Nullcline

Short description: Curves on which differential equations are zero

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

${x_1}^{\prime }=f_1(x_1,\dots ,x_n)$

${x_2}^{\prime }=f_2(x_1,\dots ,x_n)$

$⋮$

${x_n}^{\prime }=f_n(x_1,\dots ,x_n)$

where $x^{\prime }$ 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 ${x_j}^{\prime }=0$. 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 $\mathbf{𝐰}=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(P)\mathbf{𝐢}+\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(Q)\mathbf{𝐣}$, 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.

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

• "Nullcline". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}. 
• SOS Mathematics: Qualitative Analysis