dynamics

Attractors

Attractors are geometric forms that characterize long-term behaviour in the phase space. Roughly speaking, an attractor is what the behaviour of a systems settles down to, or is attracted to. They are globally stable in the sense that the system will return if perturbed off the attractor, as long as it remains within the basin of attraction.

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energy

Joule's principle of the conservation of energy is an example of the subsumption of qualitative transformations into a quantifiable entity. "Thus it is that order is maintained in the universe--nothing is deranged, nothing ever lost, but the entire machinery, complicated as it is, works smoothly and harmoniously." (Joule, quoted in Prigogine, p.108-9)

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feedback

In all feedback systems, some portion of the output system is used as input. Positive feedback adds the output to the input, leading to "vicious cycles." Negative feedback is self-regulating, inducing the system to approach equilibrium or steady-state. (In communications engineering, these two modes are also called regenerative and degenerative cycles.)

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Mandlebrot set

Mandlebrot set

The Mandlebrot set has been described as the "most complicated object in the world." The figure represents the boundary of the domain of attraction of the behaviour of a simple equation in the complex plane. It is not the domain of attraction of a single system but rather a map of a family of systems, based on a single criterion.

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non-linearity

In dynamical terms, such as in the study of chaos, a non-linear situtation is one where the result is not proportional to the cause. For instance "the straw that broke the camel's back" (eg. the elastic/plastic limit in building structures) introduces non-linearity. Up until that point, deformation had been proportional to load. Suddenly it loses all proportionality. 

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Sensitivity to Intial conditions

Sensitivity to Initial Conditions: An extremely small change in the initial conditions of a chaotic or non-linear system leads to extremely differing results. Any arbitrarily small interval of initial values will be enlarged significantly by iteration. This is the so-called "butterfly effect" in which the flapping wings of a single butterfly could theoretically make the difference whether or not a hurricane occurred in another place and time. (The title of a paper by Edward N. Lorentz was "Can the flap of a butterfly's wing stir up a tornado in Texas?" 

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