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

[math]\displaystyle{ x_1'=f_1(x_1, \ldots, x_n) }[/math]

[math]\displaystyle{ x_2'=f_2(x_1, \ldots, x_n) }[/math]

[math]\displaystyle{ \vdots }[/math]

[math]\displaystyle{ x_n'=f_n(x_1, \ldots, x_n) }[/math]

where [math]\displaystyle{ x' }[/math] here represents a derivative of [math]\displaystyle{ x }[/math] with respect to another parameter, such as time [math]\displaystyle{ t }[/math]. The [math]\displaystyle{ j }[/math]'th nullcline is the geometric shape for which [math]\displaystyle{ x_j'=0 }[/math]. 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 [math]\displaystyle{ \mathbf{w} = \mathrm{sign}(P)\mathbf{i} + \mathrm{sign}(Q)\mathbf{j} }[/math], 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