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

Basic definitions of dynamical systems: generalities and standard examples. Stability: general concepts. Assessing stability via linearization or Lyapunov functions. Periodic solutions. Poincaré-Bendixson 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.If time permits: elements of entropy in dynamical systems, concepts of predictability, and intermittency.
Throughout the course, the theoretical topics will be complemented with numerical exercises and applications, including:

• applications of linear stability analysis to geo-fluids, such as Kelvin–Helmholtz, Rayleigh–Bénard, and baroclinic instabilities;
• numerical illustration and analysis of bifurcations using the Stommel box model and energy-balance models like Budyko-Sellers;
• the Lorenz 63 model;
• the two-component fast-slow Lorenz 63 model (Pena & Kalnay);
• the Lorenz 96 model.

Readings/Bibliography

• Lecture notes and slides provided by the teachers
• S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. 3rd rd. ‎ Chapman and Hall/CRC, 2024
• Instability in Geophysical Flows, William D. Smyth and Jeffrey R. Carpenter (2019)
• Mathematics & Climate, Hans Kaper and Hans Engler (2013)

Teaching methods

Classroom lectures and python numerical illustrations and examples. 

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 Natale Alberto Carrassi

See the website of Giovanni De Cillis