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Investigation into Complex Systems

Admission Requirements
  • Knowledge of the principles of equilibrium thermodynamics
  • Knowledge of the principles of Calculus: derivatives and integrals

Learning Outcomes

Following the successful completion of the course, the following main learning outcomes are expected:


  • The principles of out-of-equilibrium thermodynamics
  • The principles of non-linear dynamics.
  • The limitations we encounter in understanding some emergent properties of Complex Systems and the reasons for these limitations.
  • The possible strategies to face these limitations of understanding emergent properties of complex systems.


  • Describe the behavior of Complex Systems.
  • Model the behavior of Complex Systems.
  • Predict the behavior of certain Complex Systems.


  • Face the challenges of the XXI century.
  • Face any global challenge.
  • Face problems involving Complex Systems.


1) Historical introduction to the investigation of Complex Systems.

2) Deepened analysis of the II Law of Thermodynamics to explain the natural irreversibility that every Complex System shows.

3) Complex Systems are out-of-equilibrium thermodynamic systems. Principles of Non-equilibrium Thermodynamics. Flows and Forces. Linear and Non-Linear regimes. Entropy Production and evolution criteria for the out-of-equilibrium systems.

4) Complex Systems exhibit non-linear dynamics. Therefore, it is mandatory to know the linear analysis of the stability of the stationary states: stable and unstable stationary states and oscillations. This analysis allows making predictions as far as the dynamical behavior of Complex Systems is concerned.

5) Oscillatory bio-chemical reactions, chemical waves, Turing structures. Periodic Precipitations. Relevance of these phenomena in nature.

6) Bifurcations and deterministic Chaos.

7) Fractal structures.

8) Natural and Computational Complexities. Strategies to promote worldwide sustainable growth and face the global challenges of this century.

The textbook used for this course is P. L. Gentili, “Untangling Complex Systems: A Grand Challenge for Science”, CRC Press, 2018, ISBN 9781466509429.

Teaching Methodology
Lectures and exercises

ECTS Credits
7.5 ECTS

II semester (STEM)

Examination methodology
The acquisition of the course topics is tested through an oral exam with questions related to the subjects presented during the lectures.

Sustainable development can be achieved if we learn how to manage Complex Systems. This course wants to teach the features of Complex Systems and some methodologies to predict their behaviour.

Existing Course but Revised
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