Nonattracting

Nonattracting is a term used in dynamical systems to describe invariant sets that do not attract nearby points under forward iteration. More precisely, a nonattracting invariant set F has no neighborhood U such that every point of U has its forward orbit approaching F in the limit. In this sense, F fails to be an attractor, even though the set remains unchanged by the dynamics.

This idea contrasts with attractors, which possess a basin of attraction: a region of initial conditions whose forward orbits converge to the attractor. Nonattracting invariant sets may still influence nearby trajectories without attracting them in the forward time direction. They often arise as natural companions to attracting sets in bifurcations and in systems with mixed dynamics.

Common examples of nonattracting invariant sets include repellers and saddle-type sets. A repeller is an invariant set that repels nearby points under forward time, so trajectories near the set tend to move away rather than toward it. A saddle is a hyperbolic invariant set with both stable and unstable directions, meaning some nearby orbits approach the set along certain directions but diverge along others. Chaotic saddles are a related concept where the set exhibits chaotic behavior but does not attract typical orbits.

In practice, identifying nonattracting sets helps describe the boundary between regions that lead to attractors and regions with more complex or transient dynamics. They play a role in the study of invariant sets, omega-limit sets, basins of attraction, and the overall structure of phase space in both discrete and continuous-time systems.