Ordinary differential equations Definition

Equilibrium point

Builds on Autonomous system Solution of an ordinary differential equation

Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function. A point $x^* \in U$ is an equilibrium point of the autonomous system $x' = f(x)$ if $f(x^*) = 0$ .

Remarks

The point $x^*$ is an equilibrium point exactly when the constant function $x(t) = x^*$ is a solution. For the logistic equation $x' = r x (1 - x/K)$ with $r, K > 0$ , the equilibrium points are $0$ and $K$ . The equilibrium points are the fixed points of the flow: $\varphi_t(x^*) = x^*$ for each $t$ . The behaviour of solutions near an equilibrium point is the subject of Lyapunov stability.

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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

Where this leads

37 pages build on this one.

Tag	`0G4X36AE0X`
URI	`m2://0G4X36AE0X`
Aliases	`critical-point`
Kind	definition
Writing	W0 stub
MSC	34C05, 34A34
Defines	Equilibrium point
Uses	Autonomous system, Solution of an ordinary differential equation

Equilibrium point

Definition

Remarks

Reading path

Where this leads

Further reading

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