Dipartimento di Sistemi e Informatica
Applied Mathematics Research Group
Resp. Prof. R. Johnson
The Applied Mathematics
Group carries out research on a wide range of problems in the area of differential
equations and dynamical systems with particular emphasis on their role in
applied situations. Many of these problems concern the properties of solutions
of non-autonomous differential equations, where it is understood that such
equations are studied using tools of ergodic theory and topological dynamics
(such as Lyapunov exponents, exponential dichotomies, and rotation numbers),
together with more classical methods such as the topological degree (when
the time-dependence happens to be periodic). Some specific themes that have
been taken up by the Applied Mathematics Group in the last three years and
which are still under active study are the following.
1.Problems of homoclinic bifurcation in singularly perturbed differential equations; these are studied using the exponential dichotomy method.
2.Periodically driven partial differential equations in thin domains, including the Navier-Stokes equations; these are studied using the topological degree.
3.Sliding mode control problems; among various methods we use the topological degree.
4.Random bifurcation theory, with particular emphasis on a random version of the Duffing-van der Pol oscillator.
5.Spectral theory of random differential operators, in particular the Cantor spectrum problem for the quasi-periodic Schroedinger operator.
6.Control theory of randomly driven systems.
7.Random polynomials, with special emphasis on a class of these polynomials which can be described using the classical theory of algebraic curves.
The Group collaborates with colleagues from the following universities: (Italy) Ancona, Milano, Siena; (foreign) Valladolid (Spain), Georgia Tech (USA), Rutgers (USA). The members of the Group at the present time are: R. Johnson (professor), R. Fabbri (post-doctoral researcher), F. Mantellini (doctoral student), E. Zoli (doctoral student).
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