Complexity_ A Guided Tour - Melanie Mitchell [17]
In short, the presence of chaos in a system implies that perfect prediction à la Laplace is impossible not only in practice but also in principle, since we can never know x0 to infinitely many decimal places. This is a profound negative result that, along with quantum mechanics, helped wipe out the optimistic nineteenth-century view of a clockwork Newtonian universe that ticked along its predictable path.
But is there a more positive lesson to be learned from studies of the logistic map? Can it help the goal of dynamical systems theory, which attempts to discover general principles concerning systems that change over time? In fact, deeper studies of the logistic map and related maps have resulted in an equally surprising and profound positive result—the discovery of universal characteristics of chaotic systems.
Universals in Chaos
The term chaos, as used to describe dynamical systems with sensitive dependence on initial conditions, was first coined by physicists T. Y. Li and James Yorke. The term seems apt: the colloquial sense of the word “chaos” implies randomness and unpredictability, qualities we have seen in the chaotic version of logistic map. However, unlike colloquial chaos, there turns out to be substantial order in mathematical chaos in the form of so-called universal features that are common to a wide range of chaotic systems.
THE FIRST UNIVERSAL FEATURE: THE PERIOD-DOUBLING
ROUTE TO CHAOS
In the mathematical explorations we performed above, we saw that as R was increased from 2.0 to 4.0, iterating the logistic map for a given value of R first yielded a fixed point, then a period-two oscillation, then period four, then eight, and so on, until chaos was reached. In dynamical systems theory, each of these abrupt period doublings is called a bifurcation. This succession of bifurcations culminating in chaos has been called the “period doubling route to chaos.”
These bifurcations are often summarized in a so-called bifurcation diagram that plots the attractor the system ends up in as a function of the value of a “control parameter” such as R. Figure 2.11 gives such a bifurcation diagram for the logistic map. The horizontal axis gives R. For each value of R, the final (attractor) values of x are plotted. For example, for R = 2.9, x reaches a fixed-point attractor of x = 0.655. At R = 3.0, x reaches a period-two attractor. This can be seen as the first branch point in the diagram, when the fixed-point attractors give way to the period-two attractors. For R somewhere between 3.4 and 3.5, the diagram shows a bifurcation to a period-four attractor, and so on, with further period doublings, until the onset of chaos at R approximately equal to 3.569946.
FIGURE 2.11. Bifurcation diagram for the logistic map, with attractor plotted as a function of R.
The period-doubling route to chaos has a rich history. Period doubling bifurcations had been observed in mathematical equations as early as the 1920s, and a similar cascade of bifurcations was described by P. J. Myrberg, a Finnish mathematician, in the 1950s. Nicholas Metropolis, Myron Stein, and Paul Stein, working at Los Alamos National Laboratory, showed that not just the logistic map but any map whose graph is parabola-shaped will follow a similar period-doubling route. Here, “parabola-shaped” means that plot of the map has just one hump—in mathematical terms, it is “unimodal.”
THE SECOND UNIVERSAL FEATURE: FEIGENBAUM’S
CONSTANT
The discovery that gave the period-doubling route its renowned place among mathematical universals was made in the 1970s by the physicist Mitchell Feigenbaum. Feigenbaum, using only a programmable desktop calculator, made a list of the R values at which the period-doubling bifurcations occur (where ≈ means “approximately equal to”):
R1 ≈ 3.0
R2 ≈ 3.44949
R3 ≈ 3.54409
R4 ≈ 3.564407
R5 ≈ 3.568759
R6 ≈ 3.569692