  Pattern formation due to oscillating reactions represents variable natural
and engineering systems, but previous studies employed only simple flow
conditions such as uniform flow and Poiseuille flow. We studied the oscillating
reaction in porous media, where dispersion enhanced the spreading of diffusing
components by merging and splitting flow channels. We considered the saddle
flow, where the stretching rate is constant everywhere. We generated patterns
with the Brusselator system and classified them by instability conditions and
P\'eclet number (Pe), which was defined by the stretching rate. The results
showed that each pattern formation was controlled by the stagnation point and
stable and unstable manifolds of the flow field due to the heterogeneous flow
fields and the resulting heterogeneous dispersion fields. The characteristics
of the patterns, such as the position of stationary waves parallel to the
unstable manifold and the size of local stationary patterns around the
stagnation point, were also controlled by Pe.
