Department assigned to the subject: Department of Mathematics
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
1.2. 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
3.2. Critical properties
3.3. Related models
Part II: Nonlinear dynamics
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)
% end-of-term-examination 60
% of continuous assessment (assigments, laboratory, practicals...) 50
G. Nicolis. Introduction to Nonlinear Science. Cambridge University Press. 1995
K. Christensen, N. R. Moloney. Complexity and Criticality . World Scientific. 2005
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.