What Is a Saddle Point?

A saddle point is an equilibrium point in a phase portrait where solutions approach the point in one direction but move away from it in another. For a linear system, it occurs when the two real eigenvalues have opposite signs.

This is HL-only content from AHL 5.17 Phase portraits and may be assessed in any Mathematics: Applications and Interpretation HL paper style.

Why a Saddle Point Occurs

Consider the coupled system

$$\frac{dx}{dt}=x, \qquad \frac{dy}{dt}=-2y.$$

Its matrix is

$$A=\begin{pmatrix}1&0\\0&-2\end{pmatrix},$$

with eigenvalues $\lambda_1=1$ and $\lambda_2=-2$. The corresponding solutions have the form

$$x=C e^{t}, \qquad y=D e^{-2t}.$$

The origin is an equilibrium because setting both derivatives to zero gives $x=0$ and $y=0$. In general, each eigenvector defines an invariant straight-line trajectory through the equilibrium. The negative eigenvalue produces decay, while the positive eigenvalue produces growth.

As $t$ increases, $e^{t}$ grows while $e^{-2t}$ approaches zero. Therefore, trajectories approach the origin along the eigenvector associated with $\lambda=-2$, called the , but move away along the eigenvector associated with , called the .

On a sketch, arrows on the stable eigendirection point toward the equilibrium, while arrows on the unstable eigendirection point away. Nearby curved trajectories show both behaviours.

A common misconception is that a saddle point is simply where two trajectories cross. This is incorrect: solution trajectories do not cross in a phase portrait. The classification comes from the signs of the eigenvalues.

Exam Technique

For an IB question asking you to determine the type of equilibrium, calculate the eigenvalues and explicitly state that they are real and have opposite signs. Conclude: “Therefore, the equilibrium is a saddle point,” and identify the stable and unstable eigendirections if required. Do not classify it from appearance alone.