K.De GREENE proposes the following classification of attractors:
"1. Points, like the static equilibrium points of catastrophe theory. Points can apply to both linear and non-linear systems.
"2. Periodic attractors, like limit cycles that apply, say, to interacting populations
"3. Strange, Lorenz or chaotic attractors: A system state can be related to a basin of attraction, but how stable the system is and where the system resides relative to the boundary of the basin may be unknown and unknowable. Indeed, the very existence of an alternative basin(s) of attraction may be unknown".(1990, p.161)
Furthermore "…minuscule differences in initial conditions may lead to the exponential expansion of these differences" (Ibid.)
A slightly different classification is possible: Fixed point: The simplest attractor. It corresponds to a generally monotonous trajectory of a non disturbed system toward a final state, for example: a real pendulum, submitted to frictions.
→ ("Fixed point")
Limit cycle: Corresponds to a closed loop within the phases space. This implies that the system's trajectory involves a series of nondisturbed periodic oscillations.
Toric attractor: Characterizes the behavior of a system simultaneously submitted to two periodic oscillations, independent from each other. The structure of these oscillations remains always predictable and possible to carry out, even when the periods are incommensurable.
→ "Numbers (Prime)"
Chaotic attractor: Corresponds to the behavior of a system simultaneously submitted to, at least, three periodic oscillations independent from each others. The simplest corresponds to systems of simple differential equations where the phases space is three-dimensional.
Even in these cases the transformations of the system are generally not predictable. An early example was the famous astronomical three bodies problem studied by POINCARÉ.