The dynamics of a physical system can typically be described as an initial value problem of finding a vector function U(t) depending on time t such that
- dU/dt + A(U) = F for t > 0 with U(0) = G,
An essential aspect of the dynamics is the perturbation dynamics described by the linearized system which is determined by the eigenvalues of its linearization matrix A, assuming for simplicity that A is diagonalizable and independent of time:
- Positive eigenvalues: Stable in forward time; unstable in backward time.
- Negative eigenvalues : Unstable in forward time; stable in backward time.
- Both positive and negative eigenvalues: Both unstable and stable in both forward and backward time.
- Imaginary eigenvalues: Wave solutions marginally stable in both forward and backward time.
- Complex eigenvalues: Combinations of 1. - 4.