Phase Lines

# Phase Lines

Sometimes we can create a little diagram known as a Phase Line that gives us information regarding the nature of solutions to a differential equation.

We have already seen from the Stable, Semi-Stable, and Unstable Equilibrium Solutions page that we can determine whether arbitrary solutions to a differential equation converge on both sides to an equilibrium solution (which we defined as an asymptotically stable equilibrium solution), whether arbitrary solutions to a differential equation converge on one side and diverge on the other side of the equilibrium solution (which we defined as a semi-stable equilibrium solution), or whether arbitrary solutions to a differential equation diverge on both sides of that equilibrium solution (which we defined as an unstable equilibrium solution). We did not need to solve the differential equation in order to determine this nature.

Phase lines are nice because they provide a nice way to diagram this information.

Consider the following differential equation:

(1)
\begin{align} \quad \frac{dy}{dt} = y(y-1)(y-2) \end{align}

The equilibrium solutions are obtained by setting $\frac{dy}{dt} = 0$, and they're precisely $y = 0$, $y = 1$, and $y = 2$.

If $-\infty < t < 0$ then $\frac{dy}{dt} < 0$, and so solutions below $y = 0$ diverge from $y = 0$.

If $0 < t < 1$ then $\frac{dy}{dt} > 0$, and so solutions between $y = 0$ and $y = 1$ converge to $y = 1$.

If $1 < t < 2$ then $\frac{dy}{dt} < 0$, and so the solutions between $y = 1$ and $y = 2$ converge to $y = 1$.

If $2 < t < \infty$ then $\frac{dy}{dt} > 0$, and so the solutions above $y = 2$ diverge to $\infty$

Therefore the equilibrium solutions $y = 0$ and $y = 2$ are both unstable, while the equilibrium solution $y = 1$ is asymptotically stable as depicted below:

To construct a phase line for this differential equation - draw a vertical line. Plot the values of the equilibrium solution, and then draw arrows indicated whether solutions between the equilibrium converge/diverge to specific equilibrium solutions. The phase line for the differential equation above is shown below:

Arrows pointing towards an equilibrium solution from both sides on a phase line indicate that equilibrium solution is asymptotically stable. One arrow pointing towards an equilibrium solution and one arrow pointing away from an equilibrium solution on a phase line indicate that equilibrium solution is semi-stable. Arrows pointing away from an equilibrium solution from both sides on a phase line indicate that equilibrium solution is unstable.