Teaching plan for the course unit

 General information

Course unit name: Dynamic Systems

Course unit code: 568178

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 -  Lecture with practical component 30
 Supervised project 20
 Independent learning 70

 Recommendations

 • Working knowledge of a programming language. • To have completed a subject in differential equations and one in complex variables.

 Competences to be gained during study

 — Capacity to determine the basic elements of the phase portrait of a dynamical system and understand the dynamic implications of their features. — Capacity to understand the basic elements of complex and dimensional dynamics. — Capacity to understand techniques of local analysis of dynamical systems. — Capacity to understand basic local bifurcations of families of systems. — Ability to calculate invariant manifolds and knowledge of the implications of the properties of their globalization.

 Learning objectives
 Referring to knowledge — To learn the different types of dynamical systems and the most common tools used to study them. — To understand what research in dynamical systems consists in. — To know and understand the basic results in each of the topics covered in the course. — To know the main conjectures in the field of dynamical systems.   Referring to skills, abilities   — To perform introductory research in dynamical systems. — To learn to analyze a specific specific dynamical system using tools studied in class. — To learn to programme numerical simulations of simple dynamical systems. — To learn to research information in databases of articles and preprints. — To know how to write a report on a mathematical topic. — To develop communication skills (oral presentations).

 Teaching blocks

1. One dimensional and complex dynamics

1.1. Introduction to dynamical systems, discrete and continuous. Basic terminology. Conjugacies.

1.2. Dynamical systems in real dimension 1. Introduction and examples. Bifurcations. Bimodal maps: the quadratic family. Circle homeomorphisms.

1.3. Dynamical Systems on the complex plane. Riemann surfaces and iteration of holomorphic functions. Normal families: The Fatou and Julia sets. Local theory: periodic points and linearization. Global theory: connected components of the Fatou set. Parameter spaces: the Mandelbrot set and main conjectures.

2. N-dimensional dynamics

2.1. Lyapunov stability.

2.2. Local theory: Hartman’s Theorem, Sternberg’s Theorem and invariant manifolds.

2.3. Normal forms and bifurcations.

2.4. Hyperbolic dynamics.

 Teaching methods and general organization

 Face-to-face learning activities include lectures and problem-solving exercises and related discussion in class; There are 30 hours of directed work assigned to the introductory research project. This project is to be presented at the end of the course in a seminar class, attendance at which is compulsory for all students. Each week, or every two weeks, a list of problems are set, to be solved and handed in within one week.

 Official assessment of learning outcomes

 Continuous assessment is based on the completion of problem-solving exercises set throughout the course, for which answers must be handed in to the teacher. The completion and presentation of a final project is also taken into account. Marks for the problem-solving exercises comprise 70% of the final grade. The marks awarded for the project and the presentation make up the remaining 30%. Students may apply for a repeat assessment. To qualify for this, the final project must have been completed and presented. Repeat assessment consists of an on-site examination. Weightings for the final grade in this case are: final project, 30% and examination, 70%.   Examination-based assessment Students who wish to opt for a single assessment must inform the Secretary by the date set in the Faculty calendar. Single assessment consists of a project (30% of the final grade) and an on-site examination (the remaining 70%).