Math Insight

The stability of equilibria of a differential equation

 

The stability of equilibria of a differential equation.

More information about video.

Imagine that, for the differential equation \begin{align*} \diff{x}{t} &= f(x)\\ x(0) &=b \end{align*} where $f(-9.3)=0$, you determine that the solution $x(t)$ approaches $-9.3$ as $t$ increases as long as $b < -8.8$, but that $x(t)$ blows up if the initial condition $b$ is much larger than $-8.8$. What can you conclude about the point $x=-9.3$?

The stability of equilibria of a differential equation, analytic approach.

More information about video.

Stability theorem

Let $\diff{x}{t} = f(x)$ be an autonomous differential equation. Suppose $x(t)=x^*$ is an equilibrium, i.e., $f(x^*)=0$. Then

  • if $f'(x^*)<0$, the equilibrium $x(t)=x^*$ is stable, and
  • if $f'(x^*)>0$, the equilibrium $x(t)=x^*$ is unstable.