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In [[mathematical analysis]], '''nullclines''', sometimes called zero-growth [[isocline]]s, are encountered in a system of [[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]] 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 [[Fixed_point_(mathematics)|fixed point]]s 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<sup>1</sup>. This article also defined 'directivity vector' as | |||
<math>\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 semiquantative results. | |||
==References== | |||
{{reflist}} | |||
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 | |||
==External links== | |||
* {{planetmath reference|id= 5972|title=Nullcline}} | |||
* [http://ocw.mit.edu/NR/rdonlyres/D369E47F-064F-48E0-83CD-3B83725B5555/0/r02sol.pdf Notes] from [[MIT OpenCourseWare]] | |||
* [http://www.sosmath.com/diffeq/system/qualitative/qualitative.html SOS Mathematics: Qualitative Analysis] | |||
[[Category:Differential equations]] | |||
{{mathanalysis-stub}} | |||
[[de:Isokline]] | |||
[[nl:nullcline]] | |||
Latest revision as of 13:11, 26 May 2013
In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations
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.
History
The definition, though with the name ’directivity curve’, was used in a 1967 article by Endre Simonyi1. 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.
References
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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