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Established in the 1960s, chaos theory (more properly called nonlinear dynamics) deals with dynamical systems that, while in principle deterministic, have a high sensitivity to initial conditions, because their governing equations are nonlinear. Examples for such systems are the atmosphere, plate tectonics, economies, and population growth.

## Description of the theory

A non-linear dynamical system can in general exhibit one or more of the following types of behaviour:

• forever at rest
• forever expanding (only for unbounded systems)
• in periodic motion
• in quasi-periodic motion
• in chaotic motion

The type of behaviour may depend on the initial state of the system and the values of its parameters, if any.

### Chaotic motion

The most famous type of behaviour is chaotic motion, a non-periodic complex motion which has given name to the theory. In order to classify the behaviour of a system as chaotic, the system must be:

• bounded
• sensitive on the initial conditions
• transitive
• the periodic orbits must be dense

Sensitivity on the initial conditions means that two such systems with however small a difference in their initial state eventually will end up with a finite difference between their states (however, two deterministic systems with identical initial conditions will remain identical).

An example of such sensitivity is the well-known butterfly effect, whereby the flapping of a butterfly's wings produces tiny changes in the atmosphere which over the course of time cause it to diverge from what it would have been and potentially cause something as dramatic as a tornado to occur. Other commonly-known examples of chaotic motion are the mixing of colored dyes and airflow turbulence.

Transitivity means that application of the transformation on any given Interval I1 stretches it until it overlaps with any other given Interval I2.

The fourth condition means that for any point in the system and any real number ε > 0 there is another point with distance d ≤ ε which is located on an periodic orbit.

### Strange attractors

One way of visualizing chaotic motion, or indeed any type of motion, is to make a phase diagram of the motion. In such a diagram time is implicit and each axis represents one dimension of the state. For instance, a system at rest will be plotted as a point and a system in periodic motion will be plotted as a simple closed curve.

A phase diagram for a given system may depend on the initial state of the system (as well as on a set of parameters), but often phase diagrams reveal that the system ends up doing the same motion for all initial states in a region around the motion, almost as though the system is attracted to that motion. Such attractive motion is fittingly called an attractor for the system and is very common for forced dissipative systems.

While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and complexity. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the eyes of an owl.

Strange attractors have fractal structure.

## History

The theory has roots back to around 1900, but progressed more rapidly after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the logistic map. The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Moore's law and the availability of cheaper computers has greatly increased the extent of chaos theory. Currently, chaos theory continues to be a very active area of research.

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a basic computer to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.

To his surprise the weather that the machine began to predict was completely different to the weather calculated before. Lorenz tracked this down to only bothering to enter 3-digit numbers in to the simulation, whereas the computer had last time worked with 5-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.

The importance of chaos theory can be illustrated by the following observations:

• In popular terms, a linear system is exactly equal to the sum of its parts, whereas a non-linear system can be more than the sum of its parts. This means that in order to study and understand the behaviour of a non-linear system one needs in principle to study the system as a whole and not just its parts in isolation.

• It has been said that if the universe is an elephant, then linear theory can only be used to describe the last molecule in the tail of the elephant and chaos theory must be used to understand the rest. Or, in other words, almost all interesting real-world systems are described by non-linear systems.

## Mathematical theory

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include:

• fractal dimension of the attractor
• Lyapunov exponents
• recurrence plots
• Poincaré maps
• bifurcation diagrams

### Minimum complexity of a chaotic system

Many simple systems can also produce chaos without relying on partial differential equations, such as the
logistic equation, which describes population growth over time.

Even discrete systems, such as cellular automata, can heavily depend on initial conditions. Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30.

## References

### Textbooks and technical works

• Chaotic and Fractal Dynamics, by Francis C. Moon, ISBN 0471545716
• The Fractal Geometry of Nature, by Benoit Mandelbrot