In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations
x
1
′
=
f
1
(
x
1
,
…
,
x
n
)
x
2
′
=
f
2
(
x
1
,
…
,
x
n
)
⋮
x
n
′
=
f
n
(
x
1
,
…
,
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 fixed 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.
The definition, though with the name ’directivity curve’, was used in a 1967 article by Endre Simonyi. This article also defined 'directivity vector' as
w
=
s
i
g
n
(
P
)
i
+
s
i
g
n
(
Q
)
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 semiquantative results.
