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This study is motivated by two different, yet, connected, motivations. Thefirst one follows the observation that the classical definition of derivativesinvolves prospective (or forward) difference quotients, not known whenever thetime is directed, at least at the macroscopic level. Actually, the availableand known derivatives are retrospective (or backward). They co\"incide wheneverthe functions are differentiable in the classical sense, but not in the case ofnon smooth maps, single-valued or set-valued. The later ones are used indifferential inclusions (and thus, in uncertain control systems) governingevolutions in function of time and state. We follow the plea of some physicistsfor taking also into account the retrospective derivatives to study prospectiveevolutions in function of time, state and retrospective derivatives, aparticular, but specific, example of historical of ''path dependent''evolutionary systems. This is even more crucial in life sciences, in theabsence of experimentation of uncertain evolutionary systems. The secondmotivation emerged from the study of networks with junctions (cross-roads intraffic networks, synapses in neural networks, banks in financial networks,etc.), an important feature of ''complex systems''. At each junction, thevelocities of the incoming (retrospective) and outgoing (prospective)evolutions are confronted. One measure of this confrontation (''jerkiness'') isprovided by the product of the retrospective and prospective velocities,negative in ''inhibitory'' junctions, positive for ''excitatory'' ones, forinstance. This leads to the introduction of the ''differential connectiontensor'' of two evolutions, defined as the tensor product of retrospective andprospective derivatives, which can be used for controlling evolutionary systemsgoverning the evolutions through networks with junctions.

Retro-Prospective Differential Inclusions and their Control by the Differential Connection Tensors of their Evolutions: The trendometer