Nullcline

In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations
 * $$x_1'=f_1(x_1, \ldots, x_n)$$
 * $$x_2'=f_2(x_1, \ldots, x_n)$$
 * $$\vdots$$
 * $$x_n'=f_n(x_1, \ldots, x_n)$$

where $$x'$$ here represents a derivative of $$x$$ with respect to another parameter, such as time $$t$$. The $$j$$'th nullcline is the geometric shape for which $$x_j'=0$$. 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.

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