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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)
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.


The definition, though with the name ’directivity curve’, was used in a 1967 article by Endre Simonyi1. 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.


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

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

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