Flow equation

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Flow equation is a differential equation to describe how additive and multiplicative generators influence the assignment in an infinitesimal step.

Formal definition

In the local polar coordinate system, it takes the form

[math]\displaystyle{ \frac{da}{ds} = \mu \cos \theta + a \lambda \sin \theta }[/math]


Or, in the local gradient-contour coordinate system, it takes the form

[math]\displaystyle{ \frac{da}{ds} = \sqrt {\mu^2 + a^2 \lambda^2} \cos \phi }[/math]



The derivation of flow equation

Considering an infinitesimal generating process over a Riemannian manifold [math]\displaystyle{ M }[/math] with one generator for additional action [math]\displaystyle{ \mu }[/math] and the other for multiplicative action [math]\displaystyle{ e^\lambda }[/math], the two generators are perpendicular. Under this generation process, we get an assignment [math]\displaystyle{ a : M \to F }[/math] over the manifold.

At any point with an assignment [math]\displaystyle{ a_0 }[/math], considering a moving with distance of [math]\displaystyle{ \epsilon }[/math], direction angle of [math]\displaystyle{ \theta }[/math], time elapse of [math]\displaystyle{ \delta }[/math], we can establish

[math]\displaystyle{ a_\delta = (a_0 + \mu \epsilon \cos \theta) e^{\lambda \epsilon \sin \theta} }[/math]



[math]\displaystyle{ a_\delta = a_0 e^{\lambda \epsilon \sin \theta} + \mu \epsilon \cos \theta }[/math]


Both can be simplified to the same

[math]\displaystyle{ a_\delta = a_0 + \epsilon (\mu \cos \theta + a_0 \lambda \sin \theta) }[/math]


Then we have

[math]\displaystyle{ \frac{1}{\delta}(a_\delta - a_0) = \frac{\epsilon}{\delta} (\mu \cos \theta + a_0 \lambda \sin \theta) }[/math]


When both [math]\displaystyle{ \delta }[/math] and [math]\displaystyle{ \epsilon }[/math] are towards zero, we get [math]\displaystyle{ \frac{da}{dt} }[/math], and hence

[math]\displaystyle{ \frac{da}{dt} = u (\mu \cos \theta + a \lambda \sin \theta) }[/math]


Or, we can change it to other forms

[math]\displaystyle{ \frac{da}{ds} = \mu \cos \theta + a \lambda \sin \theta }[/math]


As a pullback