mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, nullclines, sometimes called zero-growth

isocline image:Isocline 3.png, 300px, Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of ''y = ''xy''. Given a family of curves, assumed to be Differentiable manifold, differentiable, an isocline for that family i ...

s, are encountered in a system of

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...

s :

x_1'=f_1(x_1, \ldots, x_n)

x_2'=f_2(x_1, \ldots, x_n)

\vdots

x_n'=f_n(x_1, \ldots, x_n)

where

x'

here represents a

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

x

with respect to another parameter, such as time

t

. The

j

'th nullcline is the geometric shape for which

x_j'=0

. The equilibrium points of the system are located where all of the nullclines intersect. In a two-dimensional

linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction o ...

, 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.E. Simonyi: The Dynamics of the Polymerization Processes, Periodica Polytechnica Electrical Engineering – Elektrotechnik, Polytechnical University Budapest, 1967 This article also defined 'directivity vector' as

\mathbf =  \mathrm(P)\mathbf + \mathrm(Q)\mathbf

, 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 semi-quantitative results.

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

* {{planetmath reference, urlname=Nullcline, title=Nullcline
SOS Mathematics: Qualitative Analysis
Differential equations