But then the Poincaré–Bendixson theorem says that C is not a strange attractor at all-it is either a limit cycle or it converges to a limit cycle. By making this subset small enough, any nearby stationary points could be excluded. If a strange attractor C did exist in such a system, then it could be enclosed in a closed and bounded subset of the phase space. 1) Montrez que la probabilité quun joueur ait au moins une bonne réponse correcte est Somme ( de k1 à n) de ( (-1)k+1)/k). One important implication is that a two-dimensional continuous dynamical system cannot give rise to a strange attractor. However the theorem does not apply to discrete dynamical systems, where chaotic behaviour can arise in two- or even one-dimensional systems. In particular, chaotic behaviour can only arise in continuous dynamical systems whose phase space has three or more dimensions. On a torus, for example, it is possible to have a recurrent non-periodic orbit. The condition that the dynamical system be on the plane is necessary to the theorem. However, there could be countably many homoclinic orbits connecting one fixed point.Ī weaker version of the theorem was originally conceived by Henri Poincaré ( 1892), although he lacked a complete proof which was later given by Ivar Bendixson ( 1901). Moreover, there is at most one orbit connecting different fixed points in the same direction.