# DYNAMICAL SYSTEMS

**Academic Year 2018/2019**- 3° Year - Curriculum APPLICATIVO

**Teaching Staff:**

**Giuseppe MULONE**

**Credit Value:**6

**Scientific field:**MAT/07 - Mathematical physics

**Taught classes:**35 hours

**Exercise:**12 hours

**Term / Semester:**2°

## Learning Objectives

1. To give the base elements of continuous and discrete dynamical systems

2. Starting from the real world models to build mathematical models.

3. Analysis and interpretation of the results.

4. Study of the dynamic flow systems: equilibria and their stability, attractors.

Knowledge and understanding: students must understand theory and proofs of fundamental theorems of dynamic systems; to know and understand applications of mathematical results to problems of applied sciences; to demonstrate mathematical skills in reasoning with the simple construction of real models.

Applying knowledge and understanding: to demonstrate known mathematical results by new techniques, construction of particular dynamic systems from known models; be able to formalize mathematically problems of moderate difficulty, formulated in the natural language, and to take advantage of this formulation to clarify or solve them. The ability to apply knowledge and understanding will be achieved through a method of teaching always focused on the logical-deductive method and the presentation and analysis of the most important mathematical models of applied sciences.

- Making judgements: Students at the end of the course must be able to propose, analyze and compare mathematical models associated with concrete situations of moderate difficulty arising from other disciplines and to use such models to facilitate the study of the original situation.

- Communication skills: to be able to present materials and scientific arguments in a clear and comprehensible manner, also by means of simple multimedia tools; be able to work in groups and operate with defined degrees of autonomy.

- Learning Skills: Having developed the skills needed to build simple models with autonomy; to have a flexible mentality,.

## Course Structure

Lectures in class and exercises done by students at home and in class.

## Detailed Course Content

Introduction to discrete and continuous finite-dimensional dynamical systems.

Linear and nonlinear dynamical systems

Equilibria and their stability

Periodicity and chaos

Fractals

Complete programme here:

http://www.dmi.unict.it/~mulone/programma_sistemidinamici1819.pdf

## Textbook Information

1. E. Scheinerman, Invitation to Dynamical Systems, testo disponibile online: http://www.ams.jhu.edu/∼ers/invite/book.pdf

2. L. Perko, Differential equations and dynamical systems, 3rd ed. - New York: Springer-Verlag, 2001.

3. M. W. Hirsch, S. Smale, Differential equations, dynamical systems, and linear algebra, New York: Academic Press, 1974.

4 E. Salinelli, F. Tomarelli, Modelli dinamici discreti, Milano: Springer-Verlag Italia, 2002.

5. S.H. Strogatz, Nonlinear dynamics and caos, Westview, Cambridge, MA, 2000.

6. A. Milani comparetti, Introduzione ai sistemi dinamici, Ed. Plus, Pisa, 2002

7. G. Mulone, Appunti di sistemi dinamici, 2002.

8. S. Wolfram, Software Mathematica (downloadable for free by the students of UNICT)