Numerical Methods for Differential Equations and Dynamical Systems

Cluster

Numerical Analysis

Group's website

http://cdlab.uniud.it/

Description

The research activity is developed within the context of numerical methods for simulating differential equations and for studying the dynamical systems they generate. The classes of equations the group is interested in include not only the classical ordinary differential equations, but also functional equation having a differential, integral, or integro-differential character, as well as some partial derivative models having non-local diffusion or reaction terms.

Within this scope, methods for numerical integration in time are studied, with particular regard to the convergence and stability properties, as well as to the more typically computational aspects such as efficiency and implementation. The application scope is extended from the more industrial fields, related with processes for the control of population dynamics with reference to recently developed complex systems, both as to ecology and for the study of epidemics, e.g., resource-consumer systems.

With regard to these themes, the study of the stability of particular solutions, such as equilibria and periodic orbits plays a key role. For many of the classes of the equations listed above, the state space can be aptly represented by a Banach space of functions, thus generating dynamical systems having infinite dimensions. The problem of stability often brings about the study of spectral characteristics concerning infinite-dimensional operators, whence the interest for developing the necessary techniques for discretisation as direct methods for analysing the long-term behaviour. Discretisations for studying chaotic dynamics also fall within this approach. If the existing systems depend upon variable or uncertain parameters, efficient techniques are employed for the bifurcation analysis, such as continuation methods, or techniques based on the polynomial chaos theory.

The group activity isn't restricted to the purely theoretical aspects of the methods it develops, but it also deals with the relevant problems of a computational nature linked both with software aspects (implementation and coding, e.g., Matlab) and with application-related requests (I/O relationship and interaction with non-specialised users).

The group has well-established relations chiefly with the University of Trieste and the University of Trento and, on the international level, with the Universities of Utrecht, Helsinki and York. The research activities described above fall within the goals of the CDLab - Computational Dynamics Laboratory, whose site (http://cdlab.uniud.it/) is referred to for more information.

Research subjects

Methods for the numerical integration of differential, integral and integro-differential equations, including functional equations
Numerical methods for the spectral approximation of operators and the analysis of the stability of equilibria and periodic orbits
Methods for studying chaotic dynamics in infinite-dimensional dynamical systems
Methods for the efficient continuation of complex systems and the related bifurcation analysis
Polynomial chaos methods for the analysis of dynamics in presence of uncertain parameters
Stability analysis of complex systems in the control theory and population dynamics