### Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
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 [65/00 - Ord. 2012]  PERCORSO COMUNE 6 48
[60/68]  PHYSICS [68/00 - Ord. 2014]  PERCORSO COMUNE 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.

### 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], 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.

### 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], 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.

### Teaching Methods

Lectures will be given by using either chalk and blackboard or slides.

### Verification of learning

The course is assessed by means of an oral exam.

### Texts

Primarily the online lecture notes [krein.unica.it/~cornelis/DIDATTICA/FISMAT2/fismatdue18.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/FONDAMENTI2/fonfismat13.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].

The main tool to support teaching is the teacher personal web site (http://bugs.unica.it/~cornelis/DIDATTICA/FISMAT2). It provides information updated in real time, including: information on teaching activities, additional documents to support learning and grading tests.

In case of a difference of opinio between the italian and the english text, the italian text prevails.

In the course we deal with nonlinear methods in mathematical physics, contrary to the course of Fundamentals of Mathematical Physics where linear methods prevail.