Interpretation of eigenvalues and eigenvectors in ordinary differential equations

A system of ordinary differential equations may be viewed as a matrix of coefficients. The eigenvalues of the system give insight into its long-term behavior for the corresponding eigenvectors. This in turn provides a visual intuition for the system’s behavior for all other vector values.

For an eigenvalue of a system , we obtain the following behavior when is its corresponding eigenvector .

Real part

• Positive implies exponential growth.
• Negative implies exponential decay.
• Zero implies constant magnitude over time.

Imaginary part

• is the angular frequency of oscillation in radians per unit time.
• The period of the oscillation is .

For 2D systems, we can plot the components of and draw directed flow lines corresponding to the eigenvectors. We can then draw smaller arrows that follow these lines, indicating the rough direction of flow for all other values of .

In higher-dimension systems, the same principle applies, though visualization becomes increasingly difficult.)