Chaos and Complexity

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SHOULD WE BE SKEPTICAL?

THE BIRTH OF MODERN SCIENCE has been attributed to a variety of circumstances, events, and people, [1] but unquestionably one of the key figures in its development was Rene Descartes, the French philosopher who first articulated the fundamentals of the modem scientific method of inquiry. [2] A major tenet of Descartes' approach was the idea that complex systems can be understood by analyzing one part at a time, and then putting things back together to yield a comprehensive picture. This reductionism has been at the core of some of the most spectacular successes of the scientific endeavor, from particle physics to molecular biology. But what if some natural phenomena simply cannot be so conveniently partitioned to facilitate our comprehension? What if breaking the components apart alters their properties so much that what we learn from the separate pieces of the puzzle gives us a different and misleading idea of the system as a whole? In other words, can reductionist science study emergent properties which, by definition, are the result of complex interactions?

There has been much talk of emergent properties, especially in describing the complexity of biological development and evolution. Yet, it is hard to put a finger on what even sophisticated researchers mean when they say, for example, that human consciousness is an emergent property of the physical structure of the brain and of its interactions with the environment Perhaps the simplest way to understand emergent properties is to consider the relation between hydrogen, oxygen, and water. Although the combination of two atoms of hydrogen and one of oxygen yields water, the complex properties of water (e.g., the temperatures at which it undergoes state transitions to steam or ice) are not derivable from the individual properties of hydrogen and oxygen. In other words, knowing all we know about the structure and behavior of the atoms composing water, allows us to predict the structure but not the behavior of water. This means that complexity produces new properties specific to the new level of organization (in thi s case, molecular vs. atomic) that are due not to the sum of the parts, but to their interaction. This, it would seem, is enough to stop the Cartesian research program dead in its tracks.

Scientists from several disciplines, from astronomy to meteorology, from evolutionary biology to the social sciences, have been struggling with interactions and emergent properties without a good paradigm to guide them. That is, until Chaos Theory and its more recent derivative, Complexity Theory appeared on the scene. These novel conceptual and mathematical approaches to the study of complex systems promised to provide a way out of the thicket of emergent properties. Science, it seems, had finally cracked the next level of analysis, one that will replace the Cartesian approach and substitute a new, scientific holism for the old reductionism. More than 35 years after the publication of the first study on chaos, with an entire institute devoted to the study of complexity (http:www.santafe.edu/) headed by Nobel laureate Murray Gell-Mann and with technical journals and thousands of published papers in the offing, it is time for a skeptical evaluation of the new holism. Has chaos/complexity ("chaoplexity" as it i s sometimes called) delivered on its initial promise? Or has it fallen much short of its original goal? Does it provide a truly new set of tools and answers, or is it just a passing fad in the academic world?

WHAT Do You MEAN, CHAOS?

What is chaos? In the vernacular, the word is a synonym for randomness, completely non-deterministic and irregular phenomena. Typically it carries a negative connotation--a chaotic situation is one that we would like to avoid. In mathematical theory, however, chaos refers to a deterministic (i.e., non-random) phenomenon characterized by special properties that make the predictability of outcomes very difficult. In fact, a chaotic behavior is such that even though it does not happen randomly, it looks like a series of random occurrences.

As an example, compare the two upper graphs in FIGURE 1. While they both look random at first sight, only the one on the left is really non-deterministic. The other one is a time series generated by a set of equations describing a mathematical object called the Henon attractor; this series is chaotic, not random. The lower graphs in the figure help us discriminate between the two kinds of behavior. These representations are obtained by plotting the status of the system at time against the status of the same system at time t+1. Also known as phase plots, they reveal an orderly structure in the case of the Henon attractor (FIGURE 1d) but not in the case of the random sequence (FIGURE 1c).

Chaotic dynamics are usually, but not always the property of non-linear systems, [4] that is of systems whose behavior can be described by sets of non-linear equations. However, the converse is not true: not all non-linear dynamics generate chaotic behaviors. [5] Typically, a given system of equations can produce both non-chaotic and chaotic outcomes, depending on the range of values assumed by the parameters entered into the equations. In fact, in many systems one can increase the value of a key parameter and obtain a progression of outcomes from a steady equilibrium state to regular oscillations with two equilibria, to more complex cycles with multiple equilibria, to finally bringing about the chaotic condition. Since the latter can be thought of as an ensemble of an infinite number of equilibrium points (the so-called "strange attractor"), this process is sometimes termed the "doubling route" to chaos. [5]

Another phenomenon typically associated with chaos is the so-called "butterfly effect." The term came from a famous analogy proposed by Edward Lorenz the discoverer of the first formal system of equations that yields chaotic behavior. [6] Lorenz said that chaos is analogous to a situation in which the flapping of a butterfly's wings in Brazil ends up starting a cascade of events that results in a tornado in Texas. The technical term for this phenomenon is "sensitivity to initial conditions," and it means that a small perturbation of a system can cause a series of effects that eventually lead to macroscopic consequences later in the time sequence. Had that perturbation been of a different nature, an entirely different series of events would have unfolded.

There is a very important qualification to be made about sensitivity on initial conditions: it may happen only in some circumstances. For example, while it is conceivable that the tiny movement of air caused by flapping a butterfly's wings may indeed initiate a causal chain that affects macroscopic weather patterns, most of the time millions of butterfly wings flapping around will not alter weather forecasts one bit! If that is the case, the system (in this case the weather) will actually behave as chaotic at some moments in time, but as predictable or random at others. A more formal way to describe the butterfly effect is to state that the predictability of the system decreases exponentially with time. That is, our predictions of where the system will be are relatively good for the immediate future, but lose accuracy for slightly longer intervals of time, and pretty soon they are completely useless. (One exception is when the system is within a strange attractor, if viewed in phase space.) We are now ready to consider a formal definition of …