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77802 - Complex Systems

Academic Year 2023/2024

                
                        Docente:
                        Marco Lenci
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/07
                    
                        Language:
                        English
                    
                        Moduli:
                        
                            Natale Alberto Carrassi
                            (Modulo 1)
                        
                            Marco Lenci
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    Traditional lectures (Modulo 1)
                                
                                    Traditional lectures (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            Second cycle degree programme (LM) in
                            Science of Climate (cod. 5895)

                            Teaching resources on Virtuale
                        
                                        Course Timetable
                                    
from Oct 09, 2023 to Dec 18, 2023

                                        Course Timetable
                                    
from Sep 25, 2023 to Nov 21, 2023

Learning outcomes

At the end of the course, the student possesses the basic knowledge on complex physical, biological and social systems and on the means of analysis, predictability and control. In particular, the student is able to: - solve problems of deterministic chaos and predictability; - solve problems of emerging self-organization; - develop a project with object-oriented architecture and advanced graphics, using agent models and network theory.

Course contents

Slightly tentative syllabus:

Basic definitions of dynamical systems: generalities and standard examples. Stability: general concepts. Assessing stability via linearization or Lyapunov functions. Periodic solutions. Poincaré-Bendixon theorem. Stable and unstable manifolds. The statistical point of view for deterministic dynamical systems. Elementary ergodic theory. Hyperbolicity as the main ingredient for chaos. Attractors and basins of attraction. Structural stability. Bifurcations. In time permits: Elements of entropy in dynamical systems. Concepts of predictability. Intermittency.

Readings/Bibliography

Lenci's part:

The course will draw from no specific text. There are some reference texts for various topics and for further readings:

P. Walters, An introduction to ergodic theory, Springer
B. Hasselblatt & A. Katok, A first course in dynamics, Cambridge U. Press
V. I. Arnold & A. Avez, Ergodic problems of classical mechanics, Addison-Wesley

Teaching methods

Classroom lectures

Assessment methods

The final assessment will be in the form of an oral exam (30-60 mins) where the student will be asked a number of questions aimed at evaluating the student's degree of understanding of the concepts, methods, and problems explained in the course. The student will also be asked to solve at least a (simple) exercise.

Teaching tools

Notes handed by the instructor.

Office hours

See the website of Marco Lenci

See the website of Natale Alberto Carrassi