# Attraktor

a Attraktor is under the time development of a dynamic system invariant (i.e. itself temporally not changing) or itself this iterated approaching subset of a phase space, which will not any longer leave under the dynamics of this system. The quantity of all points of the phase space, which zustreben under the dynamics the same Attraktor, is called attraction area this Attraktors.

Dynamic systems are set up often as mathematical models physically or other procedures of the material world. Examples are the flow attitude of liquids and gases, movements of heavenly bodies under mutual influence by the gravitation, population sizes of organisms under consideration of the robber booty relationship or the development of economic characteristics under influence of the market laws. Dynamic systems defined by the description of the change in status as a function of the time t. For the mathematical definition the material system is often regarded in strongly simplified form.

One differentiates between discrete and continuous dynamic systems, according to whether the change in status is defined in firm temporal< steps> (math t \ in \ mathbb {<N>} /math) or as continuous< procedure> (math t \ in \ mathbb {<R>} /math). The condition is represented by as many as desired variables of state, these forms the dimensions of the phase space. Each condition is thereby one point in the phase space, discrete systems forms quantities of isolated points, continuous systems by lines (trajectories) is represented.

During the investigation of dynamic systems one is interested particularly in the behavior in [itex] t \ tons \ infty< /math> with a certain initial condition. The limit value in this case is called Attraktor. Typical and frequent examples from Attraktoren are:

• Fixed points: The system approaches ever more strongly, it to a certain final state, in which the dynamics come to succumbing develops on a static system. Typical example of such a system is an absorbed pendulum, which approximates the state of rest in the deepest point.
• Limit cycles: The final state is always the succession the same conditions, which will periodically go through. An example of it is the simulation of the robber booty relationship, which come down to periodic rising and sinking of the population sizes.

These examples are Attraktoren, which possess an integral dimension in the phase space. The existence structure more complicated by Attraktoren with was of course longer well-known, one regarded it however first than unstable special cases, whose occurrence is observed only with certain choice of the starting situation and the system parameters. This changed with the definition of a new, special type of Attraktor:

• Strange Attraktor: In its final state the system shows an aperiodic behavior. The Attraktor can be described not in a closed geometrical form and possesses no integral dimension, is thus a Fraktal. Important characteristic is the chaotic behavior, D. h. each still so small change of the initial condition leads in the further process to significant changes in status. Most prominent example is the Lorenz Attraktor, which was discovered with the modelling by air flows in the atmosphere. Strange Attraktoren supplies the mathematical basis for the description of chaotic procedures as for instance to turbulent currents with.

Each Attraktor has a certain catchment area in the phase space. Each course, whose initial condition lies in the catchment area, remains for increasing t in the catchment area and approaches arbitrarily strongly to the Attraktor on. The catchment area does not have to necessarily cover the whole phase space, it can in a dynamic system several Attraktoren with disjunkten catchment areas exist.

A Repellor is the opposite of a Attraktors.