Editing Helmholtz decomposition (section)

=== Dynamical systems theory ===

In the theory of [[dynamical system]]s, Helmholtz decomposition can be used to determine "quasipotentials" as well as to compute [[Lyapunov function]]s in some cases.<ref name="suda2019" /><ref name="suda2020" /><ref name="zhou2012" />

For some dynamical systems such as the [[Lorenz system]] ([[Edward N. Lorenz]], 1963<ref name="lorenz1963" />), a simplified model for [[atmosphere|atmospheric]] [[convection]], a [[closed-form expression]] of the Helmholtz decomposition can be obtained:
<math display="block">\dot \mathbf{r} = \mathbf{F}(\mathbf{r}) = \big[a (r_2-r_1), r_1 (b-r_3)-r_2, r_1 r_2-c r_3 \big].</math>
The Helmholtz decomposition of <math>\mathbf{F}(\mathbf{r})</math>, with the scalar potential <math>\Phi(\mathbf{r}) = \tfrac{a}{2} r_1^2 + \tfrac{1}{2} r_2^2 + \tfrac{c}{2} r_3^2</math> is given as:

<math display="block">\mathbf{G}(\mathbf{r}) = \big[-a r_1, -r_2, -c r_3 \big],</math>
<math display="block">\mathbf{R}(\mathbf{r}) = \big[+ a r_2, b r_1 - r_1 r_3, r_1 r_2 \big].</math>

The quadratic scalar potential provides motion in the direction of the coordinate origin, which is responsible for the stable [[fixed point (mathematics)|fix point]] for some parameter range. For other parameters, the rotation field ensures that a [[strange attractor]] is created, causing the model to exhibit a [[butterfly effect]].<ref name="glotzl2023" /><ref name="peitgen1992" />