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Chaos theory is the popular term for a branch of mathematics and physics dealing with the study of certain types of behaviours in [dynamical system]?s. |
Chaos theory is the popular term for a branch of mathematics and physics dealing with the study of certain types of behaviours in [dynamical systems]?. Description of the theory |
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The most famous type of behaviour is of course chaotic motion, which has given name to the theory. In order to classify the behaviour of a system as chaotic, the system must be what is called sensitive on 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 this 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. |
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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 |
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In addition to be sensitive on the initial condition, a system must also have a bounded state to be chaotic. |
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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. |
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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. |
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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. |
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Strange attractors have a fractal-like structure.History |
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The importance of chaos theory can be illustrated by the following points: |
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The importance of chaos theory can be illustrated by the following observations: |
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* Is 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 is described by non-linear systems. |
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* Is 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 is described by non-linear systems. |
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