Phase space: (or state space) Allows representation of the behaviour of a system in geometric form. The number of dimensions required for the phase space is a function of the "degrees of freedom" of the system. A dynamical system consists in two parts: the notions of a state (the essential information about a system) and a dynamic (a rule that describes how the state evolves with time). This evolution can be visualized in a phase space. Phase spaces can have any number of dimensions, corresponding to the “degrees of freedom” of the system. The figures drawn in the phase space that describe the system's behavior are phase portraits.
A singularity is a kind of discontinuity. It might or might not be interesting. A vaguer use of the term is simply "a point where something happens" (although this equally describes an event.) Deleuze and Guattari are fascinated by singularities because they are points of unpredictability, even when deterministic. They are thus the sites of revolutionary potential.
As used by mathematical physicists, a singularity means a place where slopes become infinite, where the rate of change of one variable with another exceeds all bounds, and where a big change in an observable is caused by an arbitrarily small change in something else. (cf sensitivity to initial conditions). It is an actual point of infinite density and energy that's kind of a rupture in the fabric of space-time.
Astrophysics describe the centers of black holes as singularities.The Big Bang is considered to be a singularity. A phase singularity is a point at which phase is ambiguous and near which phase takes on all values. Time at the poles of the earth is an example.
For Robert Smithson, architecture depends on the repression of entropy. "The dream of architecture is to escape from entropy." (Informe: Mode d'Emploi )
In the late nineteenth century speculation about entropy intersected with the culture of colonialism, with the uneasy relations between technological progress (primarily through the heat engine) and a sense of cultural pessimism. Thus for Oswald Spengler, entropy "signifies today the world's end as a completion of an inwardly necessary relation." (see Crosbie Smith and Norton Wise, Energy and Empire )
In 1852 William Thompson, Lord Kelvin, predicted the death of the earth from heat loss in an article entitled "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy." In it he wrote that "There is at present in the material world a universal tendency to the dissipation of mechanical energy. Any restoration of mechanical energy, without more than an equivalent amount of dissipation, is impossible...and is probably never effected by means of organized matter, either endowed with vegetable life or subjected to the will of an animated creature." "Within a finite period of time...the earth must again be unfit for the habitation of man as present constituted." (quoted and commented upon in Hayles, Chaos Bound, pp 39-42)
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.
The phase space of a particular dynamical system is partitioned into one or many basins of attraction , perhaps intimately intertwined, each with its own attractor . basins of attraction for a pendulum swinging over three magnets. (from Peitgen, Jürgens, and Saupe, Chaos and Fractals)
A bifurcation occurs when an attractor changes qualitatively with the smooth variation of a control parameter. Physically, bifurcations denote phase transitions from a state of equilibrium to new possible states of equilibria. (see also singularities)
a dynamical system consists of a space, or manifold, where the motion of the system takes place, and a rule of motion, or vector field. The starting point is called the initial state, and the path of motion the trajectory. The end point of a trajectory is the system's attractor.
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)
Entropy: Second law of thermodynamics: Entropy is a measure of the energy distribution through a system. As energy becomes more dispersed or more evenly distributed in a system, the possibility of that energy's being used for mechanical work is decreased, and entropy increases.
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.)
Is the concept of force a technico/scientific one or a philosophical, even metaphysical one?
By defining force purely as the product of an acceleration (a purely kinematic magnitude--from Greek kineo, referring to constrained or controlled motion) and a mass (a coefficient to be determined empirically), modern science eliminated both the metaphysical terminology and psychological origins of the concept of force. (E. J. Dijksterhuis) (see machine ) see also qualititative / quantitative .
For classical physics, force is really only explainable in terms of bodies in contact. "Action at a distance", which is also a feature of gravity, seems "occult." Yet from the work of Kepler and Newton, particles had a dual nature: on the one hand, a highly localized object, and on the other an influence extending through the whole of space. But is the concept of force to be associated primarily with Newtonian mechanics -- as opposed to Newton's alchemical writings?
Newton refused to define the nature of his fundamental gravitational force. He emphasized its heuristic importance and the possibililty of mathematizing it. On the other hand, his alchemical speculations downright invited the amplification and translation of his concept of force into the realm of living things. Could magnetism and electricity, he asked, fulfill the same role for living beings that gravitation did for inanimate matter? (see mechanism / vitalism ) Whether as quasi-mechanical attraction of molecules (as suggested by Buffon and Maupertius) or eventually as formative force (as suggested by the epigeneticists) the Newtonian concept of force supplied the debate about generation with a new energy in the eighteenth century.
Dynamical Systems can be characterized as conservative or dissipative, depending on whether their phase volume stays constant or contracts.
A linearized damped pendulum decays to a single point -- its attractor, and is said to be dissipative. (see Baker and Golub, Chaotic Dynamics)
Roughly speaking, a dissipative system is not conservative but "open," with an external control parameter that can be tuned to critical values causing the transitions to chaos. In physical terms energy flows through a dissipative system and is lost to microscopic degrees of freedom. Entropy "fans out" into irrelevant variables, while the trajectory of "relevant" variables occupies a smaller and smaller region of phase space.
Dissipative Structures: (usage in Prigogine) The interaction of a system with the outside world, its embedding in non-equilibrium conditions may become the starting point for the formation of new dynamic states of matter. A whirlpool, for example, is a dissipative structure requiring a continuous flow of matter and energy to maintain the form.
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.
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.
A phase boundary separates different areas of phase space, for example the region of ordered dynamics from the region of chaotic dynamics. The region at or near this boundary is described as the complex area or regime.
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?"
