text: In chaotic advection, a fluid particle travels within a large region, and encounters other particles that were initially far from it. One can then consider that a particle is mixed with particles that travel within the same region. However, the region covered by a trajectory does not always span the whole fluid domain. Poincaré sections are used to distinguish regions of good and bad mixing.
  The Poincaré map is defined as the transformation 
   transforms a point-like particle into the position of the particle after a time-interval T. Especially, for a time-periodic flow with period T, applying the map several times to a particle gives the successive positions of the particle period after period. A Poincaré section is built by starting from a few different initial conditions and plotting the corresponding iterates. This comes down to plotting the trajectories stroboscoped every T.
  As an example, the figure presented here (left part) depicts the Poincaré section obtained when one applies periodically a figure-eight-like movement to a circular mixing rod. Some trajectories span a large region: this is the chaotic or mixing region, where good mixing occurs. However, there are also two "holes": in these regions, the trajectories are closed. These are called elliptic islands, as the trajectories inside are elliptic-like curves. These regions are not mixed with the remainder of the fluid. For mixing applications, elliptic islands have to be avoided for two reasons :
  * Fluid particles are unable to cross the boundaries of the islands (except by slow diffusion), resulting in segregation.
  * Mixing inside these regions is not efficient because trajectories are closed and therefore not chaotic.
  Avoiding non-chaotic islands requires understanding the physical origin of these regions. Generally speaking, changing the geometry of the flow can modify the presence or absence of islands. In the figure-eight flow for instance, for a very thin rod, the influence of the rod is not felt far from its location, and almost circular trajectories exist within the loops of the figure-eight. With a larger rod (right part of the figure), particles can escape from these loops and islands do not exist any more, resulting in better mixing.
  With a Poincaré section, the mixing quality of a flow can be analyzed by distinguishing between chaotic and elliptic regions. This is a crude measure of the mixing process, however, since the stretching properties cannot be inferred from this mapping method. Nevertheless, this technique is very useful for studying the mixing of periodic flows and can be extended to a 3-D domain.