# Research Project

The purpose of the Project entitled "the Mathematical Theory for Modelling Complex Systems and Its Transdisciplinary Applications in Science and Technology"* is to develop a mathematical theory that will facilitate the modelling of complex systems and the creation of transdisciplinary applications of this theory in science and technology. This will be done through exploiting advances in mathematical engineering (a discipline originally developed in Japan) and chaos engineering (a field aimed at creating various applications based on deterministic chaos, fractals and complex networks).

This project is being conducted under the leadership of Prof. Kazuyuki Aihara, the director of the Collaborative Research Center for Innovative Mathematical Modelling (established in March 2010 at the Institute of Industrial Science, University of Tokyo) in close collaboration with researchers affiliated with various research organizations both in Japan and overseas. These organizations include not only institutions within the University of Tokyo (such as the Graduate School of Information Science and Technology, the Graduate School of Mathematical Sciences, the Graduate School of Engineering, and the Graduate School of Frontier Sciences), but also the following external organizations: Tokyo Institute of Technology, Tokyo Denki University, Tokyo City University, Tokyo University of Science, Teikyo University of Science, Kyoto University, Osaka University, University of Tokushima, Kyushu University, Kyushu Institute of Technology, Riken, Chinese Academy of Sciences, and Cambridge University.

* This project is supported by the Japan Society for the Promotion of Science (JSPS) through the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program), initiated by the Council for Science and Technology Policy (CSTP).

#### Sub-Project 1: Mathematical Theory for Modelling Complex Systems

(Sub-project leader: Jun-ichi Imura)

The aim of this sub-project 1 is to integrate dynamical systems theory and control theory—two theories that were originally developed independently of each other in the fields of mathematics and engineering, respectively. The end-goal of this research is to develop an advanced control theory of complex systems based primarily on the theory of hybrid dynamical systems. To this end, we are studying hybrid dynamical systems (which are a representative example of complex systems), network systems, systems of partial differential equations, systems of hybrid partial differential equations, and systems of delay differential difference equations. We have also identified sub-themes associated with the three basic components that are required to develop a control theory of complex systems (i.e. fundamentals of systems, system analysis, and prediction/control), with a view to promoting research through close collaboration among researchers studying the various sub-themes.

#### Sub-Project 2:Engineering Applications for Modelling Complex Systems

(Sub-project leader: Yoshihiko Horio)

In order to apply the mathematical theory of modelling complex systems developed in our project to actual engineering systems, we fully exploit the results of our theoretical research and conduct applied engineering studies focused on the following sub-themes: information processing through complex systems, novel A/D and D/A converter circuits based on β-maps, engineering applications of complex networks, and applications of innovative mathematical modelling to industrial processes. Once we have obtained the results from these specific studies and extracted intrinsic universalities of complex engineering systems from applied research, we then feed them back into the theoretical research and promote our research project in close collaboration with the other sub-groups associated with the specific sub-themes.

#### Sub-Project 3:Integration of Theories and Applications for Modelling Complex Systems

(Sub-project leader: Kazuyuki Aihara)

We are conducting theoretical research with a view to developing various theories of mathematical analysis (such as a theory of complex networks and a theory of nonlinear time series analysis), while at the same time promoting applied research in a variety of disciplines, such as nonlinear science, life science, information science, engineering, medical science and economics. By integrating theoretical and applied research, we aim to develop and then systematize the mathematical theory for modelling complex systems. In order to achieve this aim, we have identified five sub-themes (the theory of complex networks, the theory and application of time series analysis, mathematical modelling of the brain and the life systems, mathematical modelling for diseases, and systematization of mathematical modelling for complex systems), and are now conducting research by maintaining close collaboration among researchers working on the various sub-themes.