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Figure 1: A single trajectory of the Lorenz attractor.

Chaos describes a system that is predictable in principle but unpredictable in practice. In other words, although the system follows deterministic rules, its time evolution appears random.

In dynamical systems theory, the term chaos is applied to deterministic systems that are aperiodic and that exhibit sensitive dependence on initial conditions. Sensitivity means that a small change in the initial state will lead to progressively larger changes in later system states. Because initial states are seldom known exactly in real-world systems, predictability is severely limited.

The concept of chaos has been used to explain how systems that should be subject to known laws of physics, such as weather, may be predictable in the short term but are apparently random on a longer time scale.

## Examples

Chaos has been routinely observed in controlled laboratory experiments. Electric circuits and analog computers were some of the first examples, perhaps due to the fidelity of deterministic models to these systems. Chemical reactions, optical systems such as lasers, and fluid dynamics experiments have also been designed to exhibit chaotic dynamics. The frictionless double pendulum has chaotic trajectories, and an approximate version can be built with low enough friction to show long chaotic transients in a tabletop demonstration.

The existence of chaos in natural phenomena is more controversial. Celestial mechanics can be approximated using Newton's laws to high precision. Since n-body gravitational systems contain chaotic trajectories for $$n\ge 3\ ,$$ chaotic orbits of celestial bodies are probably pervasive in the solar system. In other areas such as population dynamics, firing of neural ensembles, and weather and climate, the issue is clouded due to the question whether the process can be convincingly modeled as a deterministic system.

## History

The conventional wisdom of the early 19th century was articulated by Laplace, who postulated a great intellect that "at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed" and concluded that "nothing would be uncertain and the future just like the past would be present before its eyes." However, Laplace's enthusiasm for the predictability of deterministic systems was not shared by Maxwell, who a few decades later made the uncertainty of particle velocities after collisions a fundamental part of his kinetic theory of gases. By the end of the 19th century, Poincare had discovered homoclinic orbits, which would later be recognized as the telltale signs of chaos, in the planar three body problem.

In 1927, Van der Pol (Van der Pol, 1927) reported "irregular noise" in a radio circuit driven at certain frequencies, but considered it a subsidiary phenomenon. Cartwright and Littlewood, alerted to a model of Van der Pol's circuit by the British Radio Research Board, identified "random-like" dynamics in the equations (Cartwright and Littlewood, 1945), clarified shortly after in (Levinson, 1949). These papers came to the attention of Smale, who developed the Smale horseshoe as a simplest reduction of Levinson's observations. Smale's important insight unified the homoclinic behavior found by Poincare in celestial mechanics with Van der Pol's noisy oscillator.

At about the same time, in the early 1960s, the meteorologist Lorenz was trying to understand the failures of linear prediction techniques for weather forecasts. Using one of the world's first mass-produced computers to simulate atmospheric dynamics, he found that long aperiodic trajectories could be produced quite robustly. Then, in one of history's most serendipitous episodes of computer rounding error, he found that the aperiodicity was paired with sensitive dependence on initial conditions.

With the help of Saltzman, he reduced the atmosphere simulation to a differential equation in three variables that produced the Lorenz attractor (Lorenz, 1963). Lorenz later gave a lecture entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?, which caused the concept of sensitive dependence on initial conditions to become popularly known as the "butterfly effect".

Also in the 1960s, Ueda posed a mathematical model on an analog computer that displayed chaotic dynamics. However, this work was not published until years later (Ueda, 1970).

In 1975, Yorke and his Ph.D. student Li showed that sustained aperiodic behavior could be found in one-dimensional maps. Their article (Li and Yorke, 1975) coined the term chaos for the various phenomena that showed aperiodicity along with sensitive dependence on initial conditions. In addition to showing that the existence of a period-three orbit in a one-dimensional continuous map implies sensitive dependence, they showed another remarkable consequence: the existence of infinitely many other periodic orbits. The latter part of the result had been preceded by several years in (Sharkovsky, 1964), implied by what is now known as the Sharkovsky ordering. (May, 1976) presented the logistic map as a plausible population model with a period-doubling cascade of bifurcations and chaotic trajectories.

Around the same time, (Gollub and Swinney, 1975) reported aperiodic dynamics in a flow between rotating cylinders, in an attempt to explain the transition from laminar flow to turbulence. Toward the same goal, Libchaber demonstrated a period-doubling cascade in a convective Rayleigh-Benard experiment a few years later. Since then, experiments in a wide array of scientific and engineering disciplines have been designed that clearly display the effects of deterministic chaos.

## Definition

A bounded trajectory of a dynamical system is said to be chaotic if it has sensitive dependence on initial conditions and is not quasiperiodic.

## Chaotic attractors

Lorenz, Rossler, Double pendulum, Chua-Matsumoto circuit, Logistic map, Henon, Standard map, billiards

## Implications for modeling

The existence of chaos has important consequences for modeling real-world systems. Even though a system exhibits random behavior, modeling strategies need not be restricted to stochastic models.

Most properly, chaos is a concept that is applied to models, not to natural or experimental systems. The question is often asked of a system exhibiting complicated motion, whether the system is random or chaotic. A more useful question is whether the system is better approximated using a deterministic model that allows chaotic dynamics, or alternatively by a stochastic, or mixed deterministic/stochastic model.

## References

• M. L. Cartwright and J. E. Littlewood (1945) On non-linear differential equations of the second order: I. The equation $$\ddot y - k(1 - y^2 )\dot y + y = b\lambda k cos (\lambda t + a); k$$ large, Journal of the London Mathematical Society, 20:180-189
• J. P. Gollub and H. L. Swinney (1975). Onset of turbulence in a rotating fluid. Physical Review Letters 35: 927–930.
• N. Levinson (1949) A second order differential equation with singular solutions. Ann. Math. 50:127-153
• T.Y. Li, J.A. Yorke (1975) Period three implies chaos. Amer. Math. Monthly 82:985-992
• E. Lorenz (1963) Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20:130–141
• R.M. May (1976) Simple mathematical models with very complicated dynamics. Nature 261:459.
• A.N. Sharkovsky (1964) Co-existence of cycles of a continuous mapping of a line onto itself. Ukrainian Math. Z. 16:61-71
• S. Smale (1967) Differentiable dynamical systems. Bull. Amer. Math. Soc. 73:747-817.
• Y. Ueda, C. Hayashi, N. Akamatsu, and H. Itakura (1970) On the Behavior of Self-Oscillatory Systems with External Force. Electronics & Communication in Japan, 53:31-39
• Van der Pol, B. and Van der Mark, J. (1927) Frequency demultiplication. Nature, 120:363-364

## Recommended reading

• Alligood, K. T., Sauer, T.D., Yorke, J.A. (1996) Chaos: an introduction to dynamical systems.Springer-Verlag New York. ISBN 0-387-94677-2
• Devaney, R. L. (2003) An Introduction to Chaotic Dynamical Systems, 2nd ed, Westview Press. ISBN 0-8133-4085-3
• Gollub, J. P.; Baker, G. L. (1996) Chaotic dynamics. Cambridge University Press ISBN 0-521-47685-2
• Gutzwiller, M. (1990) Chaos in Classical and Quantum Mechanics. Springer-Verlag New York ISBN 0-387-97173-4
• Moon, F. (1990) Chaotic and Fractal Dynamics. Springer-Verlag New York ISBN 0-471-54571-6
• Ott, E. (2002) Chaos in Dynamical Systems. Cambridge University Press New York ISBN 0-521-01084-5
• Strogatz, S. (2000) Nonlinear Dynamics and Chaos. Perseus Publishing ISBN 0-7382-0453-6
• Tufillaro, N., Abbott, T., Reilly, J. (1992) An experimental approach to nonlinear dynamics and chaos. Addison-Wesley New York ISBN 0-201-55441-0
• Zaslavsky, G. M. (2005) Hamiltonian Chaos and Fractional Dynamics. Oxford University Press ISBN 0-198-52604-0