We have described a decirculation process which marks perturbations of network structure that are necessary for nonlinear network dynamics to proceed from one circulating state (a limit cycle) to another stable state (a limit cycle or a fixed point). Armed with the decirculation process, a sort of decirculating maps and their structural properties have also been built, dedicated to showing that circulation breaking taking place in nonlinear network dynamics can collaborate harmoniously toward the completion of network structure that generates attractors (equilibrium states). Here we wish to extend the notion of decirculating maps to the notion of depathing maps. The extension allows us to reshape network structure not only on the occasion of circulating states but on the occasion of any required path states. This gives a crucial improvement in generating circulating state shifts more feasibly.
MSC: 47H10, 37F20, 92B20, 00A71, 68T05, 91E40.