Checking date: 09/07/2020

Course: 2020/2021

Modeling and simulation of complex systems
Study: Master in Mathematical Engineering (70)

Coordinating teacher: CUERNO REJADO, RODOLFO

Department assigned to the subject: Department of Mathematics

Type: Electives
ECTS Credits: 6.0 ECTS


Competences and skills that will be acquired and learning results.
* Global vision over complex systems and emergent behavior. * Ability to model complex phenomena in simple terms which allow to capture their main qualitative aspects. * Become acquainted with standard tools employed in interdisciplinar research. * Understand the relation between a system complexity and that of the models employed to study it. * Understand basic concepts of thermodynamics and statistical mechanics as a framework of choice to study systems composed by a large numbers of agents. * Understand the concept of emergent properties or global behavior which can not be directly inferred from single agent properties. * Familiarity with basic notions and tools on critical phenomena as a paradigm of transitions among different emergent behaviors. * Familiarity with basic phenomenology of nonlinear systems, in particular with the ideas of stability and bifurcation. * Understand the implications of deterministic chaos as long-term umpredictability, different from stochasticity. * Employ fractal dimensions as set-characterizing tools. * Familiarity with the use of basic numerical tools for simulation.
Description of contents: programme
Part I: Large number of agents in equilibrium 1. Introduction: Thermodynamicas and Statistical Mechanics 1.1. Thermodynamics 1.2. Phase transitions 2. Critical phenomena 2.1. Ising model and related systems. 2.2. Continuum descriptions 2.3. Mean-field and Gaussian approximations 2.4. Scaling theories and the renormalization group 3. Heterogeneity 3.1. Percolation 3.2. Critical properties 3.3. Related models Part II: Nonlinear dynamics 4. Introduction 4.1. Paradigmatic models 5. Finite number of degrees of freedom 5.1. Phase portrait 5.2. Linerized stability 5.3. Non-linear behavior: bifurcations 6. Infinite number of degrees of freedom 6.1. Pattern formation 6.2. Reaction-diffusion systems 6.3. Bifurcation analysis 7. Chaotic dynamics 7.1. Recurrences in one dimension 7.2. Routes to chaos 7.3. Probabilistic descriptions
Learning activities and methodology
Lecture hours (1.4 ECTS) * Theory sessions. * Practical sessions: hands-on demonstrations, exercise solving, etc. Tutoring, mentoring, etc. (1.4 ECTS) Autonomous student work (3.2 ECTS)
Assessment System
  • % end-of-term-examination 60
  • % of continuous assessment (assigments, laboratory, practicals...) 50
Basic Bibliography
  • G. Nicolis. Introduction to Nonlinear Science. Cambridge University Press. 1995
  • K. Christensen, N. R. Moloney. Complexity and Criticality . World Scientific. 2005
Additional Bibliography
  • D. Stauffer, A. Aharony. Introduction to Percolation Theory. Taylor and Francis. 1994
  • G. Nicolis, C. Nicolis. Foundations of Complex Systems. World Scientific. 2007
  • J. H. Holland. Complexity: A Very Short Introduction. Oxford University Press. 2014
  • N. Goldenfeld. Lectures on Phase Transitions and the Renormalization Group. Addison Wesley. 1993
  • S. Strogatz. Nonlinear Dynamics and Chaos . Perseus Books. 1994
  • S. Thurner, R. Hanel, P. Klimek. Introduction to the Theory of Complex Systems. Oxford University Press. 2018
  • Y. Bar-Yam. Dynamics of Complex Systems. Addison-Wesley. 1997

The course syllabus and the academic weekly planning may change due academic events or other reasons.