Strangeness of attractors sensu stricto depicts their fractal nature of self-similarity, though attractor shouldn’t necessarily possess strangeness to be a set the system tends to evolve whither.

While trajectories are basinbound, the choice of initial conditions for a participant is nonetheless crucial, if most other orbits that pass close to the one we are following at some point do not remain close to it as time advances.

Except on rare strange non-chaotic attractors, that is, where Lyapunov exponents are non-positive. I, for one, am more interested in local instability than in global fate; not all values of parameters the maps depend on give birth to chaos.

Our cliodynamical orbit may be periodic while not chaotic, nested stable limit cycles may be hidden attractors, as could be seen, e.g., in Chinese dynastic cycles, where translatio imperii reattests itself time and again through something akin to delay embedding theorem, without spill over basin’s edges (China is not an expansionistic civilization), while clearly unable to surmount the living standards gap between sea shores and heartland.

The big picture can be likened roughly to the following, from Chua’s circuit: