like surface, i.e., high fitness is portrayed as mountainous-like peaks, and low fitness is depicted as lower peaks or valleys Such a display provides an indication of the degree to which various combinations add or detract from the system s survivability or sustainability. The use of fitness landscapes in understanding the behavior of complex, adaptive systems has been pioneered by Stuart Kauffman in his study of random boolean networks. An important implication from studying fitness landscapes is that there may be many local peaks or "okay" solutions instead of one, perfect, optimal solution. Thinking in terms of fitness landscapes can point to foolish adaptation, i.e., a downward trend on the slopes of the peaks. Moreover, studies of N/K models using fitness landscapes demonstrates that there is a decreasing rate of finding fitter adaptable configurations as one travels uphill on a fitness landscape. The use of fitness landscapes can be applied to gain insight into various organizational issues including which innovative organizational designs, processes, or strategies promise greater potential.

Sensitive Dependence on Initial Conditions (SIC):

The property of chaotic systems in which a small change in initial conditions can have a hugely disproportionate effect on outcome. SIC is popularly captured by the image of the Butterfly Effect. SIC makes chaotic systems largely unpredictable because measurements at initial conditions always will contain some amount of error, and SIC exponentially increases this error.

A type of mathematical pattern in which the frequency of an occurrence of a given size is inversely proportionate to some power (or exponent) of its size. For example, in the case of avalanches or earth quakes, large ones are fairly rare, smaller ones are much more frequent, and in between are cascades of different sizes and frequencies. Power laws define the distribution of catastrophic events in Self-organized Critical systems. According to Stuart Kauffman, systems at the "edge of chaos" will show a power law distribution, therefore, having this type of distribution can be evidence that systems are at this particularly "pregnant" phase.

Processes of self-organization and emergence occur within bounded regions, e.g., the container holding the Benard System so that the liquid is intact as it undergoes far-from- equilibrium conditions. In cellular automata the container is the electronic network itself which is "wrapped around" in that cells at the outskirts of the field are hooked back into the field. These boundaries or containers act to demarcate a system from its environment, and, thereby, maintain the identity of a system as it changes. Furthermore, boundaries channel the nonlinear processes at work during self-organization. In human systems, boundaries can refer to the actual physical plant, organizational policies, "rules" of interaction, and whatever serves to underlie an organization’s identity and that distinguishes an organization from its boundaries. Boundaries need to be both permeable in the sense that they allow exchange between a system and its environments as well as impermeable in so far as they circumscribe the identity of a system in contrast with its environments.

Self-organized Criticality (SOC):

Formulated by the physicist Per Bak, a phenomena of sudden change in physical systems in which they evolve naturally to a critical state at which abrupt changes can occur. That is, when these systems are not in a critical state, i.e., they are characterized by instability, output follows from input in a linear fashion, but when in the critical state, systems characterized by self- organized criticality act like nonlinear amplifiers, similar to but not as extreme as the exponential increase in chaos due to sensitive dependence on initial conditions. That is, the nonlinear amplification in a self-organized, critical system follows a power law instead of an exponential law. SOC systems are self-organized in the sense that they reach a critical state on their own. Examples of such systems include avalanches, plate tectonics leading to earthquakes or stock market systems leading to crashes. Because SOC systems follow power laws, and because fractals also show a similar mathematical pattern then it may be the case that many naturally occurring fractals, such as tree growth, the structure of the lungs, and so on, may be generated by some form of self-organized criticality.