# 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:

Title nullcline Nullcline 2013-03-22 14:27:14 2013-03-22 14:27:14 Daume (40) Daume (40) 5 Daume (40) Definition msc 34C99