# 1.6 Systems of ODE's and Eigenvalue Stability

## 1.6.4 Imaginary Eigenvalues

Measurable Outcome 1.2, Measurable Outcome 1.9

What is the behaviour associated with eigenvalues whose real part is zero and imaginary part is nonzero imaginary part. Which of the following numerical methods is consistent?

Exercise 1
&nbsp; Varying or constant amplitude oscillations &nbsp;
&nbsp; Only constant amplitude oscillations &nbsp;
&nbsp; Only decaying amplitude oscillations &nbsp;
&nbsp; No oscillations &nbsp;

Answer: Eigenvalues with nonzero imaginary part have oscillatory behavior. Purely imaginary eigenvalues correspond to oscillations with constant amplitude. Generally complex eigenvalues have oscillations that either grow (positive real part) or decay (negative real part) in amplitude.