Complexity_ A Guided Tour - Melanie Mitchell [19]
There is one more remarkable fact to mention about this story. Similar to many important scientific discoveries, Feigenbaum’s discoveries were also made, independently and at almost the same time, by another research team. This team consisted of the French scientists Pierre Coullet and Charles Tresser, who also used the technique of renormalization to study the period-doubling cascade and discovered the universality of 4.6692016 for unimodal maps. Feigenbaum may actually have been the first to make the discovery and was also able to more widely and clearly disseminate the result among the international scientific community, which is why he has received most of the credit for this work. However, in many technical papers, the theory is referred to as the “Feigenbaum-Coullet-Tresser theory” and Feigenbaum’s constant as the “Feigenbaum-Coullet-Tresser constant.” In the course of this book I point out several other examples of independent, simultaneous discoveries using ideas that are “in the air” at a given time.
Revolutionary Ideas from Chaos
The discovery and understanding of chaos, as illustrated in this chapter, has produced a rethinking of many core tenets of science. Here I summarize some of these new ideas, which few nineteenth-century scientists would have believed.
Seemingly random behavior can emerge from deterministic systems, with no external source of randomness.
The behavior of some simple, deterministic systems can be impossible, even in principle, to predict in the long term, due to sensitive dependence on initial conditions.
Although the detailed behavior of a chaotic system cannot be predicted, there is some “order in chaos” seen in universal properties common to large sets of chaotic systems, such as the period-doubling route to chaos and Feigenbaum’s constant. Thus even though “prediction becomes impossible” at the detailed level, there are some higher-level aspects of chaotic systems that are indeed predictable.
In summary, changing, hard-to-predict macroscopic behavior is a hallmark of complex systems. Dynamical systems theory provides a mathematical vocabulary for characterizing such behavior in terms of bifurcations, attractors, and universal properties of the ways systems can change. This vocabulary is used extensively by complex systems researchers.
The logistic map is a simplified model of population growth, but the detailed study of it and similar model systems resulted in a major revamping of the scientific understanding of order, randomness, and predictability. This illustrates the power of idea models—models that are simple enough to study via mathematics or computers but that nonetheless capture fundamental properties of natural complex systems. Idea models play a central role in this book, as they do in the sciences of complex systems.
Characterizing the dynamics of a complex system is only one step in understanding it. We also need to understand how these dynamics are used in living systems to process information and adapt to changing environments. The next three chapters give some background on these subjects, and later in the book we see how ideas from dynamics are being combined with ideas from information theory, computation, and evolution.
CHAPTER 3
Information
The law that entropy increases—the Second Law of Thermodynamics—holds, I think, the supreme position among the laws of Nature… [I] f your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.
—Sir Arthur Eddington, The Nature of the Physical World
COMPLEX SYSTEMS ARE OFTEN said to be “self-organizing”: consider, for example, the strong, structured bridges made by army ants; the synchronous flashing of fireflies; the mutually sustaining markets of an economy; and the development of specialized organs by stem cells—all are examples of self-organization. Order is created out of disorder, upending the usual turn of events in which order decays and disorder (or entropy)