= Solution
Use a <Poincaré map> sampled once per forcing period. A hyperbolic periodic trajectory has <stable manifolds> containing parcels that approach it under forward iteration and <unstable manifolds> containing parcels that approach it under backward iteration. In the integrable limit the relevant branches coincide along a <heteroclinic orbit> and form a barrier between neighboring cells.
A transverse crossing in the perturbed map means that this barrier has split: a parcel on the intersection belongs to both asymptotic manifolds, and adjoining branches enclose fluid lobes transported across the former boundary. Iteration stretches and folds those lobes. In the connected heteroclinic network this creates complicated intercell transport and <chaotic advection>. The crossing is between invariant curves at the same forcing phase; it does not mean that different trajectories cross at one time in violation of uniqueness.
The <incompressible flow> gives an <area-preserving map>, so lobe exchange preserves parcel area. Without <diffusion>, an advected <passive scalar> also keeps its value on each trajectory. Therefore manifold intersections can generate fine filaments and efficient advective exchange but do not by themselves produce molecular homogenization, nor prove that every region of the flow becomes chaotic.
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