DYNAMICAL SYSTEMS: THEORY AND NUMERICAL METHODS
Stampa
Enrollment year
2020/2021
2020/2021
Regulations
DM270
MAT/08 (NUMERICAL ANALYSIS)
Department
DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING
Course
BIOENGINEERING
Curriculum
Sensoristica e strumentazione biomedica
Year of study
Period
1st semester (28/09/2020 - 22/01/2021)
ECTS
6
Lesson hours
56 lesson hours
Language
Italian
Activity type
WRITTEN TEST
Teacher
PAVARINO LUCA FRANCO (titolare) - 6 ECTS
Prerequisites
Differential and integral calculus for function of many variables, vector calculus, matrices. MATLAB programming
Learning outcomes
The course is divided in two parts. DYNAMICAL SYSTEMS: theory and numerical methods (6CFU) and FINITE ELEMENT METHOD AND APPLICATIONS (3CFU).

The first part of 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 contents
DYNAMICAL SYSTEMS: theory and numerical methods
- Basic notion of linear algebra and analysis. Vector spaces, matrices, eigenvalues, eigenvectors.
- Linear differential equations, differential and integral calculus, vector Taylor series.
- 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.
- Complements of numerical analysis for ordinary differential equations.
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 and extensions.
Teaching methods
Lectures +
programming labs using MATLAB and xppaut