Lecturer: Piero Colli Franzone  

Course name: Dynamical Systems: theory and numerical methods
Course code: 502886
Degree course: Bioingegneria
Disciplinary field of science: MAT/08
University credits: CFU 6
Course website: n.d.

Specific course objectives

The course introduces the main concepts related to qualitative and quantitative study of solutions of ordinary differential systems providing the main analytical and numerical methods for the investigation of the dynamics of mathematical models and the critical interpretation of the numerical results.

Course programme

The course is an introduction to the solvability of initial value problem for ordinary differential systems and to the investigation of the qualitative properties of solutions and of equilibrium points with their asymptotic behaviour. The course develops the numerical methods for the numerical simulation of dynamical systems with applications to population dynamics and bistable models.

Basic notion of linear algebra and analysis
Vectorial spaces, matrices, eigenvalues, eigenvectors, linear differential equations, differential and integral calculus, vectorial Taylor development.

Introduction to initial value problems for ordinary differential equations
Local and global solvability, continuous dependence on the initial data, parameters and right hand side perturbations.

Asymptotic Stability
Stability of solutions and of equilibrium points. Linear systems. Stability of the linear autonomous systems based on the spectral abscissa. Nonlinear system: linearization. Nonlinear system: Liapunov function. Two dimension linear system and global analysis of the phase plane.

Basic notions of numerical analysis
Polynomial interpolation and remainder terms. Numerical integration: Newton-Cotes formulae and Gausian quadrature. Functional iteration for a system of nonlinear equations: explicit iteration scheme and Newton method.

Numerical methods for ordinary differential systems
One step methods: consistency, zero-stability and convergence. Runge-Kutta methods based on numerical quadratures, Runge-Kutta methods based on collocation methods. Linear multistep methods: consistency, zero-stability and convergence. Adams Bashforth and Moulton methods, Predictor-Corrector methods, backwords differentiation formulae. Estimators of the local discretization error and adative strategy of the time step. Test problems and region of absolute stability. Stiff problems.

Introduction to bifurcation involving fixed points and limit cycles in biological systems.

Analysis and Simulation of dynamical systems: Lotka-Volterra model, FitzHugh-Nagumo model.

Course entry requirements

Basic mathematical courses of the "laurea triennale" or " undergraduate degree" and or "bachelor degree"

Course structure and teaching

Lectures (hours/year in lecture theatre): 36
Practical class (hours/year in lecture theatre): 12
Practicals / Workshops (hours/year in lecture theatre): 12

Suggested reading materials

F. Verhulst.. Nonlinear differential equations and dynamical systems.. Springer-Verlag,Heidelberg, 2006..

R. Mattheij, J. Molenaar . Ordinary differential equations in theory and practice.. SIAM, Philadelphia, 2002..

M. Crouzeix, A.L. Mignot . Analyse Numèriques des Èquations Diffèrentielles. Masson, Paris 1984..

A.M. Stuart , A.R. Humphries . Dynamical Systems and Numerical Analysis. . Cambridge University Press 1998..

A. Quarteroni, R. Sacco, F. Saleri. . Matematica Numerica.. Springer 3ra ed., 2008..

Testing and exams

Oral examination with discussion and interpretation of the models simulations developed in the laboratory.