“Quanto mais leio sobre o caos, mais me convenço que aquilo se aplica a tudo”
Colocado por Paulo Sacramento em Geral“This constant, called Feigenbaum’s number, crops up repeatedly in self-similar figures and has an approximate value of
4.
669201609102990671853203820466201617258185577475768632745651
343004134330211314737138689744023948013817165984855189815134
408627142027932522312442988890890859944935463236713411532481
714219947455644365823793202009561058330575458617652222070385
410646749494284981453391726200568755665952339875603825637225
Not only does Feigenbaum’s constant reappear in other figures, but so do many other characteristics of the bifurcation diagram. In fact, remarkably similar diagrams can be generated from any smooth, one-dimensional, non-monotonic function when mapped on to itself. A circle, ellipse, sine, or any other function with a local maximum will produce a bifurcation diagram with period-doublings who’s ratios approach “d” [delta]. Together with a second constant “a” [alpha], the scaling factor “d” [delta] demonstrates a universality previously unknown in mathematics: metrical universality. The behavior of the quadratic map is typical for many dynamical systems. One year after their discovery, the period-doubling route to chaos and the constants “a” [alpha] and “d” [delta] appeared in an unruly mess of equations used to describe hydrodynamic flow. This might not be so amazing if it weren’t for the fact that Feigenbaum’s constants were originally derived from a mathematical model of animal populations. In the segmented, fragmented world of modern science hydrodynamicists and population biologist rarely interact with one another. The realization that a set of five coupled differential equations describing turbulence could exhibit the same fundamental behavior as the one-dimensional map of the parabola on to itself was one of the key events in the history of mathematics.“
Tópicos (RSS)
Senti exactamente o mesmo quando andei a ‘brincar’ com esses sistemas dinâmicos não lineares, dimensões fraccionárias e afins. Infelizmente a minha deficiente formação em matemáticas não me permitiu ir muito longe
Contudo, o fascínio por encontrar comportamentos imprevisíveis mas não aleatórios em múltiplas manifestações do mundo físico ficou. É como se finalmente uma teoria conseguisse explicar o comportamento das coisas do dia-a-dia e não apenas o das estrelas e dos átomos.