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