Assuming the interaction matrix is diagonizable, the system state can be represented as a linear combination of the eigenvectors. The eigenvector with the largest positive eigenvalue grows the fastest under the system dynamics. Therefore, the respective compontent of the system state will become the dominating component, much larger than the others. (The growth of the components is exponential.) Ult... (read more)
I wondered the same thing. The explanation I've come up with is the following:
See https://en.wikipedia.org/wiki/Linear_dynamical_system for the relevant math.
Assuming the interaction matrix is diagonizable, the system state can be represented as a linear combination of the eigenvectors. The eigenvector with the largest positive eigenvalue grows the fastest under the system dynamics. Therefore, the respective compontent of the system state will become the dominating component, much larger than the others. (The growth of the components is exponential.) Ult... (read more)