Nullcline

In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations

${\displaystyle x_{1}'=f_{1}(x_{1},\ldots ,x_{n})}$

${\displaystyle x_{2}'=f_{2}(x_{1},\ldots ,x_{n})}$

${\displaystyle \vdots }$

${\displaystyle x_{n}'=f_{n}(x_{1},\ldots ,x_{n})}$

where ${\displaystyle x'}$ here represents a derivative of ${\displaystyle x}$ with respect to another parameter, such as time ${\displaystyle t}$. The ${\displaystyle j}$'th nullcline is the geometric shape for which ${\displaystyle x_{j}'=0}$. The equilibrium points of the system are located where all of the nullclines intersect. In a two-dimensional linear system, the nullclines can be represented by two lines on a two-dimensional plot; in a general two-dimensional system they are arbitrary curves. Nullclines are useful for visualization in phase plane plot analysis. Nullclines split the plot into regions of potentially similar dynamics¹.

The definition, though with the name 'directivity curve', was used in a 1967 article by Endre Simonyi.² This article also defined 'directivity vector' as ${\displaystyle \mathbf {w} =\mathrm {sign} (P)\mathbf {i} +\mathrm {sign} (Q)\mathbf {j} }$, where ${\displaystyle P}$ and ${\displaystyle Q}$ are the ${\displaystyle dx/dt}$ and ${\displaystyle dy/dt}$ differential equations, and ${\displaystyle i}$ and ${\displaystyle j}$ are the ${\displaystyle x}$ and ${\displaystyle y}$ direction unit vectors.

Simonyi developed a new stability test method from these new definitions, and with it he studied differential equations. This method, beyond the usual stability examinations, provided semi-quantitative results.