Course teached as: B018801 - EQUAZIONI DIFFERENZIALI ORDINARIE Second Cycle Degree in MATHEMATICS Curriculum DIDATTICO
Teaching Language
Italian
Course Content
Asymptotic behavior of the solutions of a ODE.Oscillation theory for 2^order ODE's. Sturm's separation
and comparison theorems. Sturm-Liouville problems. Autonomous and nonautonomous linear systems in R^n. Hill's equation. Floquet theory. Quasi-periodic case. Schroedinger equation. Rotation number and oscillation theory. Nonautonomous dynamics. Skew-product flows, exponential dichotomy. Dichotomy spectrum. Spectral Theorem (Sacker Sell)
W. Boyce-R. Di Prima Elementary differential equations and boundary value problems. Wiley.
E. Coddington, N. Levinson,"Theory of Ordinary Differential Equations", McGraw-Hill 1955 (capitolo VIII).
W. Magnus, S.Winkler, "Hill's Equations", Interscience Publishers, New York- London-Sidney 1966.
F. Colonius, W. Kliemann, "Dynamical Systems and linear Algenra", Graduate Studies in Mathematics, vol. 158, AMS, Providence, Rhode Island, 2014.
Learning Objectives
Basic concepts of the theory of differential equations.
Knolewdge of the main results in the qualitative theory of planar dynamical systems.
Capability of analyzing a mathematical model. Skills necessary for the study of dynamical planar systems.
Prerequisites
Courses to be used as requirements (required and/or recommended)
Courses required: Mathematical Analysis II.
Courses recommended: None
Teaching Methods
Lectures.
9CFU
Contact hours for: Lectures (hours): 72
Further information
Frequency of lectures, practice and lab: Recommended
Teaching Tools UniFi E-Learning: http://e-l.unifi.it
Office hours:
two hours a week to fix with the students or by appointment
Dipartimento di Matematica e Informatica "Ulisse Dini",V.le Morgani 65 , 50134 Firenze. Tel: 055 2751469
roberta.fabbri@unifi.it
Type of Assessment
There will be an oral examination concerning the subjects treated in the course. Students are requested to show their effective understanding on the topics of qualitative theory of dynamical systems. The critical discussion of the presented results, as well as their proves, will be a crucial part of the examination. Analysis of a particular topic of the whole course will be discussed in details in order to verify the autonomy in treating a mathematical subject.
Course program
Oscillation theory for second order ODE's. Sturm's separation and comparison theorems. Sturm-Liouville problems. Stability zones for II order periodic equations. Mathieu's equation.Hill's equation Floquet theory.Hill's equations with quasi-periodic forcing. Schroedinger equation. Rotation number and oscillation theory. Quasi-periodic linear systems. Elements of topological dynamics and dynamical systems. Flows, skew-product flows. Lyapunov exponents. Exponential dichotomy: hyperbolic decomposition of the product space. Dynamical spectrum, Sacker-Sell Spectrum. Spectral Theorem of Sacker Sell.