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 order periodic equations. Mathieu's equation.Hill's equation
Floquet theory.Hill's equations with quasi-periodic forcing.
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
CFU: 9
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 4237117 Fax: 055 4237165
gabriele.villari@unifi.it, gabriele.villari@math.unifi.it
* Roberta Fabbri
Thuesday 15:00-17:00, or by appointment
Dipartimento di Matematica e Informatica "Ulisse Dini",Via Santa Marta, 3 50139-Firenze. Tel: 055 4796496
roberta.fabbri@unifi.it
Type of Assessment
Oral
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.