An attractor or attractor state in mathematical terms is a closed invariant set toward which system trajectories converge over time. Formally, it's a set A where all nearby trajectories approach A as t → ∞. In social systems, attractors represent stable configurations of organization - the cultural norms, institutional arrangements, and power structures that societies naturally evolve toward and maintain.
Visually, one can think of a Fixed Point Attractor as a sink (like a marble settling in a bowl), a Limit Cycle Attractor as a closed track (like a train orbiting a looped rail, always returning to start), and a Strange Attractor as a fractal “tangle” or strange landscape in which the system roams indefinitely.
Behaviorally, as we move from fixed to limit cycle to strange attractors, the system's long-term behavior becomes more complex: from static, to periodic, to chaotic. This progression often occurs via parameter changes – e.g. a simple steady-state can give way to oscillations and then to chaos as a control parameter passes critical thresholds (as seen in the logistic map or Lorenz system, where a single system can exhibit all three types of attractors depending on parameters)