Academic Year 2022/2023 - Teacher: Andrea RAPISARDA

Expected Learning Outcomes

The course aims to present a broad overview of models and of statistical and numerical techniques for the study and characterization of complex phenomena, of physical, biological and socioeconomic kind.

More specifically  the objectives of the course are:

Critical understanding of the most advanced developments of Modern Physics, both theoretical and experimental, and their interrelations, also across different subjects.

Adequate knowledge of advanced mathematical and numerical tools, currently used in both basic and applied research.

Remarkable acquaintance with the scientific method, understanding of nature, and of the research in Physics.

Ability to identify the essential elements in a phenomenon, in terms of orders of magnitude and approximation level, and being able to perform the required approximations

Ability to use analytical and numerical tools, or science computing, including the development of specific software.

Ability to discuss about advanced physical concepts, both in Italian and in English.

Ability to present one's own research activity or a review topic both to an expert and to an non-expert audience.

Ability to acquire adequate tools for the continuous update of one's knowledge.

Ability to access to specialized literature both in the specific field of one's expertise, and in closely related fields.

Ability to exploit databases and bibliographical and scientific resources to extract information and suggestions to better frame and develop one's study and research activity.

Course Structure

Lectures and excercises in the classroom

Note: should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

Required Prerequisites

No one

Attendance of Lessons

It is compulsory to attend the lessons in the classroom

Detailed Course Content

Determinism and predictability. Deterministic chaos and sensitivity to initial conditions. Iterative maps and Hamiltonian systems. Lyapunov exponents. Kolmogorov-Sinai entropy. Strange attractors and fractal dimensions. KAM theorem. Chaos and complexity. Emergency, interdependence and self-organization. Examples of complex systems of various kinds: turbulent fluids, financial and economic systems, biological, geological and social systems. Models and numerical techniques for a quantitative study. Generalized Statistics. Superstatistics. Self-organized criticality. Methods of time series analysis. Cellular automata. Agent-based models. Models of opinion dynamics and synchronization. Efficiency of random strategies. Techniques and algorithms for numerical simulations. Complex networks. Random networks, small-world and scale-free. Characterization and main measures of centrality of complex networks.

Textbook Information

R.C. Hilborn Chaos and Nonlinear Dynamics Oxford University Press (1994)

J.C. Sprott: Chaos and Time-series Analysis,, Oxford University Press (2003)

E. Ott: Chaos in Dynamical systems, Cambridge University Press (1993)

F. R. Badii e A. Politi: Complexity, Cambridge University Press (1997)

Y. Bar-Yam: Dynamics of Complex systems, Westview press (1997)

Z. R.N. Mantegna e H.E. Stanley: An introduction to Econophysics, Cambridge University Press (2000)

H. Kantz e T. Schreiber : Nonlinear Time Series Analysis, Cambridge University Press (2000) S.N. Dorogovtsev e J.F.F. Mendes: Evolution of Networks,, Oxford University Press (2003) 

L. Barabasi, Network Science, Cambridge University Press (2016)

Course Planning

 SubjectsText References
1Deterministic chaos R.C. Hilborn : Chaos and Nonlinear Dynamics Oxford University Press (1994); J.C. Sprott: Chaos and Time-series Analysis,, Oxford University Press (2003)
2Emergence and self-organization in complex systemsY. Bar-Yam: Dynamics of Complex systems, Westview press (1997)
3Cellular automata and agent-based modelsOriginal papers available in Studium
4Complex Networks L. Barabasi, Network Science, Cambridge University Press (2016)


Learning Assessment Procedures

Preparation of a short written dissertation on one of the topics of the program for a general oral discussion on the main topics presented in class

The criteria adopted for the evaluation are: the relevance of the answers to the questions asked, the level of in-depth analysis of the contents presented, the ability to connect with other topics covered by the program and with topics already acquired in previous years' courses, the ability to report examples, language properties and expository clarity.

Note: Verification of learning can also be carried out electronically, should the conditions require it.

Examples of frequently asked questions and / or exercises

The following questions are only a few examples. 

  • Discuss deterministic chaos and explain Lyapunov's exponents
  • Explain the self-organized criticality
  • Discuss the difference between chaos and complexity
  • Explain the phenomenon of synchronization
  • Explain the phenomenon of emergence in complex systems
  • Discuss the difference between a random network and one with no scale or a small world network