[Math] Stability of linear systems with complex eigenvalues

matricesordinary differential equationsstability-in-odes

I'm very confused with stability of linear systems, especially when they have complex eigenvalues. Suppose I have the matrix

$$\begin{bmatrix}-5 & 3\\-1&1\end{bmatrix}$$

whose eigenvalues are $-2+i\sqrt{6}$ and $-2-i\sqrt{6}$. This is an unstable system, but can someone explain why? My book has a theorem but I don't understand what it's saying.

Also, if I have a system with one negative eigenvalue and one that is zero, does that make it stable? I know negative eigenvalues are stable but not sure about the zero. Thanks!

Best Answer

  • If any eigenvalue has a positive real part, the system will tend to move away from the fixed point (unstable system).
  • If any eigenvalue has a negative real part, the system will tend to move back to steady state (stable system).
  • If any eigenvalue has an imaginary part, the system oscillate around the steady state.
  • If eigenvalue is zero, the system remains position or amplitude constant.

This is because find eigenvalues is similar to find roots of auxiliar equation, where roots are arguments of exponential function times independent value.

$$ f(t) = \sum_{i=0}^n A_{i}\exp(\lambda_{i}t). $$

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