The Chaotic Attractor

A third structural aspect or underlying framework of chaos is the chaotic attractor. During a chaotic system's long-term evolution, the variables don't haphazardly occupy every possible location within the phase space. Instead, the same chaotic attractor eventually takes shape, regardless of where the iterations began (value of x₀). Although different values of x₀ do lead to different trajectories on the attractor, the trajectories never leave the attractor, and the attractor occupies only part of the phase space.