A non-linear dynamical system can in general exhibit one or more of the following types of behaviour:
The type of behaviour may depend on the initial state of the system and the value 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 and have what is called sensitivity on the initial conditions. This means that two such systems with however small a difference in their initial state eventually will end up with a finite difference between their state.
An example of such sensitivity is the well-known butterfly effect, whereby the flapping of a butterflys wings produces tiny changes in the atmosphere which over the course of time 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 includes mixing of colored die and airflow turbulence.
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 attrated 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 is known as strange attractors which is attractors that can have very much detail and complexity. For instance, a simple three-dimensional model of the Lorenz? weather system gives rise to the famous Lorenz attractor.
Strange attractors have a fractal-like structure.
The theory has roots back to around 1950 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. However, major parts of the theory has only been developed since around 1980 and only recently has the theory been accepted by the scientific community as a whole.
An early pioneer of the theory was [Edward Lorenz]? whos interest in chaos came about accidently through his work on weather prediction in the 1960s. Lorenz was using a basic computer to run his simulation of the weather. 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 print-out 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 a previous time. 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 concensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changed in initial conditions produced large changes in the long-term outcome.
The importance of chaos theory can be illustrated by the following observations:
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