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

Abstract

In this chapter, first it is shown how any (smooth) adaptive dynamical systemAdaptive dynamical system can be modeled in a unique, canonical way as a self-modeling networkSelf-modeling network model. In this way, any adaptive dynamical systemAdaptive dynamical system has its canonical representation as a self-modeling networkSelf-modeling network model and can be analysed based on this canonical representation. This is applied in particular to perform equilibrium analysisEquilibrium analysis for adaptive dynamical systemsAdaptive dynamical system. Addressing equilibrium analysisEquilibrium analysis for a self-modeling networkSelf-modeling network model can relate to any of its networkNetwork characteristics, such as its connectivity characteristicsConnectivity characteristic and aggregation characteristicsAggregation characteristic. For aggregation characteristics, it is shown how, in contrast to often held beliefs, certain classes of nonlinearNonlinear functions used for aggregationAggregation in networkNetwork models enable equilibrium analysisEquilibrium analysis of the emerging dynamicsDynamics within the networkNetwork like linear functionsLinear do. For connectivity characteristicsConnectivity characteristic, it is shown by introducing a form of stratification how specifically for acyclic networksNetwork the equilibriumEquilibrium values of all nodesNode can be directly computed (by the functions used to specify aggregationAggregation) from the equilibriumEquilibrium values of the (independent) nodesNode without incoming connectionsConnection. Moreover, for any type of (cyclic) connectivityConnectivity, by introducing a form of stratification for the network’sNetwork strongly connected componentsStrongly connected component, similar equilibrium analysisEquilibrium analysis results can be obtained relating equilibrium values in any component to equilibriumEquilibrium values in (independent) components without incoming connectionsConnection.