1.6 Systems of ODE's and Eigenvalue Stability | 1.6 Systems of ODE's and Eigenvalue Stability | Unit 1: Numerical Integration of Ordinary Differential Equations | Computational Methods in Aerospace Engineering | Aeronautics and Astronautics

<Eigenvalue Stability for a Linear ODE
1.6.1Nonlinear Systems
1.6.2Linear Constant Coefficient Systems
1.6.3Eigenvalue Stability for a Linear ODE
1.6.4Imaginary Eigenvalues
>Stiffness and Implicit Methods

1.6.4 Imaginary Eigenvalues

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?

Varying or constant amplitude oscillations

Only constant amplitude oscillations

Only decaying amplitude oscillations

No oscillations

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.