Nullcline

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In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations

${\displaystyle x_{1}'=f_{1}(x_{1},\ldots ,x_{n})}$

${\displaystyle x_{2}'=f_{2}(x_{1},\ldots ,x_{n})}$

${\displaystyle \vdots }$

${\displaystyle x_{n}'=f_{n}(x_{1},\ldots ,x_{n})}$

where ${\displaystyle x'}$ here represents a derivative of ${\displaystyle x}$ with respect to another parameter, such as time ${\displaystyle t}$. The ${\displaystyle j}$'th nullcline is the geometric shape for which ${\displaystyle x_{j}'=0}$. 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. Nullclines are useful for visualization in phase plane plot analysis. Nullclines split the plot into regions of potentially similar dynamics.

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 ${\displaystyle \mathbf {w} =\mathrm {sign} (P)\mathbf {i} +\mathrm {sign} (Q)\mathbf {j} }$, where ${\displaystyle P}$ and ${\displaystyle Q}$ are the ${\displaystyle dx/dt}$ and ${\displaystyle dy/dt}$ differential equations, and ${\displaystyle i}$ and ${\displaystyle j}$ are the ${\displaystyle x}$ and ${\displaystyle y}$ 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.

See also

• Critical point (mathematics)

References

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". PlanetMath.
• SOS Mathematics: Qualitative Analysis