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.E. Simonyi: The Dynamics of the Polymerization Processes, Periodica Polytechnica Electrical Engineering – Elektrotechnik, Polytechnical University Budapest, 1967 This article also defined 'directivity vector' as
$\mathbf = \mathrm(P)\mathbf + \mathrm(Q)\mathbf$,
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.

See also

* Critical point (mathematics)

References

Notes

* E. Simonyi – M. Kaszás: Method for the Dynamic Analysis of Nonlinear Systems, Periodica Polytechnica Chemical Engineering – Chemisches Ingenieurwesen, Polytechnical University Budapest, 1969

External links

* {{planetmath reference, urlname=Nullcline, title=Nullcline

SOS Mathematics: Qualitative Analysis

Differential equations