77802 - Complex Systems

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

The teacher will also hand out some lecture notes.

Carrassi's part:

• Lecture notes and slides provided by the teacher
• S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. 3rd rd. ‎ Chapman and Hall/CRC, 2024

 

Teaching methods

Classroom lectures

Assessment methods

The final assessment will be in the form of an oral exam (30-60 mins), whose date is to be stipulated between the student and the teachers on a case-by-case basis.

The student will be asked a number of questions aimed at evaluating

• their understanding of the concepts, methods, and problems explained in the course;
• their critical re-elaboration of the material, e.g., their ability to connect different topics among themselves and with other subjects learnt in the Study Program;  
• the clarity and scientific precision of their presentation.

The student will also be asked to solve at least a (simple) exercise.

 

Students with specific learning disabilities (SLD) or other temporary or permanent disabilities should contact the University Office for Students with Disabilities and SLD (https://site.unibo.it/studenti-con-disabilita-e-dsa/en) in a timely manner. The office is responsible for proposing any necessary accommodations to students. These accommodations must be submitted to the instructor for approval at least 15 days in advance and will be evaluated based on the course's learning objectives.

Teaching tools

Notes handed by both teachers.

Office hours

See the website of Marco Lenci

See the website of Natale Alberto Carrassi