On Flows I

"In planning the idea of decentralization must be connected with routines of linking plans of rather autonomous parts of the whole system. Here one can use a conditional separation of the system by means of fixing values of flows and parameters transmitted from one part to another. One can use an idea of sequential recomputation of the parameters, which was successfully developed by many authors for the scheme of Dantzig-Wolfe and for aggregative linear models." (Leonid V Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives," 1975)

"All nature is a continuum. The endless complexity of life is organized into patterns which repeat themselves - theme and variations - at each level of system. These similarities and differences are proper concerns for science. From the ceaseless streaming of protoplasm to the many-vectored activities of supranational systems, there are continuous flows through living systems as they maintain their highly organized steady states." (James G Miller, "Living Systems", 1978)

"The study of changes in the qualitative structure of the flow of a differential equation as parameters are varied is called bifurcation theory. At a given parameter value, a differential equation is said to have stable orbit structure if the qualitative structure of the flow does not change for sufficiently small variations of the parameter. A parameter value for which the flow does not have stable orbit structure is called a bifurcation value, and the equation is said to be at a bifurcation point." (Jack K Hale & Hüseyin Kocak, "Dynamics and Bifurcations", 1991)

"Dynamical systems that vary continuously, like the pendulum and the rolling rock, and evidently the pinball machine when a ball’s complete motion is considered, are technically known as flows. The mathematical tool for handling a flow is the differential equation. A system of differential equations amounts to a set of formulas that together express the rates at which all of the variables are currently changing, in terms of the current values of the variables." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Dynamical systems that vary in discrete steps […] are technically known as mappings. The mathematical tool for handling a mapping is the difference equation. A system of difference equations amounts to a set of formulas that together express the values of all of the variables at the next step in terms of the values at the current step. […] For mappings, the difference equations directly express future states in terms of present ones, and obtaining chronological sequences of points poses no problems. For flows, the differential equations must first be solved. General solutions of equations whose particular solutions are chaotic cannot ordinarily be found, and approximations to the latter are usually determined by numerical methods." (Edward N Lorenz, "The Essence of Chaos", 1993)

"When the pinball game is treated as a flow instead of a mapping, and a simple enough system of differential equations is used as a model, it may be possible to solve the equations. A complete solution will contain expressions that give the values of the variables at any given time in terms of the values at any previous time. When the times are those of consecutive strikes on a pin, the expressions will amount to nothing more than a system of difference equations, which in this case will have been derived by solving the differential equations. Thus a mapping will have been derived from a flow." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Knowledge, truth, and information flow in networks and swarm systems. I have always been interested in the texture of scientific knowledge because it appears to be lumpy and uneven. Much of what we collectively know derives from a few small areas, yet between them lie vast deserts of ignorance. I can interpret that observation now as the effect of positive feedback and attractors. A little bit of knowledge illuminates much around it, and that new illumination feeds on itself, so one corner explodes. The reverse also holds true: ignorance breeds ignorance. Areas where nothing is known, everyone avoids, so nothing is discovered. The result is an uneven landscape of empty know-nothing interrupted by hills of self-organized knowledge." (Kevin Kelly, "Out of Control: The New Biology of Machines, Social Systems and the Economic World", 1995) 

"In the network society, the space of flows dissolves time by disordering the sequence of events and making them simultaneous in the communication networks, thus installing society in structural ephemerality: being cancels becoming." (Manuel Castells, "Communication Power", 2009)

"Another property of bounded systems is that, unless the trajectory attracts to an equilibrium point where it stalls and remains forever, the points must continue moving forever with the flow. However, if we consider two initial conditions separated by a small distance along the direction of the flow, they will maintain their average separation forever since they are subject to the exact same flow but only delayed slightly in time. This fact implies that one of the Lyapunov exponents for a bounded continuous flow must be zero unless the flow attracts to a stable equilibrium." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"In contrast to flow maps, origin-destination maps’ paths are highly structured, and do not use arrowheads to indicate direction. Both types of maps illustrate the volume of flow by varying the thickness of the path line’s shaft, some by gradually trimming the thickness of the shaft, others by splitting the shaft into sections and giving each section its own uniform thickness." (Menno-Jan Kraak, "Mapping Time: Illustrated by Minard’s map of Napoleon’s Russian Campaign of 1812", 2014)