Research on Dynamics of Complex Systems at Politecnico di Milano concerns several aspects of the theoretical and numerical analysis of nonlinear dynamical systems, both as individual units and as networks of interconnected units.
In the case of isolated systems, the focus is on the use of nonlinear systems theory, in particular bifurcation analysis, to classify the behaviours of the system and to understand the critical transitions occurring when parameters are varied.
In the case of networks of systems, the focus is on the mutual interactions between the topological structure and the emerging collective dynamics. Applications in very diverse fields are considered, including e.g. biology, epidemiology, social sciences, and vehicle dynamics.

Bifurcations in smooth and non-smooth systems: theory and numerical tools
The analysis of the transitions occurring in a system’s behaviour when parameters are varied is a fundamental step in the understanding of its dynamics. It allows one to list the catalogue of qualitatively similar behaviours, providing a powerful tool for analysis, design, and control. Bifurcation theory is the main tool of investigation. The group’s research achievements include significant theoretical advances and software implementations of numerical algorithms.

Modelling and analysis of innovation and competition processes
Innovation and competition processes are ubiquitous in many fields of science.
They are responsible for evolutionary dynamics entrained by innovative changes in the characteristics of individual agents and by competitive interactions promoting the best performances. Genetic mutations and natural selection play such roles in biology, but the scope of the evolutionary paradigm well embraces social sciences, economics, engineering, and computer science.
Methodological and applied contributions have been given to Adaptive Dynamics, one of the most flexible modelling approaches to innovation and competition processes, and the first comprehensive book on the subject has been written.

Interaction of topology and dynamics in complex networks
The collective dynamics of systems over networks may be strongly influenced by the topology of the interconnections. An effective method has been devised for finding and testing community structures (i.e., dense clusters of nodes), with effective applications in economics and finance. Subtle and unexpected interactions between topology and dynamics also characterize contact processes, i.e., epidemic spreading over networks.

Modelling and control of vehicular traffic
Modern cars are equipped with loads of sensors, can control brake, accelerator, and sometimes steering via software, and more and more frequently can communicate with nearby vehicles and roadside infrastructure.  Technology is ripe for attacking the city traffic problem with more efficient tools than the time-space diagrams that have been the basis of most traffic management strategies in the past decades.   We have developed new modelling and control strategies to harness this opportunity.  We are learning to efficiently interact with traffic using vehicles sensors and actuators.  Some of our results have been tested on the field, in collaboration with other research centres worldwide.