Generalized Dynamical Systems Part I: Foundations

In the first of three works we consider a generalized dynamical system (GDS) extended from that initially proposed by [25, 24], where a data structure is mapped to itself and the space of such mappings is closed under composition. We argue that GDS is the natural environment to consider questions arising from the computational implementation of autonomous and semi-autonomous decision problems with one or more constraints, nesting into one framework well-studied models of optimal control, system dynamics, agent-based modeling, and networks, among others. Particular attention is paid to mathematical constructions which support applications in mechanism design. The contingent derivative approach is defined, along with an associated metric, for which a GDS admits the study of existence of state trajectories that satisfy system constraints. The system may also be interpreted as a discretized version of a differential inclusion, allowing the characterization of the reachable subspaces of the state space, and locally controllable trajectories. The second and third parts in the three-part series are briefly described and cover, respectively, applications and implementations, with the latter demonstrating explicitly how a GDS can be implemented as software using Complex Adaptive Dynamics Computer Aided Design (cadCAD) [30].