First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Engineering
MATHEMATICAL ENGINEERING
Course unit
DYNAMICAL SYSTEMS (MOD. B)
INP5070521, A.A. 2017/18

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course Second cycle degree in
MATHEMATICAL ENGINEERING (Ord. 2017)
IN2191, Degree course structure A.Y. 2017/18, A.Y. 2017/18
N0
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Degree course track MATHEMATICAL MODELLING FOR ENGINEERING AND SCIENCE [001PD]
Number of ECTS credits allocated 6.0
Type of assessment Mark
Course unit English denomination DYNAMICAL SYSTEMS (MOD. B)
Department of reference Department of Civil, Environmental and Architectural Engineering
Mandatory attendance No
Language of instruction English
Branch PADOVA

Lecturers
Teacher in charge MASSIMILIANO GUZZO MAT/07

Integrated course for this unit
Course unit code Course unit name Teacher in charge
INP5070520 MATHEMATICAL PHYSICS (C.I.) FRANCO CARDIN

Mutuating
Course unit code Course unit name Teacher in charge Degree course code
SCN1032593 MATHEMATICAL PHYSICS MASSIMILIANO GUZZO SC1173

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/07 Mathematical Physics 6.0

Mode of delivery (when and how)
Period Annual
Year 1st Year
Teaching method frontal

Organisation of didactics
Type of hours Credits Hours of
teaching
Hours of
Individual study
Shifts
Lecture 6.0 48 102.0 No turn

Calendar
Start of activities 02/10/2017
End of activities 15/06/2018

Examination board
Examination board not defined

Syllabus

Common characteristics of the Integrated Course unit

Prerequisites: None
Target skills and knowledge: Objective
Introduce the students to mathematical tools in continuum mechanics and dynamical systems.
Outcomes
A student who has met the objectives of the course will have a basic knowledge of :
• advanced topics in the mathematical description of continuous mechanics
• fundamentals of ODEs and dynamical systems
Examination methods: Final examination based on: Written and oral examination.
Assessment criteria: Critical knowledge of the course topics. Ability to present the studied material. Discussion of the student project.

Specific characteristics of the Module

Course unit contents: 1. Ordinary differential equations: Cauchy theorem, phase-space flow, dependence on the initial conditions; linear equations; phase-portraits, first integrals; equilibrium points; linearizations, stable, center and unstable spaces.

2. Integrable systems: elementary examples from population dynamics, from Mechanics and from Astronomy; integrability of mechanical systems, action-angle variables, examples.

3. Non-integrable Systems: discrete dynamical systems, Poincare' sections; bifurcations, elementary examples. Stable and Unstable manifols, homoclinic chaos; Lyapunov exponents, the forced pendulum and other examples; Center manifolds and partial hyperbolicity. The three body-problem, the Lagrange equilibria,
Laypunov orbits, the tube manifolds.

THE FOLLOWING TOPICS (4) AND (5) ARE ONLY IN THE PART FOR THE STUDENTS OF THE SECOND CYCLE DEGREE IN ASTRONOMY

4. Linear PDEs of first and second order, well-posed problems,
the vibrating string, 1-dimensional wave equation, normal modes of vibrations, heat equation, Fourier series, 2-dimensional wave equation, Laplace operator and polar coordinates, separation of variables, Bessel functions, eigenfunctions of the Laplacian operator.

5. Laplace operator and spherical coordinates, separation of variables, Legendre polynomials and associate functions, Spherical harmonics, multipole expansions, L2 operator-eigenvalues and eigenfunctions, complete solution of the wave equation in space, Schrodinger polynomials.

THE FOLLOWING TOPICS (6) ARE ONLY IN THE PART FOR THE STUDENTS OF THE SECOND CYCLE DEGREE IN MATHEMATICAL ENGINEERING

6. Examples and Applications: examples of analysis of three and four dimensional systems; limit cycles; the Lorenz system, the three-body problem; examples from fluid dynamics, non autonomous dynamical systems, chaos indicators, Lagrangian Coherent Structures.
Planned learning activities and teaching methods: Classroom lectures and exercises.
Additional notes about suggested reading: Lecture notes.
Textbooks (and optional supplementary readings)