Course teached as: B018801 - EQUAZIONI DIFFERENZIALI ORDINARIE Second Cycle Degree in MATHEMATICS Curriculum DIDATTICO
Teaching Language
Italian
Course Content
Cauchy problem.Systems of differerential equations The phase
plane.Linear systems. Qualitative theory.The logistic, competition and
predator-prey models.Models in epidemiology: SIS, SIR and mixed. The
Van Der Pol and Duffing equations. Liénard equation and his
phase-portrait.Massera's Theorem.
Asymptotic behavior of the solutions of a differential
equation.Oscillation theory for second order ODE's. Sturm's separation
and comparison theorems. Sturm-Liouville problems. Stability zones
for II ord
W. Boyce-R. Di Prima Elementary differential equations and boundary value problems. Wiley.
M. Iannelli Appunti di dinamica di popolazioni. Università di Trento.
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.
Learning Objectives
Basic concepts of the theory of differential equations.
Knolewdge of the standard models in dynamic of populations and epidemiology. 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:
Gabriele Villari
Thursdays, 15:00-17:00, or by appointment.
Dipartimento di Matematica "Ulisse Dini"
Viale Morgagni, 67/a
50134 - Firenze (FI)
Tel: 055 4237165
gabriele.villari@unifi.it
* Roberta Fabbri
Thuesday 15:00-17:00, or by appointment
Dipartimento di Matematica e Informatica "Ulisse Dini",V.le Morgani 67/A, 50134 Firenze. Tel: 055 2751430
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 some mathematical model will be part of this process, and students may choose a particular topic of the whole course, which will be discussed in details in order to verify their autonomy in treating a mathematical subject.
Course program
Cauchy problem. Non uniqueness of solutions. Peano example. Persistence of solutions. Maximal solutions Gronwall’s lemma. Systems of differerential equations The phase plane. Singular points. Quasi linear systems. Qualitative theory. Dynamic of populations. The logistic model. Competition. The principle of exclusion. Predator-prey model Volterra’s theorem. Models in epidemiology: SIS, SIR and mixed. The Van Der Pol equation and his phase-portrait. Existence and uniqueness of limit cycles. Liénard equation and his phase-portrait. Massera’s Theorem.
Asymptotic behavior of the solutions of a differential equation.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.