This diagram reveals not one, but rather an infinite number of period doubling bifurcations. As is increased a period two orbit becomes a period four orbit, and this in turn becomes a period eight orbit, and so on. This sequence of period doubling bifurcations is known as a period doubling cascade. This process appears to converge at a finite value of around 3.57, beyond which a nonperiodic motion appears to exist. This period doubling cascade often occurs in nonlinear systems. For instance, a similar period doubling cascade occurs in the bouncing ball system (Figure 1.16). The period doubling route is one common way, but certainly not the only way, by which a nonlinear system can progress from a simple behavior (one or a few periodic orbits) to a complex behavior (chaotic motion and the existence of an infinity of unstable periodic orbits).
In 1976, Feigenbaum began to wonder about this period doubling cascade. He started playing some numerical games with the quadratic map using his HP65 hand-held calculator. His wondering soon led to a remarkable discovery. At the time, Feigenbaum knew that this period doubling cascade occurred in one-dimensional maps of the unit interval. He also had some evidence that it occurred in simple systems of nonlinear differential equations that model, for instance, the motion of a forced pendulum. In addition to looking at the qualitative similarities between these systems, he began to ask if there might be some quantitative similarity–that is, some numbers that might be the same in all these different systems exhibiting period doubling. If these numbers could be found, they would be “universal” in the sense that they would not depend on the specific details of the system.
Feigenbaum was inspired in his search, in part, by a very successful theory of universal numbers for second-order phase transitions in physics. A phase transition takes place in a system when a change of state occurs. During the 1970s it was discovered that there were quantitative measurements characterizing phase transitions that did not depend on the details of the substance used. Moreover, these universal numbers in the theory of phase transitions were successfully measured in countless experiments throughout the world. Feigenbaum wondered if there might be some similar universality theory for dissipative nonlinear systems .
By definition, such universal numbers are dimensionless; the specific mechanical details of the system must be scaled out of the problem. Feigenbaum began his search for universal numbers by examining the period-doubling cascade in the quadratic map. He recorded, with the help of his calculator, the values of at which the first few period-doubling bifurcations occur. We have listed the first eight values (orbits up to period ) in Table 2.1.
Table 2.1: Period doubling bifurcation values for the quadratic map.
While staring at this sequence of bifurcation points, Feigenbaum was immediately struck by the rapid convergence of this series. Indeed, he recognized that the convergence appears to follow that of a geometric series, similar to the one we saw in equation (1.35) when we studied the sticking solutions of the bouncing ball.
Let be the value of the nth period doubling bifurcation, and define as . Based on his inspiration, Feigenbaum guessed that this sequence obeys a geometric convergence,; that is,
where c is a constant, and is a constant greater than one. Using equation (2.26) and a little algebra it follows that if we define by
then is a dimensionless number characterizing the rate of convergence of the period doubling cascade.
The three constants in this discussion have been calculated as
The constant is now called “Feigenbaum’s delta,” because Feigenbaum went on to show that this number is universal in that it arises in a wide class of dissipative nonlinear systems that are close to the single-humped map. This number has been measured in experiments with chicken hearts, electronic circuits, lasers, chemical reactions, and liquids in their approach to a turbulent state, as well as the bouncing ball system .
To experimentally estimate Feigenbaum’s delta all one needs to do is measure the parameter values of the first few period doublings, and then substitute these numbers into equation (2.27). The geometric convergence of is a mixed blessing for the experimentalist. In practice it means that converges very rapidly to , so that only the first few ‘s are needed to get a good estimate of Feigenbaum’s delta. It also means that only the first few ‘s can be experimentally measured with any accuracy, since the higher ‘s bunch up too quickly to . To continue with more technical details of this story, see Rasband’s account of renormalization theory for the quadratic map .
Feigenbaum’s result is remarkable in two respects. Mathematically, he discovered a simple universal property occurring in a wide class of dynamical systems. Feigenbaum’s discovery is so simple and fundamental that it could have been made in 1930, or in 1830 for that matter. Still, he had some help from his calculator. It took a lot of numerical work to develop the intuition that led Feigenbaum to his discovery, and it seems unlikely that the computational work needed would have occurred without help from some sort of computational device such as a calculator or computer. Physically, Feigenbaum’s result is remarkable because it points the way toward a theory of nonlinear systems in which complicated differential equations, which even the fastest computers cannot solve, are replaced by simple models–such as the quadratic map–which capture the essence of a nonlinear problem, including its solution. The latter part of this story is still ongoing, and there are surely other gems to be discovered with some inspiration, perspiration, and maybe even a workstation.