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In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form geometric shapes that resemble tongues, in which case they are called Arnold tongues.

Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and cardiac electric waves. Sometimes the frequency of oscillation depends on, or is constrained (i.e., phase-locked or mode-locked, in some contexts) based on some quantity, and it is often of interest to study this relation. For instance, the outset of a tumor triggers in the area a series of substance (mainly proteins) oscillations that interact with each other; simulations show that these interactions cause Arnold tongues to appear, that is, the frequency of some oscillations constrain the others, and this can be used to control tumor growth. (Full article...)

The Lorenz attractor is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its butterfly shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.

The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.

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W. Ross Ashby (1948)

W. Ross Ashby (6 September 1903 – 15 November 1972) was an English psychiatrist and a pioneer in cybernetics, the study of the science of communications and automatic control systems in both machines and living things. His first name was not used: he was known as Ross Ashby.

His two books, Design for a Brain and An Introduction to Cybernetics, introduced exact and logical thinking into the brand new discipline of cybernetics and were highly influential. These "missionary works" along with his technical contributions made Ashby "the major theoretician of cybernetics after Wiener". (Full article...)

• ... that American systems theorist Debora Hammond in the new millennium explores new ways of thinking about complex systems that support more participatory forms of social organization?
• ... * and next a science of action, called "Interactive management".
• ... that the Austrian American Heinz von Foerster in 1960 in Science magazine stated, that the human population would reach "infinity" and he proposed a formula for predicting future population growth.
• ... * then a science of complexity,