Study plan Study plan



We can find complex systems in every scientific field, from non-linear chemical kinetics, to physical and biological systems. Disciplines such as biomedical engineering, systems and synthetic biology, and both natural and artificial evolution, are characterized by complex networks displaying many, strong non-linear interactions.  In this course, we introduce contents from complex networks theory. This is a powerful framework to understand self-organization and emergent phenomena. The other main component is an introduction to the theory of dynamical systems, which allows us to characterize and investigate the dynamics of complex systems. We will focus on both continuous and discrete dynamical systems, as well as in the mechanisms behind the transitions in complex systems. Mathematical and computational modeling provide the basic tools required to understand the behavior of complex systems. This subject has a strong theoretical part and development and analysis of mathematical models will be performed. We will carry out some practical exercises that give students the basic skills on computer simulation and numerical integration of differential equations.


Students who have successfully completed first-year mathematics courses in algebra and calculus, such as those offered in BSc degrees in mathematics, physics, engineering, or life sciences, should be able to follow the course. Basic knowledge in computer programming is required, preferably in Netlogo, C/C++, Python, and Matlab.



1. Introduction to nonlinear dynamics. History of non-linear dynamics. Determinism: the Laplace´s devil. Stochastic dynamics. Continuous vs discrete systems. Chaos: the paradigm of complexity science. Examples of nonlinear dynamics in real systems.

2. Continuous dynamical systems: Mean field models. Ordinary Differential Equations (ODEs). Linear systems. Nonlinear Dynamics. Equilibria and linear stability analysis. Local bifurcations and normal forms. Universality and scaling in bifurcations.

3. Discrete dynamical systems: Simple models with complex dynamics. Equilibria and stability. Iteration. Period-doubling. Chaos and routes to chaos. Feigenbaum scenario. Fractals.

4. Fractals: Self-similarity. Geographic scaling. von Koch curve. Fractal dimension. Box-counting algorithm. Chaos game and Sierpinsky attractor.

5. Evolutionary algorithms: Biomimicry, Genetic algorithms, Fitness landscape, Quasispecies equation, Error threshold, Learning, Ant colony optimization.

6. Spatially-extended systems. Cellular automata (CA) models. Examples of transitions in spatial systems. Cancer spatial dynamics and ecological systems. Diffusion-induced chaos. Coupled-map lattices.

7. Complex networks I & II: Network properties. Random graph. Percolation transition. Hubs, connectors and paths. Small-worlds, scale-free networks. Modularity.


Practical exercises will take place along with the lectures. Students will be asked to bring their own computers, or to team up with partners that brigs a laptop. The chosen computational language will be Netlogo and C (and Matlab in some cases). However, some of the practicals can be implemented using any language, and we will encourage students to use their favorite tools.


Final exam: 30%

Practical exercises: 30%

Final individual project: 40%


Melanie Mitchell (2009) Complexity: a guided tour, Oxford University Press.

Steven H. Strogatz (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering  (studies in nonlinearity), Westview Press.

Ricard V. Solé and Susanna C. Manrubia (1996) Orden y caos en sistemas complejos, Politex UPC.

Ricard V. Solé and Brian Goodwin (2001) Signs of life: how complexity pervades biology, Basic Books.

Benoit B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Company.

Przemylslaw Prusinkiewicz and Aristid Lindenmayer (2004) The algorithmic beauty of plants, Springer-Verlag.

Michael Barnsley (1988) Fractals Everywhere, Academic Press. (chapter IX is online:

James Gleick (1987) Chaos: making a new science, Viking Books.

Stuart Kauffman (2000), Investigations, Oxford University Press.

Stuart Kauffman, Antichaos and adaptation.

Goldberger and B. West (1973), Fractals in physiology and medicine, J. Biol. Med.

John Koza et al (2003), Evolving inventions, Scientific American.

Martin Gardner (1970), Mathematical games - the fantastic combinations of John Conway’s new solitaire game “life”, Scientific American 223: 120-123.

Stephen Wolfram (1984) Cellular automata as models of complexity, Nature 311(5985), 419-424.

Stephen Wolfram (2002) A new kind of science, Wolfram research.

Ricard V. Solé (2009) Redes complejas: del genoma a internet, Tusquets (Metatemas).

Ricard V. Solé (2011) Phase transitions, Princeton University Press.

Duncan J. Watts and Steven H. Strogatz (1998) Collective dynamics of ‘small-world’ networks, Nature 393 (6684).

Albert-Laszlo Barabasi and Reka Albert (1999) Emergence of scaling in random networks, Science 286 (5439).

Mark Newman (2010) Networks: an introduction, Oxford University Press.

Geoffrey West (1999) The origin of universal scaling laws in biology, Physica A.

Sergi Valverde and Jordi Garcia-Ojalvo (2016) Hacia una teoría unificada de la criticalidad biológica. Investigación y Ciencia, Marzo 2016 (N. 474).

Sergi Valverde and Ricard V. Solé (2013) Networks and City, 83(4): 112-119.