Flow (mathematics)
The concept of a (phase) flow in the mathematics allows the description of time-dependent (system) states. It is therefore especially for the analysis of ordinary differential equations of meaning and thus finds application in many areas of mathematics and physics. Formally, the river is an operation of a parameter semigroup on a set.
Definition
Be a lot, a set of parameters. A picture
Ie flow if the following conditions are met:
And
So we have a semigroup action.
The amount
Ie orbit.
If the mapping is differentiable, one also speaks of a differentiable flow.
Local River
For a set of parameters, a local river for an open subset is generally defined with open intervals, if the conditions
And
Is satisfied. A local river with a ( global ) flow with.
Discussion
With regard to the analysis of dynamic systems of flow describes the movement in the phase space in the course of time. This is called as a function of the parameter set from a continuous dynamic system (), or a discrete dynamic system ().
Consider a system of ordinary differential equations
So are the solutions of this system specified with or an open subset of it by the phase flow, depending on the initial state. One then often chooses an implicit form of the flow specification and writes