Dynamical Systems and Complexity
Code: 45559 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| Modelización para la Ciencia y la Ingeniería / Modelling for Science and Engineering | OP | 1 |
Contact
- Name:
- Jordi Villadelprat Yague
- Email:
- jordi.villadelprat@uab.cat
Teachers
- Jose Sardañes Cayuela
- Daniel Campos Moreno
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
Students must have mathematical skills at a graduate level of a scientific degree.
Objectives and Contextualisation
The course aims to develop the students’ ability to systematically analyze deterministic nonlinear dynamical models and to elaborate mathematical models of real systems.
Learning Outcomes
- CA09 (Competence) Devise models based on dynamical systems and complex systems to solve specific practical problems.
- CA09 (Competence) Devise models based on dynamical systems and complex systems to solve specific practical problems.
- CA10 (Competence) Communicate to an expert audience the results obtained from the analysis of models based on dynamic and complex systems incorporating ethical, sustainability and gender equality criteria.
- CA10 (Competence) Communicate to an expert audience the results obtained from the analysis of models based on dynamic and complex systems incorporating ethical, sustainability and gender equality criteria.
- CA10 (Competence) Communicate to an expert audience the results obtained from the analysis of models based on dynamic and complex systems incorporating ethical, sustainability and gender equality criteria.
- CA11 (Competence) Assess, using specific metrics and mathematical tools, the level of complexity of a set of data obtained through experimentation and/or observations.
- KA09 (Knowledge) Recognise the main analysis techniques to study dynamical systems, as well as the theoretical scope of application of each of them.
- KA09 (Knowledge) Recognise the main analysis techniques to study dynamical systems, as well as the theoretical scope of application of each of them.
- KA09 (Knowledge) Recognise the main analysis techniques to study dynamical systems, as well as the theoretical scope of application of each of them.
- KA10 (Knowledge) Recognise the different criteria that can be used to quantify and/or measure the complexity of a system.
- KA10 (Knowledge) Recognise the different criteria that can be used to quantify and/or measure the complexity of a system.
- SA09 (Skill) Formulate dynamical systems and complex models capable of capturing essential dynamic features of specific applications.
- SA09 (Skill) Formulate dynamical systems and complex models capable of capturing essential dynamic features of specific applications.
- SA09 (Skill) Formulate dynamical systems and complex models capable of capturing essential dynamic features of specific applications.
- SA10 (Skill) Solve, either analytically or computationally, complex dynamic models using the appropriate mathematical tools for each situation.
- SA10 (Skill) Solve, either analytically or computationally, complex dynamic models using the appropriate mathematical tools for each situation.
- SA11 (Skill) Implement tools and methodologies to study emerging behaviours in reference models in the field of complex systems.
- SA11 (Skill) Implement tools and methodologies to study emerging behaviours in reference models in the field of complex systems.
- SA11 (Skill) Implement tools and methodologies to study emerging behaviours in reference models in the field of complex systems.
- SA11 (Skill) Implement tools and methodologies to study emerging behaviours in reference models in the field of complex systems.
Content
1. Introduction to Dynamical Systems
Types and characteristic properties. Related concepts.
2. One-Dimensional Discrete Dynamical Systems
Graphical and analytical study. Fixed points. Linear stability. Bifurcations. The logistic map.
3. Two-Dimensional Dynamical Systems
Classification of linear systems. Phase portrait. Limit cycles. Bifurcations. Biological models.
4. Chaotic Dynamical Behavior
Deterministic chaos. Definition. Examples.
5. Introduction to numerical methods
Numerical methods: error sources. Euler and Runge-Kutta methods.
6. Spatio-temporal dynamics
Metapopulation models. Coupled map lattices. Cellular autonomy. Reaction-diffusion equations.
7. Complexity
Systems with novel organized topology. Basic elements of complex systems. Emergent behaviors. Case studies. Measures of complexity.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Theory and problem-solving classes | 38 | 1.52 | CA09, CA10, CA11, KA09, KA10, SA09, SA10, SA11, CA09 |
| Type: Supervised | |||
| Problem sets and projects | 40 | 1.6 | CA09, CA10, CA11, KA09, KA10, SA09, SA10, SA11, CA09 |
| Type: Autonomous | |||
| Independent study | 69 | 2.76 | CA09, CA11, KA09, KA10, SA09, SA10, SA11, CA09 |
The methodology is based on lectures that include some practical exercises (either written or computational). Most of the exercises will be solved and submitted periodically by students through the Virtual Campus. Afterwards, any doubts regarding these exercises will be discussed in class.
Note: 15 minutes of one class session, within the schedule established by the department or degree program, will be reserved for students to complete surveys evaluating the teaching performance and the course/module.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Exams | around 50% | 3 | 0.12 | CA09, CA11, KA09, KA10, SA09, SA10 |
| Projects and worked-out exercises | around 50% | 0 | 0 | CA09, CA10, CA11, KA09, KA10, SA09, SA10, SA11 |
Continuous Assessment
Final grades will be based on:
- Submission of solved problems, simulations, reports, and presentations.
- At least two written exams, which will account for approximately 50% of the final grade.
To pass the course:
- The average grade of the exams must be greater than 4 (on a scale of 10), and
- The final average grade (including exams and other assessment components) must be greater than 5.
Single Assessment
Students who opt for the single assessment modality must take a final exam consisting of problem-solving and some theoretical questions. Upon completion, they must also submit all required exercises and project reports.
The final grade will be calculated in the same way as in continuous assessment: the exam will account for approximately 50% of the final grade, and the submissions will account for the remaining 50%.
To pass the course:
- The exam grade must be greater than 4 (on a scale of 10), and
- The final average grade (exam and submissions) must be greater than 5.
If the exam grade is below 4 or the final average is below 5, students will have a second opportunity to pass the course through a resit exam. The same recovery system as in continuous assessment will apply: students may retake the part corresponding to theory and problem-solving (approximately 50%). The submission component is not recoverable.
Bibliography
- S.H. Strogatz. Nonlinear Dynamics and Chaos. Second Edition. Perseus Books, Westview Press, Boulder, 2014.
- R.V. Solé y S.C. Manrubia, Orden y caos en sistemas complejos, ediciones UPC, Barcelona, 2001.
- S.H. Strogatz. SYNC. Rythms of nature, rythms of ourselves, Penguin, 2004.
- S. Parker , Leon O. Chua. Practical Numerical Algorithms for Chaotic Systems (1989).
- B.C. Goodwin, How the Leopard Changed Its Spots: Evolution of Complexity. Prentice Hall, 1994.
− Hanski, I. Metapopulation Ecology Oxford University Press. 1999.
− J.D. Murray. Mathematical Biology I: An introduction. Interdisciplinary Applied Mathematics 2002
− W. A. Strauss, Partial Differential Equations: An Introduction, John Wiley & Sons, 1992.
− K. Kaneko. Theory and Applications of Coupled Map Lattices (Nonlinear Science: Theory and Applications) 1st Edition, 1993
− A. Ilachinski. Cellular Automata: A Discrete Universe, 2001
− U. Dieckmann, R. Law, J.A.J. Metz. The Geometry of Ecological Interactions: Simplifying Spatial Complexity: 1 (Cambridge Studies in Adaptive Dynamics, Series Number 1), 2000
- R. Clark Robinson, An introduction to Dynamical Systems: Continuous and Discrete, Pure and Applied undergraduate texts, American Mathematical Society, 2012
- Robert L. Devaney, An introduction to Chaotic Dynamical Systems, Westview Press, 2003
- Stefan Thurner, Peter Klimek, Rudolf Hanel, Introduction to the Theory of Complex Systems, Oxford University Press, 2018
- Introduction to Complexity (online). Complexity Explorer, Santa Fe (https://www.complexityexplorer.org/courses/185-introduction-to-complexity#gsc.tab=0)
Software
There is no specific software for the subject.
Groups and Languages
Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (TEm) Theory (master) | 1 | English | first semester | afternoon |