IB Maths AI 5.9 Notes

Phase portraits

In this section, differential equations are extended to coupled systems and second order equations. Phase portraits are used to understand how solutions behave over time, and Euler's method is used to generate approximate numerical solutions when an exact solution is difficult or unnecessary.

A phase portrait is a diagram showing the paths traced by solutions of a coupled system. It is drawn in the $xy$-plane, where each point represents the state of the system.

An easier way to think about this is to imagine pushing a swing and graphing the line of its position and speed. A phase portrait essentially shows all possible forms that line can take based on how hard you push the swing, or any other parameter that can change. The phase portrait would look similar to the diagram below:

For a linear system of the form $\frac{dx}{dt}=ax+by$ and $\frac{dy}{dt}=cx+dy$, the behaviour of the solutions depends on the eigenvalues of the matrix:

$A=\begin{pmatrix}a&&b\\c&&d\end{pmatrix}$

The origin is an equilibrium point because when $x=0$ and $y=0$, both derivatives are zero. The type of equilibrium depends on the eigenvalues.