Nonlinear dynamics

Research focus

The Peer review has evaluated this group as Good

The research activity has been mainly performed through application-oriented studies in the broad area of nonlinear dynamics. From a technical point of view, the focus is on the study of the attractors of parameterized families of finite-dimensional nonlinear systems. The target is typically to classify all the qualitatively different system behaviors obtainable by varying the parameters. This is accomplished by performing the so-called bifurcation analysis of the differential or difference equations describing the system. Original contributions have been obtained on the theoretical aspects of bifurcation analysis and related numerical methods, as well as on the use of special properties of nonlinear dynamics for solving problems of analysis, identification, and control. However, the greatest value of the contributions in this area stands in the specific results obtained in each application. Although the applications fall in areas of radically different nature, ranging from chemistry and biology to economics and social sciences, from an abstract point of view each contribution is characterized by the study of one or two of the following general aspects of nonlinear dynamical systems.
Analysis of slow-fast systems: Systems with compartments characterized by strongly diversified dynamics call for singular perturbation approaches, in which the fast compartment is analyzed for frozen values of the slow variables. This is nothing but the bifurcation analysis of the fast compartment with respect to the slow variables treated as parameters.
Order reduction of chaotic systems: Close to the critical bifurcations separating chaotic from regular behaviors, successive output peaks of n-dimensional chaotic systems are simply related one to each other. More precisely, any peak can be approximately computed from the previous peak using a first-order nonlinear discrete-time model. This order reduction greatly simplifies the analysis of the system without any serious loss, since in most applications the output peaks are indeed the variables of greatest concern.
Switching between alternative attractors: Multiple attractors with radically different associated performances characterize many nonlinear systems. The problem of finding the control actions that force the switch from a poor to a high-quality attractor is of crucial importance. This is a typical nonlinear control problem “in the large” that can be effectively solved through bifurcation analysis.
Discontinuous systems: Nonlinear discontinuous systems (Filippov, impact, hybrid, and switching systems) are of relevant interest in applications. They raise a number of still open problems, like the classification of the bifurcations critically involving the discontinuities.
Evolution of adaptive systems: Adaptive systems are composed of units whose performance is determined by unit-specific characteristic traits. Competition among units selects the bestperforming traits, while innovation inject new traits in the system. Fast competition processes and relatively low innovation rates result in the slow evolution of the traits. All this can be formally described by suitable nonlinear systems, where fast compartments describe competition and slow compartments trace evolution. Bifurcation analyses of these two compartments allow explaining evolutionary phenomena of paramount interest in applications, such as the diversification between coexisting innovative and resident traits and the disappearance of low-performing traits.
Networks and emergence of collective dynamics: The study of networks composed of similar interconnected systems is of crucial interest in applications, where the emergence of collective 260 dynamics (such as clusters and pattern formation, synchronization, network coherence) has relevant consequences. All above network properties depend on the parameters of the single system as well as on the parameters describing the network topology and the coupling strength, so that complex bifurcation problems are involved. Part of the research activity has been carried out in cooperation with foreign institutions, including: International Institute for Applied Systems Analysis, Laxenburg, and Technical University of Vienna (Austria); Universities of Utrecht and Wageningen (The Netherlands); Ecole Normale Supérieure, Paris (France); Cornell University, NY, and Princeton University, NJ (USA).

Dipartimento di afferenza

Dipartimento di Elettronica e Informazione (DEI)

Docenti afferenti

Carlo Piccardi (full professor)
Sergio Rinaldi (full professor)
Renato Casagrandi (associate professor)
Fabio Dercole (assistant professor)
Alessandra Gragnani (assistant professor)
Massimo Miari (assistant professor)