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 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.

History

The definition, though with the name ’directivity curve’, was used in a 1967 article by Endre Simonyi.^[1] 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 semiquantative results.

References

1. E. Simonyi: The Dynamics of the Polymerization Processes, Periodica Polytechnica Electrical Engineering – Elektrotechnik, Polytechnical University Budapest, 1967

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

External links

• Nullcline at PlanetMath.org.
• SOS Mathematics: Qualitative Analysis