where here represents a derivative of with respect to another parameter, such as time . The 'th nullcline is the geometric shape for which . 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 Simonyi. This article also defined 'directivity vector' as , 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.
- 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