How Do Eigenvalues Determine Whether Solutions Move Toward or Away From the Origin?

The eigenvalues of the system matrix determine whether exponential factors decrease or increase with time. Negative real parts move solutions toward the origin, while positive real parts move them away.

The Mechanism

For a linear coupled system

$$\frac{d\mathbf{X}}{dt}=A\mathbf{X},$$

an eigenvalue $\lambda$ produces a solution component proportional to $e^{\lambda t}$. Therefore:

• if $\lambda<0$, then $e^{\lambda t}\to 0$, so that component approaches the origin;
• if $\lambda>0$, then grows, so that component moves away;

For example, eigenvalues $-2$ and $-5$ give factors $e^{-2t}$ and $e^{-5t}$. Both tend to zero, so every solution approaches the origin, although one component decays faster.

The corresponding eigenvectors give the directions in which these components act. For two real eigenvalues, trajectories align with eigendirections; near a stable node, the slower-decaying component usually determines the final direction of approach because its exponential factor remains dominant for longer.

For instance, with eigenvalues $-1$ and $-4$, both components decay, but $e^{-t}$ decays more slowly than $e^{-4t}$. The trajectory therefore tends to approach the origin tangent to the eigendirection associated with .

A common misconception is that any negative eigenvalue makes the origin stable. This is false: eigenvalues $-2$ and $3$ produce a saddle because the positive eigenvalue creates an outward direction.

Exam Technique

This is HL-only content, AHL 5.17 Phase portraits, and may be tested across HL paper styles. Calculate the eigenvalues, inspect their signs or real parts, classify the equilibrium, and show the correct arrow direction; a GDC result without interpretation is incomplete.