Degree | Type | Year | Semester |
---|---|---|---|
4313136 Modelling for Science and Engineering | OT | 0 | 1 |
Students must have mathematical skills at a graduate level of a scientific degree.
The course aims to develop the students’ ability to systematically analyze deterministic nonlinear dynamical models and to elaborate mathematical models of physical systems.
1.- Introduction to dynamical systems
Introduction.- Characteristic properties of nonlinear dynamical systems.- Examples of nonlinear dynamical behaviors.- Classification of dynamical systems.- Dynamical systems according to their dynamics.
2.- Discrete dynamical systems.
Maps.- Logistic map.- Fixed points. Stability.- Universality.
3.- Dynamical systems in one dimension.
Graphical Solution. Fixed-Points.- Analytical solution. Linear stability.- Numerical Solution.- Bifurcations.- Flows on the circle.- Synchronization of fireflies.
4.- Dynamical Systems in 2 dimensions. Oscillations.
Introduction. Dynamic Behaviors in 2 dimensions.- Linear stability.- Population dynamics.- Bifurcations.- Oscillations. Biological Rhythms.
5.- Dynamical Systems in 3 dimensions. Chaos.
Deterministic Chaos.- Lorenz Equations.- Rossler system.- Applications.- Chaos descriptors.- Epidemics.
6.- First order partial differential equations
Definitions. Transport equation.- Travelling waves.- Characteristics method. Application to structured population dynamics.- Conservation laws.- Weak solutions and shock waves.- Burgers equation.- Traffic equation.
The methodology is based on lectures that include some exercises. Most exercises will be solved by the students and delivered periodically. After that, any doubt about them will be discussed in class.
If the sanitary situation derived from Covid-19 required teaching to be virtual, our intention is to keep as much presenciality as possible, especially for exams. However, if necessary, teaching will be given by electronic means, either uploading the registered class so that you can visualize it at your convenience, or by sinchronous classes through some videoconference platform (Zoom, Teams,...). In any case, some hours would be reserved weekly for tutorials via videoconference to solve doubts. Exercises done by students would be delivered either through the Virtual Campus or by email as usual.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Theory and exercise classes | 38 | 1.52 | 2, 1, 4, 6, 5, 7, 9, 8, 3, 10, 11 |
Type: Supervised | |||
Assesments and projects | 40 | 1.6 | 2, 1, 4, 6, 5, 7, 9, 8, 3, 10, 11 |
Type: Autonomous | |||
Personal study | 69 | 2.76 | 2, 1, 4, 5, 7, 9, 8, 3, 11 |
Grades will be obtained from:
1) deliveries of solved problems, simulations, reports and presentations.
2) at least two written exams, weighing around 50% of the final grade.
To pass the course:
- the average mark of the exams must be greater than 4 (on a scale of 10), and
- the final mark (exams and other evaluation tests) must be greater than 5.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Exams | close to 50% | 3 | 0.12 | 2, 4, 6, 9, 8 |
Projects and solved exercises | close to 50% | 0 | 0 | 2, 1, 4, 6, 5, 7, 9, 8, 3, 10, 11 |
- 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.
- B.C. Goodwin, How the Leopard Changed Its Spots: Evolution of Complexity. Prentice Hall, 1994.
- I. Peral, Primer Curso de EDPs, Addison-Wesley/UAM, 1995.
- R. Haberman. Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow. 1998.
- W. A. Strauss, Partial Differential Equations: An Introduction, John Wiley & Sons, 1992.