The 2,040th Meeting of the Society

February 17, 1995

Making Use of Chaos

Louis M. Pecora

Naval Research Laboratory


The President, Mr. Ohlmacher, called the 2040th meeting to order at 8:22 p.m. on February 17, 1995. The Recording Secretary read the minutes of the 2039th meeting and they were approved. The President then read a portion of the minutes of the 432nd meeting February 16, 1895. The President introduced Mr. Louis M. Pecora of the Naval Research Laboratory to discuss “Making Use of Chaos”. At first, some of the scientists studying chaos were undoubtedly attracted because of the pleasures that can be derived from exploring a new theory, from detecting patterns in seemingly random phenomena. After the development of mathematical theories of mechanics and dynamics by Newton and Laplace, it eventually became evident that there were some simple physical systems whose motions were neither simple nor easily predictable. Poincare was able to demonstrate that there are systems of motion in which errors are continually amplified until prediction becomes impossible. These are the systems that are now called chaotic. Mechanical systems may be analyzed either in what is referred to as state space or phase space. A state space description is developed by following the system in its changes and finding those state variables that change independently of the path the system follows. For example, the total energy of a closed system, even a chaotic one, is constant. A phase space description is developed by following the frequency of changes in system and the correlations in those frequencies. Using a video produced by Roland Tang at the University of Colorado, the different descriptions of state space and phase space were illustrated by plotting the motion of pendulums, both normal and chaotic, as a time series in state space and in phase space. Regular pendulum motion was graphed as an oscillating curve with regular period and diminishing magnitude in state space, and as an inwardly spiraling curve in phase space; chaotic pendulum motion is graphed as an oscillating curve with irregular period and magnitude in state space and as a limiting-case closed loop in phase space. Tom Caroll and Jim Heagy have shown that the phase space descriptions of chaotic systems have continuous instabilities and are continually difficult to predict. Chaotic systems are now characterized by being (1) extremely sensitive to initial conditions, (2) deterministically random and hence ultimately unpredictable, and (3) continually unstable, displaying no true cyclic behavior. When disturbed nonchaotic systems eventually return to the same fixed point in state space, or in phase space. This is referred to as a fixed point attractor. A chaotic system exhibiting a closed loop in phase space has no such fixed point attractors. Chaotic systems do have what are called “strange” or chaotic attractors. Chaotic attractors were found by Lorentz in 1963 in the unstable periodic orbits of weather models. Because of this chaotic behavior in model systems for weather, Lorentz thought that the best weather predictions would be possible for periods of about 2 weeks. In contrast the solar system is only mildly nonlinear; it is thought to have only about a 1 million year limit for predictability. Within the last few years researchers have been able to shift their attention toward devising ways of controlling chaotic behavior and putting it to work. The most productive approaches have been not to avoid chaos, but to synchronize the behavior of chaotic systems. Tom Carroll at the Naval Research Laboratory, “a very stable guy working with unstable systems”, has developed applications for synchronizing chaotic circuits. Yorke and his co-workers at the University of Maryland pioneered methods for balancing chaotic systems by controlling their motion around saddle points. Spano at the White Oak Laboratory, Naval Surface Warfare Center (who addressed the Society on October 14, 1994) has been using similar approaches to control biological chaotic systems. Yet another approach has been used by Scott Hayes in an attempt to build secure communications systems with synchronized chaos. [1] W. Ditto and L. M. Pecora, “Mastering Chaos”, Scientific American, 269(2), August 1993, pp. 78-84. The President thanked the speaker on behalf of the Society. The President presented the name of two new members and two new life-members. The President then announced the speaker for the next meeting, reiterated the statement about parking, and adjourned the 2040th meeting at 9:43 p.m. Attendance: 99 Temperature: +5.6°C Weather: partly cloudy Respectfully submitted, John S. Garavelli Recording Secretary