Edward Lorenz's Groundbreaking Study of Weather Patterns Leads in Part to the Development of Chaotic Dynamics

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Edward Lorenz's Groundbreaking Study of Weather Patterns Leads in Part to the Development of Chaotic Dynamics

Overview

In 1961 Edward Norton Lorenz (1917- ) demonstrated that as nonlinear deterministic systems evolve, they exhibit sensitive dependence on their initial conditions. This means that small changes in those conditions have large, unpredictable consequences. Lorenz was the first to emphasize the importance of identifying and studying such systems. His original research initiated the development of chaos theory, which has applications in fields ranging from astronomy and engineering to economics and medicine.

Background

Physical systems whose later states evolve from earlier ones according to fixed laws are deterministic systems. Isaac Newton's (1642-1727) Principia Mathematica embodied the belief that the natural world was a deterministic system—one whose behavior was governed by his equations of motion and law of gravitation. When Newtonian mechanics were successfully applied to natural phenomena in the eighteenth and nineteenth centuries, their predictive accuracy was directly related to the accuracy of initial conditions. Minor inaccuracies in the initial states translated into small prediction errors. Consequently, extremely weak effects could be neglected without significantly affecting predictability.

Certain phenomena, though, proved particularly recalcitrant to Newtonian methods, the most notable being weather and turbulence, both of which are complex and apparently random processes. Meteorologists did develop a set of equations they believed governed changes in weather patterns, but employing them successfully required a vast number of computations. Only with the introduction of self-programming digital computers in the 1940s could the solutions to these governing equations be approximated.

Even with the power of computers, numerical weather forecasts were only accurate for a few days. These limitations were thought to be caused by insufficient knowledge of initial conditions and the physical approximations dictated by the computational constraints of available computers. But in 1961 Edward Lorenz demonstrated Newtonian predictability to be a fantasy.

Lorenz formulated a simple atmospheric model that allowed him to simulate recognizable weather patterns. To track variations in these patterns, he printed out the values of prognostic variables after each simulated day. During the winter of 1961 he decided to investigate a particular computer run in greater detail. He used a line of numbers from a previous printout as the initial conditions for the new run. The new results differed significantly from the original. Lorenz quickly traced the problem to his truncation of the initial condition values. The numbers stored in the computer were accurate to six decimal places, but Lorenz had them rounded off to three decimal places before printing. Assuming that such small differences could have no significant effect, Lorenz used the printout values. What he found was that extremely small differences can generate widely varying outcomes, making long-term prediction impossible. Lorenz referred to this as sensitive dependence on initial conditions—known today as chaos.

Sensitive dependence is a direct result of the nonlinearity of chaotic systems. In linear systems, changes to initial states result in proportional changes to later states. Nonlinear systems like the atmosphere are not so well behaved. Variables in these systems are coupled such that changes result in complex cumulative feedback effects. For example, increases in temperature raise humidity. This causes more solar radiation to be absorbed, which raises the temperature further, which then increases humidity and so on.

Lorenz studied the behavior of nonlinear equations in greater detail. In particular, he investigated the attractor for such a system. An attractor is a set of states a system can occupy once transient effects have dissipated. The simplest attractor is a fixed point that represents the stable states of a pendulum in an unwound clock. Regardless of how it is set in motion, the pendulum will always come to rest in the same position. The attractor for a pendulum in a continuously wound clock will be a closed elliptical curve. The points of the curve represent the states of the system. In both cases future states can be accurately predicted.

The attractor Lorenz discovered proved to be the first strange attractor. As with any attractor there is a set of states to which the nonlinear system is attracted. However, once the system is in the strange attractor, nearby states diverge from each other exponentially fast, making long-range prediction impossible. Strange attractors, however, are stable geometric objects with definite structure. This gives order to the apparent random behavior of nonlinear systems. For instance, the underlying order of the weather's strange attractor—the climate—explains why similar weather patterns occur over and over.

Impact

Many physical phenomena besides weather are nonlinear deterministic systems and exhibit chaotic behavior. While this is generally viewed as undesirable, chaotic properties can be exploited to analyze unresolved problems and generate practical applications.

As already mentioned, the most immediate consequence of sensitive dependence is the impossibility of ever making perfect or even approximate long-range predictions. However, by studying the structure of strange attractors it may be possible to determine the probability distribution of allowed events. This would then allow long-term probabilistic weather and other predictions to be made with mathematical precision.

Economics is another area where forecasting is of primary importance. Some economists now believe many economic data series—such as gross national product, unemployment, stock indexes, and industrial production—are chaotic. Chaos theory has given insight into the forces driving economic fluctuations, allowing better model specification and improved predictive capabilities.

Transition to chaos is another area of rapidly growing research. This is particularly so in fluid dynamics, where turbulence is of great concern, because smooth-flowing fluids often break-up into wild swirls and eddies. Fluid dynamicists try to minimize this turbulence, which controls drag around ships and planes and affects turbine engine and propeller efficiency. Chaotic attractors are now believed to underlie turbulence. Investigating these attractors has increased our understanding of how and when turbulence emerges. This in turn may lead to better designs.

Chaotic transition is also important in physiology and medicine. At present scientists are debating whether healthy biological systems are regular and predictable or chaotic. Interestingly, periodic and chaotic phenomena are present in both healthy and diseased conditions. Research is trying to detect transitions from one regime to the other in the hopes of developing predictive and diagnostic tools. Heart rhythms have been studied under normal and abnormal conditions in an attempt to identify impending arrhythmia and cardiac arrest. Brain wave activity has also been scrutinized in an attempt to understand epilepsy, manic depression, and other illnesses.

Chaotic dynamics might also be exploited in situations where great flexibility is beneficial. Small changes in control variables could be used to change system dynamics rapidly. This would allow quick responses to changing conditions. Practical applications are yet to be realized, but this idea has been used to explain the evolution of life on Earth.

Control through small changes also suggests connections between the theory of chaotic systems and information theory. For instance, by manipulating various control variables it might be possible to encode information into chaotic oscillator signals. Chaotic dynamics may also be exhibited by the brain's electrical activity. Tantalizing research in this area suggests that the cortex supports a global attractor that provides for rapid dissemination of information.

Chaos theory is also extensively used in astronomy, because the dynamical structures and evolution of the solar system exhibit chaos. The Kirkwood gaps in the asteroid belts are generally thought to be chaotic regions where asteroid orbital elements are altered drastically. The Cassini divide in Saturn's rings is similarly thought to be a chaotic region, and the spin-orbit of Saturn's satellite Hyperion displays characteristically chaotic behavior. The orbits of the planets also show some evidence of chaotic motion over extremely long periods of time. In addition, chaos theory has been fruitfully applied to the analysis of galactic dynamics, stellar oscillations, and sun spots.

Electrical engineers confront chaos regularly. Many electrical systems involved in the generation and transmission of power are susceptible to chaotic responses. Voltage collapse, the principal threat to the stability and reliability of power grid systems, is believed to be chaotically induced. Flow-induced vibrations in condenser tubes and transmission wires are also chaotic. Chaos is believed to occur at the physical data level in computer networks as well. Efforts remain focused on developing procedures for ensuring optimal operation of electrical systems and electronic devices fundamentally plagued by chaotic dynamics.

This review is only a glimpse of the wide range of fields in which chaos theory is used. The explosive growth this discipline experienced during the last two decades of the twentieth century shows no signs of abating, and many important applications are as yet expected.

STEPHEN D. NORTON