In linear dynamics, one seeks the fundamental solutions from which one can build all other solutions. The appearance and disappearance of a non-wandering set is called a bifurcation . Change of stability and bifurcation always coincide. Stability A non-wandering set may be stable or unstable.
Every orbit starting in a neighborhood of the non-wandering set remains in a neighborhood.
Asymptotically stable non-wandering sets are also called attractors . The basin of attraction is the set of all initial states approaching the attractor in the long time limit. It is asymptotically stable if any infinitesimal small perturbations u decays. These linear equations of motion are justified as long as the orbit doesn't go too far away from u 0 . The fixed point u 0 is asymptotically stable if all eigenvalues s are inside a stability area of the complex plane.