Continuum State Cellular Automata and Random Differential Equations
The research project described in this proposal aims to start the development of a new approach to the mathematical modeling of complex systems. The main ideas are based on an extension of the approach to urban dynamics developed by the researchers during the previous SNF project nr. 2100-067032 "Mathematical modeling of urban growth processes: a cellular automata and statistical mechanics based approach" to a wider class of complex systems. The above mentioned model for urban dynamics is based on a continuum valued stochastic cellular automaton (CA) which reduces in the limit for the time step going to zero to a Markov jump process (MJP) described by a Markov semigroup. It is possible to derive a system of approximated ordinary differential equations (ODE) for the mean values of the dynamical variables from the Kolmogorov forward equation (KFE) associated to the semigroup, assuming the approximation corresponding to infinitesimal fluctuations of the dynamical variables with respect to the mean values. The genaralization we aim to in this reseach project constitutes a wide extension of our approach for urban systems with applications to several complex systems. We will call this generalization of our automaton Interaction Space (IS). For its description we use a random differential equation (RDE) based on the concept of forward mean derivative (see E. Nelson, 1985). Such kind of equations can be derived without introducing any Markov assumption and in a very general context. It describes the full stochastic behavior of the system without any approximation, unlike the previous case of the ODE. Main aim of the research project from the theoretical point of view will be the development of a precise mathematical definition of IS, the derivation of the system of RDE (one for each moment of the extensive observable unknown of the RDE), and the development of numerical methods for their solution. This enable in turn to investigate both from the theoretical and the computational point of view some relevant phenomena like bifurcations, chaotic phenomena and phase transitions in complex systems described by IS. On the side of applications we will construct an improved model for urban dynamics starting from the theoretical setting of the preceding SNSF research project and using the new ideas made available by the development of our IS. The evolution rules of the IS will be written systematically exploiting method of fuzzy decision and control theory. The model will be applied to "model situations" as well as to real case studies both in regions of Ticino, Switzerland, and in the City Weihai, China together with the Harbin Institute of Technology. The computer simulations will be performed using numerical methods to solve RDE which will be developed during the first part of the research. Problems of parameters calibration and model validation will be faced as problems of goal attaining under constraints using method of fuzzy programming. A part of the project will be devoted also to computer simulations on idealized urban systems with the aim to investigate general laws about urban growth processes like the Zipf´s law. In this part of the research we aim on the one hand to improve our knowledge about these relevant aspects of urban dynamics and on the other hand to perform the first empirical and computational steps toward the study of Zipf´s law in a generic IS.