nullcline

Let

${\dot{x}}_1$$=$$f_1(x_1,\mathrm{\dots },x_n)$

$\mathrm{⋮}$

${\dot{x}}_n$$=$$f_n(x_1,\mathrm{\dots },x_n)$

be a system of first order ordinary differential equation. The $x_j$ nullcline is the set of points which satisfy $f_j(x_1,\mathrm{\dots },x_n)=0$. Note that at an intersection point of all the nullclines implies that

$0$$=$$f_1(x_1,\mathrm{\dots },x_n)$

$\mathrm{⋮}$

$0$$=$$f_n(x_1,\mathrm{\dots },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