Flow (math)

The concept of a (phase) flow in mathematics enables the description of time-dependent (system) states. It is therefore of particular importance for the analysis of ordinary differential equations and is therefore used in many areas of mathematics and physics . Formally, the flow is an operation of a parameter half group on a set . Mostly, especially in the theory of ordinary differential equations , a flow is understood to be an operation of the semigroup . ${\ displaystyle (\ Gamma, +)}$ ${\ displaystyle X}$ ${\ displaystyle (\ mathbb {R} _ {\ geq 0}, +)}$

definition

Be a set, a set of parameters. An illustration ${\ displaystyle X}$ ${\ displaystyle \ Gamma}$

{\ displaystyle \ varphi \ colon X \ times \ Gamma \ to X}

is called a flow if the following conditions are met:

{\ displaystyle \ varphi (x, 0) = x \ \ forall \ x \ in X}

and

{\ displaystyle \ varphi (\ varphi (x, s), t) = \ varphi (x, s + t) \ \ forall \ x \ in X, \ s, t \ in \ Gamma}

So we have a semi-group effect .

The amount

{\ displaystyle {\ mathcal {O}} (x, \ varphi): = \ left \ {\ varphi (x, t) | t \ in \ Gamma \ right \}}

is called orbit of . ${\ displaystyle x}$

If the mapping is differentiable, one speaks of a differentiable flow . ${\ displaystyle \ varphi \ colon X \ times \ Gamma \ to X}$

Local flow

For a parameter set, a local flow for an open subset with open intervals is more generally defined, if the conditions ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ varphi \ colon U \ to X}$ ${\ displaystyle U = \ bigcup \ nolimits _ {x \ in X} \ {x \} \ times I_ {x} \ subseteq X \ times \ mathbb {R}}$ ${\ displaystyle 0 \ in I_ {x} \ subseteq \ mathbb {R}}$

{\ displaystyle \ varphi (x, 0) = x \ \ forall \ x \ in X}

and

{\ displaystyle \ varphi (\ varphi (x, s), t) = \ varphi (x, s + t) \ \ forall \ x \ in X, \ s, s + t \ in I_ {x}, \ t \ in I _ {\ varphi (x, s)}}

is satisfied. A local flow with is a (global) flow with . ${\ displaystyle U = X \ times \ mathbb {R}}$ ${\ displaystyle \ Gamma = \ mathbb {R}}$

discussion

With regard to the analysis of dynamic systems , the flow describes the movement in phase space over time. Depending on the set of parameters, one speaks of a continuous dynamic system ( ) or a discrete dynamic system ( ). ${\ displaystyle \ Gamma}$ ${\ displaystyle \ Gamma = \ mathbb {R}}$ ${\ displaystyle \ Gamma = \ mathbb {N}}$

Let us consider a system of ordinary differential equations

{\ displaystyle {\ dot {\ mathbf {x}}} = \ mathbf {F} (t, \ mathbf {x})}

with or an open subset of it, the solutions of this system are given by the phase flow depending on the initial state . You then often choose an implicit form of flow specification and write ${\ displaystyle \ mathbf {x} \ in \ mathbb {R} ^ {n}}$

{\ displaystyle \ mathbf {x} (t) {\ text {or}} \ mathbf {x} (0)}

.

example

For example, you can assign a flow to each vector field . This is given by the maximum integral curve of the vector field. In fact, every flow on a differentiable manifold is the flow of a vector field obtained through . ${\ displaystyle F (x) = {\ frac {d} {dt}} \ vert _ {t = 0} \ Phi (t, x)}$

The Ricci flow plays a central role in the now proven Thurstonian geometry conjecture , which is a generalization of the Poincaré conjecture .

Individual evidence

↑ Theodor Bröcker, Klaus Jänich: Introduction to the differential topology . Springer, Berlin 1973, ISBN 3-540-06461-3 , pp. 80 (§ 8. Dynamic Systems).

literature

Manfred Denker: Introduction to the Analysis of Dynamic Systems . Springer Verlag, Berlin, Heidelberg, New York 2005, ISBN 3-540-20713-9
Werner Krabs: Dynamic systems: controllability and chaotic behavior . BGTeubner, Leipzig 1998, ISBN 3-519-02638-4 .

[1] Theodor Bröcker, Klaus Jänich: Introduction to the differential topology . Springer, Berlin 1973, ISBN 3-540-06461-3 , pp. 80 (§ 8. Dynamic Systems).