Ordinary differential equations Definition

Nullcline

Builds on Autonomous system Equilibrium point

Definition

Let $U$ be an open subset of $\mathbb{R}^n$ , let $f = (f_1, \ldots, f_n) : U \to \mathbb{R}^n$ be a function, and consider the autonomous system $x_j' = f_j(x_1, \ldots, x_n)$ for $j = 1, \ldots, n$ . For $j \in \{1, \ldots, n\}$ , the $j$ -th nullcline of the system is the set $N_j = \{x \in U : f_j(x) = 0\}.$

Remarks

On the $j$ -th nullcline, the velocity has no component in the direction $x_j$ . In the plane, trajectories cross the $x$ -nullcline vertically and the $y$ -nullcline horizontally. The equilibrium points are the points that lie on all the nullclines. The nullclines divide the phase plane into regions where each component of the velocity has a constant sign. For a planar linear system, the nullclines are lines through the origin, or the whole plane when a row of the matrix is zero.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Autonomous system definition
Equilibrium point definition
Nullcline definition

Tag	`0WW060JR6P`
URI	`m2://0WW060JR6P`
Aliases	`zero-growth-isocline`
Kind	definition
Writing	W0 stub
MSC	34C05
Defines	Nullcline
Uses	Autonomous system, Equilibrium point

Nullcline

Definition

Remarks

Reading path

Further reading

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