
Course unit
DYNAMICAL SYSTEMS (MOD. B)
INP5070521, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2017/18
Integrated course for this unit
ECTS: details
Type 
ScientificDisciplinary 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 
Start of activities 
02/10/2017 
End of activities 
15/06/2018 
Examination board
Examination board not defined
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, phasespace flow, dependence on the initial conditions; linear equations; phaseportraits, 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, actionangle variables, examples.
3. Nonintegrable 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 bodyproblem, 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, wellposed problems,
the vibrating string, 1dimensional wave equation, normal modes of vibrations, heat equation, Fourier series, 2dimensional 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 operatoreigenvalues 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 threebody 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) 


