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.
Table of contents |
2 History 3 Mathematical theory 4 Other examples of chaotic systems 5 See also 6 References 7 External links |
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 I_{1} stretches it until it overlaps with any other given Interval I_{2}.
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.
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.
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:
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.