ATTRACTOR (Chaotic) 1)2)
Also called "strange attractors"
D. BROOM HEAD describes the chaotic attractor in these terms: "(It)… can be thought of as a bag in the abstract phase space defined by the variables of position and momentum; the bag contains an infinite number of periodic states, all of which are unstable. 'Unstable' in this sense means that if, at some time, the behavior of the system can be approximated by one of the periodic states, the approximation error grows exponentially with time. The interesting thing – and the resolution of the chaotic paradox – is that the bag, as a whole, is stable. Once the system is in the 'bag', it cannot leave; rather it wanders, moving close to one periodic state as it diverges from another" (1990, p.23)
It is thus quite clear that randomness in chaotic systems is confined within a global determinism.
I. PRIGOGINE and I. STENGERS observe: "Chaotic attractors are not characterized by whole dimensions, as a line or a surface, but by fractionary dimensions. They are what is called , since MANDELBROT, fractal varieties" (1992, p.73) (see: "Fractal dimensions")
J. CASTI makes this clearer as follows: "(in classical attractors) regardless of the dimension of the overall state space, the fixed point has dimension 0, while the limit cycle, being a simple closed curve, has dimension 1 These numbers are the geometric dimension of the attractor. At the other end of the scale is the situation in which the system is truly random. In this case, every point of the state space is eventually visited, leading to the attractor's having the same dimension as the state space itself. Chaotic systems with their strange attractors lie somewhere in between. For such systems, the attractor is clearly not such a primitive geometrical object as a point or a simple curve. Yet it is still a proper subset of the overall set of states" (1994, p.101) (and, consequently has a fractionary, or fractal dimension)
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Bertalanffy Center for the Study of Systems Science(2020).
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