Math Insight

The stability of equilibria of a differential equation

 

The stability of equilibria of a differential equation.

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Imagine that, for the differential equation \begin{align*} \diff{x}{t} &= f(x)\\ x(0) &=b \end{align*} where $f(-8.5)=0$, you determine that the solution $x(t)$ approaches $-8.5$ as $t$ increases as long as $b < -7.4$, but that $x(t)$ blows up if the initial condition $b$ is much larger than $-7.4$. What can you conclude about the point $x=-8.5$?

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

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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.