Modeling and Analysis of Adaptive Dynamical Systems via Their Canonical Self-modeling Network Representation

In this chapter, first it is shown how any (smooth) adaptive dynamical system can be modeled in a unique, canonical way as a self-modeling network model. In this way, any adaptive dynamical system has its canonical representation as a self-modeling network model and can be analysed based on this canonical representation. This is applied in particular to perform equilibrium analysis for adaptive dynamical systems. Addressing equilibrium analysis for a self-modeling network model can relate to any of its network characteristics, such as its connectivity characteristics and aggregation characteristics. For aggregation characteristics, it is shown how, in contrast to often held beliefs, certain classes of nonlinear functions used for aggregation in network models enable equilibrium analysis of the emerging dynamics within the network like linear functions do. For connectivity characteristics, it is shown by introducing a form of stratification how specifically for acyclic networks the equilibrium values of all nodes can be directly computed (by the functions used to specify aggregation) from the equilibrium values of the (independent) nodes without incoming connections. Moreover, for any type of (cyclic) connectivity, by introducing a form of stratification for the network’s strongly connected components, similar equilibrium analysis results can be obtained relating equilibrium values in any component to equilibrium values in (independent) components without incoming connections.