Dynamical Systems and Applications

Cluster

Dynamic Systems

Description

The research activity is characterised by the multi-disciplinary nature of its diverse components, which at the same time amounts to a factor for synergy and innovation. The individual competences realise a potential non-trivial intersection of tools and results, an intersection developing along the main lines described below, ranging from the more theoretical and abstract aspects to the more numerical and application-related ones.

In the systems generated by differential equations, interesting for wide areas of application in physics, engineering, biology and economics, it is important to determine the presence of stationary and periodic solutions, as well as to study their stability properties. The research in the field of fixed points and periodic points, therefore, has a significant impact on those sectors, and it avails itself both of rigorous tools of mathematical analysis having a topological and variational nature, and of efficient numerical methods for the simulation and the analysis of robustness when uncertain parameters vary. Within this context, finite dimensional problems are studied, e.g. extensions of the Poincaré-Birkhoff Theorem, and complex systems on Banach spaces, e.g. delay differential equations and physiologically structured populations.

Alongside these problem types, there is that of the presence of deterministic chaos, that is the possibility that a mathematical model that is entirely determined in the short term evolves in an essentially unpredictable way in the long term. A classical example, which also is a subject of study by the group, is that of the Gauss-type maps induced by continued fraction algorithms.

Within the more topological scope, after the measure-theoretic entropy of Kolmogorov and Sinai in the ergodic theory, several notions of topological entropy have been introduced in topological dynamics and of algebraic entropy for endomorphisms of topological groups. Our interest is focused, therefore, on studying such entropies, in particular on researching a general approach unifying the different entropies by using tools of the category theory. Then, with an orientation towards a field more closely linked with applications in mechanics and industry, particular attention is paid, on the one hand, to studying non-local constants of motion in the spirit of Noether's Theorem for variational and non-variational Lagrangian systems, and on the other hand to structurally studying dynamical networks, that is complex systems whose dynamical behaviour is strongly affected by the type, quantity and intensity of the interactions among the various state variables.

Research subjects

Fixed points, periodic points and chaos dynamics for low-dimensional (2 or 3) dynamical systems
Numerical methods for studying infinite-dimensional dynamical systems
Topological and variational approaches for the generalisation of the Poincaré-Birkhoff Theorem
Study of entropies in mathematics and unifying approach through the category theory
Gauss-type maps induced by continued fraction algorithms
Study of non-local constants for variational and non-variational lagrangian systems
Structural study of dynamical networks in complex systems