nullcline

Let

$\displaystyle\dot{x}_{1}$	$\displaystyle=$	$\displaystyle f_{1}(x_{1},\ldots,x_{n})$
	$\displaystyle\vdots$
$\displaystyle\dot{x}_{n}$	$\displaystyle=$	$\displaystyle f_{n}(x_{1},\ldots,x_{n})$

be a system of first order ordinary differential equation. The $x_{j}$ nullcline is the set of points which satisfy $f_{j}(x_{1},\ldots,x_{n})=0$ . Note that at an intersection point of all the nullclines implies that

$\displaystyle 0$	$\displaystyle=$	$\displaystyle f_{1}(x_{1},\ldots,x_{n})$
	$\displaystyle\vdots$
$\displaystyle 0$	$\displaystyle=$	$\displaystyle f_{n}(x_{1},\ldots,x_{n}).$

Hence the intersection point of all the nullclines is an equilibrium point of the system.

example:

•

see some qualitative analysis of FitzHugh-Nagumo equation using nullclines

Title	nullcline
Canonical name	Nullcline
Date of creation	2013-03-22 14:27:14
Last modified on	2013-03-22 14:27:14
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	5
Author	Daume (40)
Entry type	Definition
Classification	msc 34C99