### Teachings

Select Academic Year:     2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022
Professor
CORNELIS VICTOR MARIA VAN DER MEE (Tit.)
Period
Second Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/40 - Ord. 2020]  MATEMATICA PURA 6 48

### Objectives

To acquire an operational knowledge of the nonlinear partial differential equations of stationary or evolution type relevant to mathematical physics, in particular dynamical systems and integrable equations. For this reason some of the arguments are illustrated with examples and exercises without formal distinction between the two.

### Prerequisites

The course requires a good knowledge of the basic concepts of mathematical analysis and linear algebra of the first two years of the undergraduate degree.

### Contents

1. Hamilton equations: Symplectic formulation, canonical transformations, Poisson and Lagrange brackets, lagrangian and hamiltonian for continuous systems.

2. Equilibrium points: Autonomous systems, Lie derivative, classification of equilibrium points, examples [damped pendulum, etc.].

3. Lyapunov stability: Definition, stability of solutions of y'=Ay. Lyapunov and Perron theorems, examples [pendulum, rotating pendulum, oscillations with dissipation, Lotka-Volterra].

4. Stability of discrete systems: cycles, contraction mapping theorem, examples [Newton-Raphson, logistic map, Bernouilli shift, Mandelbrot set, Collatz conjecture], billiards, Sarkovskii theorem.

5. Bifurcations and limit cycles: Bendixson criterion,
Poincare-Bendixson theorem, Hopf bifurcations, examples [Van der Pol, Verhulst logistic model, Lorenz oscillator].

6. Fractals: Cantor set and variants, characteristics of fractals, Hausdorff dimension.

7. Integrable equations: History, AKNS pairs, inverse scattering transform for the Korteweg-de Vries and nonlinear Schroedinger equations, transformations between certain integrable equations.

### Teaching Methods

Lectures, problem sessiona, and correction of problems. Online using Microsoft Teams and Samsung Notes.
Team: Sistemi Dinamici, codice: 2rirqca

### Verification of learning

The course is assessed by means of an oral exam to be held online using Microsoft Teams or by means of a meeting teacher-student.

### Texts

Primarily the online lecture notes [krein.unica.it/~cornelis/DIDATTICA/FISMAT2/fismatdue20.pdf]. For additional consultation,
1. C. Lanczos, The variational Principle of Mechanics, fourth ed., Dover Publ., New York, 1970.
2. Notes from the teacher [available at the following link: http://krein.unica.it/~cornelis/DIDATTICA/FISMAT2/fonfismat20.pdf].
3. C. van der Mee, Nonlinear Evolution Models of

Integrable Type, SIMAI e-books, Vol. 11, 2013 [http://krein.unica.it/~cornelis/RICERCA/PAPERS/190.pdf].