# Nullcline

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{{Multiple issues|
{{context|date=May 2025}}
{{technical|date=May 2025}}
}}{{Short description|Curves on which differential equations are zero}}
In [mathematical analysis](/source/mathematical_analysis), '''nullclines''', sometimes called zero-growth [isocline](/source/isocline)s, are encountered in a system of [ordinary differential equation](/source/ordinary_differential_equation)s
:<math>x_1'=f_1(x_1, \ldots, x_n)</math>
:<math>x_2'=f_2(x_1, \ldots, x_n)</math>  
::<math>\vdots</math>
:<math>x_n'=f_n(x_1, \ldots, x_n)</math> 

where <math>x'</math> here represents a [derivative](/source/derivative) of <math>x</math> with respect to another parameter, such as time <math>t</math>.  The <math>j</math>'th nullcline is the geometric shape for which <math>x_j'=0</math>.   The equilibrium points of the system are located where all of the nullclines intersect.
In a two-dimensional [linear system](/source/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.
Nullclines are useful for visualization in phase plane plot analysis. Nullclines split the plot into regions of potentially similar dynamics<ref>[https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Chemical_Process_Dynamics_and_Controls_(Woolf)/10%3A_Dynamical_Systems_Analysis/10.05%3A_Phase_Plane_Analysis_-_Attractors_Spirals_and_Limit_cycles LibreTexts "Phase Plane Analysis - Attractors, Spirals, and Limit cycles"]</ref>.

== History ==

The definition, though with the name 'directivity curve', was used in a 1967 article by Endre Simonyi.<ref>E. Simonyi: The Dynamics of the Polymerization Processes, Periodica Polytechnica Electrical Engineering – Elektrotechnik, Polytechnical University Budapest, 1967</ref>  This article also defined 'directivity vector' as
<math>\mathbf{w} =  \mathrm{sign}(P)\mathbf{i} + \mathrm{sign}(Q)\mathbf{j}</math>,
where <math>P</math> and <math>Q</math> are the <math>dx/dt</math> and <math>dy/dt</math> differential equations, and <math>i</math> and <math>j</math> are the <math>x</math> and <math>y</math> direction [unit vector](/source/unit_vector)s.

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.

== See also ==

* [Critical point (mathematics)](/source/Critical_point_(mathematics))

==References==
{{reflist}}

==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==
* {{planetmath reference|urlname=Nullcline|title=Nullcline}}
* [http://www.sosmath.com/diffeq/system/qualitative/qualitative.html SOS Mathematics: Qualitative Analysis]

Category:Differential equations

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