Adaptive models of learning in complex physical and social systems

This thesis is developed in the framework of the PhD school in complexity science at the University of Urbino. As a consequence of the interdisciplinary nature of such PhD program, the present work deals with different research areas: it includes economic modeling and physical system control design through reinforcement learning techniques. In the first Chapter a nonlinear discrete-time dynamic system proposed in economic literature, as a market share attraction model, is studied. A rich scenario of local and global bifurcations is obtained even with just two competing firms and the effects of heterogeneities between them are investigated. In the second Chapter the view point on economic modeling is enlarged to critically discuss about assumptions of synchronization of economic agent allowing the description in terms of a representative one stressing the role of heterogeneities even when these are very small. In the subsequent two Chapters the analysis of two evolutionary games in which players in the population choose between two different strategies according to a profitdriven evolutionary selection rule is carried out. The resulting discrete dynamical systems, represented by two and three-dimensional nonlinear maps respectively, are characterized by the presence of invariant manifolds on which the dynamics are governed by the restrictions of evolution equations and represent synchronized populations of players that adopt the same strategies. Finally, in the last Chapter an adaptive nonlinear feedback controller is designed based on reinforcement learning techniques, implemented through a neural network and applied to a coupled two-pendulum system in order to improve stability and performance of classical PID linear controller. This kind of approach improves performances in particular when the available model of the real plant is wrong, a circumstance that reduces benefits of optimal control design and provides superior limits for feedback values avoiding problems of reliability of real feedback actuators.