Stable, Semi-Stable, and Unstable Equilibrium Solutions

# Stable, Semi-Stable, and Unstable Equilibrium Solutions

Recall that if $\frac{dy}{dt} = f(t, y)$ is a differential equation, then the equilibrium solutions can be obtained by setting $\frac{dy}{dt} = 0$. For example, if $\frac{dy}{dx} = y(y + 2)$, then the equilibrium solutions can be obtained by solving $y(y + 2) = 0$ for $y$. We hence see that $y = 0$ and $y = -2$ are the equilibrium solutions.

We will now look at classifying these equilibrium solutions.

 Definition: An equilibrium solution is said to be Asymptotically Stable if on both sides of this equilibrium solution, there exists other solutions which approach this equilibrium solution. An equilibrium solution is said to be Semi-Stable if one one side of this equilibrium solution there exists other solutions which approach this equilibrium solution, and on the other side of the equilibrium solution other solutions diverge from this equilibrium solution. An equilibrium solution is said to be Unstable if on both sides of this equilibrium solution other solutions diverge from this equilibrium solution.

The following image is the slope field of the differential equation $\frac{dy}{dx} = (y - 1)^2(y - 2)(y- 3)$ which has three equilibrium solutions, $y = 1$, $y = 2$, and $y = 3$.

• The equilibrium solution $y = 1$ is green and is a semi-stable equilibrium solution.
• The equilibrium solution $y = 2$ is yellow and is an asymptotically stable equilibrium solution.
• The equilibrium solution $y = 3$ is red and is an unstable equilibrium solution.