Teaching plan for the course unit

 Catalą English Close

 General information

Course unit name: Mathematical Models and Dynamical Systems

Course unit code: 360153

Coordinator: Ernest Fontich Julia

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 Estimated learning time Total number of hours 150

 Face-to-face learning activities 60
 -  Lecture 30 -  Problem-solving class 15 -  Laboratory session 15
 Supervised project 30
 Independent learning 60

 Competences to be gained during study

 - Basic knowledge and understanding of mathematics. - Ability to use physical and electronic bibliographical resources.
 - Understanding of and capacity to apply the language of mathematics. - Understanding of the applications of mathematics in other branches of science and technology. - Capacity to assimilate new mathematical concepts through the prior understanding of other concepts. - Capacity to construct mathematical models of simple, real-life situations. - Capacity to select and apply the most suitable mathematical process for each problem encountered. - Ability to identify flaws in scientific reasoning.
 Learning objectives
 Referring to knowledge — Understand and interpret various mathematical models from different scientific fields (biology, physics, economics, etc.).   — Understand some of the key results of dynamic systems theory.  Referring to abilities, skills — Solve difference equations and elementary differential equations.   — Qualitatively analyse the dynamics of low-dimensional discrete and continuous systems.   — Apply dynamic systems tools in the analysis of mathematical models.

 Teaching blocks

1. Discrete dynamical systems

1.1. Discrete evolutionary processes: application iteration

1.2. Linear difference equations

1.3. One-dimensional applications: qualitative analysis, phase portraits and bifurcation

1.4. Discrete linear systems: resolution, qualitative analysis and classification

1.5. Introduction to qualitative theory: stationary solution and stability

1.6. Applications

2. Fractals and complex dynamics

2.1. Iterated function systems

2.2. Fractal dimension

2.3. Fractals and dynamic systems

2.4. Julia and Mandelbrot sets

3. Continuous dynamic systems

3.1. Continuous evolution process: ordinary differential equations

3.2. Standard methods of solving ordinary differential equations

3.3. One-dimensional systems: qualitative analysis, phase portraits and bifurcation

3.4. Linear differential equation: resolution, qualitative analysis and classification

3.5. Introduction to qualitative theory: stationary solution and stability

3.6. Applications

 Teaching methods and general organization

 The weekly timetable consists of three hours of theory-practical classes and one hour of laboratory exercises and practical work. The theory-practical classes are devoted to presenting the subject matter, which will be illustrated through examples and practical applications. Laboratory classes are dedicated to problem-solving exercises, and the occasional use of computer tools to further understanding of the concepts introduced in the theory-practical classes.

 Official assessment of learning outcomes

 Continuous assessment is based on a mid-semester examination (P1) and a final examination, of two parts, at the end of the semester, from which two scores are generated (F1 and F2) corresponding to each of the parts, respectively. There is the possibility of sitting the first part of the final examination to increase the grade. A minimum grade of 3 out of 10 is required in each section in order to pass the subject. Also, part of the grade corresponds to an evaluation of the problem-solving work done in the laboratory (L) during the course. The teacher will specify the value of this component at the beginning of the course. The final grade is obtained by the formula NF = 0.8 * NE + 0.2 * L, where NE = 0.5 * (max [P1, F1] + F2). Once the grading period has passed, there is the possibility of repeat assessment (R), which consists of a single examination covering the entire subject. The definitive grade is obtained by the formula ND = 0.8 * max [NE, R] + 0.2 * L. Students who do not obtain a score for each of the two parts of the course will be marked as absent.   Examination-based assessment Single assessment consists of a final examination (F). This option must be requested of the Secretary’s Office at the Faculty before the date set by the Board of Studies. There is the possibility of repeat assessment (R). The final score is obtained by the formula ND = max [F, R].