In a system of difference equations...
$\textbf{x}(n+1) = \textbf{A}\textbf{x}(n) + \textbf{f}$
... I am given the following rules to determine the nature of the fixed point (note that the analogies below refer to continuous equations):
- Case 1) $\lambda_1$ and $\lambda_2$ are complex, and further $\lambda_1 = \overline{\lambda_2}$. In this case, as for continous systems, there is a spiralling effect - towards the fixed point if $|\lambda| < 1$ (stable case), and away from it if $|\lambda| > 1$ (unstable case)
- Case 2) $\lambda_1$ and $\lambda_2$ are real and greater than 1. The behavior is analogous to an unstable node
- Case 3) $\lambda_1$ and $\lambda_2$ are real and $0 < \lambda_1, \lambda_2 < 1$. The behavior is analogous to that of a stable node
- Case 4) $\lambda_1$ and $\lambda_2$ are real and $0<\lambda_1 < 1 < \lambda_2$. The behavior is analogous to that of a saddle point
- Case 5) Either $\lambda_1$ or $\lambda_2$ is negative, or both are. There is no analogy with the continuous situation. The solution keeps jumping from one side of the fixed point to the other. If $|\lambda_1| < 1$ and $|\lambda_2| < 1$ then the behavior is stable, and if either $|\lambda_1| > 1$ or $|\lambda_2| > 1$, then it is unstable.
Clearly, the above rules apply to a system of 2 difference equations. How are the above rules written if I extend and generalize the system of difference equations to any number, say, n?
For example, I've just calculated three eigenvalues $\lambda_1 = 6$, $\lambda_2 = -5$ and $\lambda_3 = 3$ for a system of difference equations and am unsure of the nature of the fixed point given that the above rules assume two eigenvalues are derived.