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Chaotic Systems William Ditto and Toshinori Munakata Principles and Applications of Chaotic Systems William Ditto and Toshinori Munakata This overview of There lies a behavior between rigid regularity and randomness based on pure the behavioral ele- chance. It’s called a chaotic system, or chaos for short [5]. Chaos is all around us. ments of chaos Our notions of physical motion or dynamic systems have encompassed the pre- cise clock-like ticking of periodic systems and the vagaries of dice-throwing introduces a wide chance, but have often been overlooked as a way to account for the more com- array of commer- monly observed chaotic behavior between these two cial applications to extremes. When we see irregularity we cling to random- ness and disorder for explanations. Why should this be enhance, manipu- so? Why is it that when the ubiquitous irregularity of late, or better con- engineering, physical, biological, and other systems are trol many current studied, it is assumed to be random and the whole vast technical functions. machinery of probability and statistics is applied? Rather recently, however, we have begun to realize that the tools of chaos theory can be applied toward the understand- ing, manipulation, and control of a variety of systems, with many of the practi- cal applications coming after 1990. To understand why this is true, one must start with a working knowledge of how chaotic systems behave---profoundly, but sometimes subtly different, from the behavior of random systems. As with many terms in science, there is no standard definition of chaos. The typical features of chaos include: • Nonlinearity. If it is linear, it cannot be chaotic. • Determinism. It has deterministic (rather than probabilistic) underlying rules every future state of the system must follow. • Sensitivity to initial conditions. Small changes in its initial state can lead to radi- cally different behavior in its final state. This “butterfly effect” allows the pos- sibility that even the slight perturbation of a butterfly flapping its wings can 96 November 1995/Vol. 38, No. 11 COMMUNICATIONS OF THE ACM emerging technologiesAI dramatically affect whether sunny or Table 1. Major Historical Developments in the Study of Chaos cloudy skies will predominate days later. • Sustained irregularity in the behavior of 1890 King Oscar II of Sweden. Announced a the system. Hidden order including a prize for the first person who could solve large or infinite number of unstable the n-body problem to determine the orbits periodic patterns (or motions). This of n-celestial bodies and thus prove the hidden order forms the infrastruture stability of the solar system. As of 1995, no one has solved the problem. of irregular chaotic systems---order in disorder for short. 1890 Henri Poincare´. Won first prize in King • Long-term prediction (but not control!) Oscar's contest by being closest to solve the is mostly impossible due to sensitivity to n-body problem. Discovered that the orbit initial conditions, which can be known of three or more interacting celestial bodies can exhibit unstable and unpredictable only to a finite degree of precision. behavior. Thus is chaos born (but not yet named!). A simple example of a chaotic system in computer science is a pseudo-random 1963 Edward Lorenz. Irregularity in a toy model number generator. The underlying rule in of the weather displays first chaotic or strange attractor [9]. this case is a simple deterministic formula (e.g., xn+1= cxn modm). However, the 1975 Tien-Yien Li and James A. Yorke. In paper resulting solutions, such as the pseudo-ran- “Period Three Implies Chaos,” introduced dom numbers are very irregular and the term “Chaos Theory.” unpredictable (the more unpredictable, 1976 Robert M. May. Application of the logistic the closer to random). We also note that a equation to ecology, showing chaotic small change in the initial condition (seed) population behavior. can yield a significantly different sequence of random numbers. These pseudo-ran- 1978 Mitchell Feigenbaum. Universal number dom number generators are chaotic but associated with the way systems approach chaos. also periodic with certain periods. Such generators viewed carefully yield the hid- 1980 Benoit Mandelbrot. Fractal geometry with den order characteristic of chaos. (For application to computer graphics and more on fundamentals, see [1, 2, 14]). image compression. While this random number generator 1990 Ed Ott, Celso Grebogi and James Yorke. is an artificial chaotic system, there are Beginning of chaos control theory. numerous chaotic systems in nature. While engineers typically shun chaos and 1990 Lou Pecora. Synchronization of chaotic irregularity, nature may indeed treasure systems. and exploit it. For example, normal brain activity may be chaotic, and pathological order may indeed be the cause of such diseases as tems. One key element of this recent interest is the epilepsy. It has been speculated that too much peri- power of easily accessible computer hardware for its odicity in heart rates might indicate disease. Perhaps speed and memory capacity. Although the history of the chaotic characteristics of the human body are bet- chaotic systems research is not new, the computer ter adapted to its chaotic environment. It is even sus- revolution gave life to their practical applications. pected that biological systems exploit chaos to store, There are various types of potential commercial and encode, and decode information. industrial applications, based on different aspects of Generally, the boundary between deterministic chaotic systems (see Tables 2 and 3). For simplicity, chaos and probabilistic random systems may not we classify the application types into the categories always be clear since seemingly random systems could stabilization, synthesis, and analysis [11]. involve deterministic underlying rules yet to be found. Stabilization and control. The extreme sensitivity Nonlinear problems are very difficult to solve; various of chaotic systems to tiny perturbations can be manip- approximation techniques have been employed. ulated to stabilize or control the systems. The funda- These techniques include linearization, which is a sim- mental idea is that tiny perturbations can be plification of the problem, and fuzzy theory, which artificially incorporated either to keep a large system can be used as an approximation method. stable (stabilization) or to direct a large chaotic sys- Historically, the study of chaos started in mathe- tem into a desired state (control). For example, matics and physics. It then expanded into engineer- although it is still problematic, small, carefully chosen ing, and more recently into information and social chaos control interventions might lead to more effi- sciences (see Table 1). For the past five years or so in cient airplane wings, power delivery systems, turbines, particular, there has been growing interest in com- chemical reactions in industrial plants, implantable mercial and industrial applications of chaotic sys- defibrillators, brain pacemakers, conveyer belts, eco- COMMUNICATIONS OF THE ACM November 1995/Vol. 38, No. 11 97 Table 2. Potential Application Types of Chaos ural for human comfort. (See Aihara and Katayama in this issue.) Whether the chaotic Category Applications output outperforms random output in these consumer products has yet to be studied, but Control First application of chaos is control of it appears to be better than homeostasis in irregular behavior in devices and systems. certain applications. List of applications is included in Table 3. Two identical sequences of chaotic sig- nals (called “synchronized”) can be used for Synthesis Potential control of epilepsy, improved encryption by superposing a message on dithering of systems, such as ring laser gyroscopes. Switching of packets in one sequence. Only a person with the other computer networks. sequence can decode the message by sub- tracting the chaotic masking component. In Synchronization Secure communications, chaotic broad communications, artificially generated band radio, encryption. chaotic signals can follow a prescribed Information Processing Encoding, decoding, and storage of sequence, thus enabling them to transmit information in chaotic systems, such as information. Artificially generated chaotic memory elements and circuits. Better fluctuations can be used to stimulate performance of neural networks. Pattern trapped solutions so they escape from local recognition. minima for optimization problems or for Short Term Prediction Contagious diseases, weather, economy. learning, as in neural networks. Analysis and prediction of chaotic systems. How do you tell if a system is random or chaotic? The study of chaos may lead to bet- Table 3. Sample Potential Application Areas of Chaos ter detection and prediction algorithms for chaotic systems. Amazingly, the lack of long- Category Applications term predictability in chaotic systems does not imply that short-term prediction is impos- Engineering Vibration control, stabilization of circuits, sible. In counterpoint to purely random sys- chemical reactions, turbines, power grids, tems, chaotic systems can be predicted for a lasers, fluidized beds, combustion, and many more. short interval into the future. For example, many people believe that financial markets Computers Switching of packets in computer networks. may exhibit chaotic behavior. Obviously, if Encryption. Control of chaos in robotic the market were chaotic rather than random, systems. there would exist the possibility of reliably predicting market behavior in the
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