• Description

In this subject will enter the basic concepts of a dynamic system generated by the iteration of a function.  The study of these dynamic systems will enter us in the world of the groups fractales, conjoint autosimilares, chaotic groups, groups of Cantor, groups of Julia and Fatou. Three fundamental examples in the study of the chaotic dynamic systems will be the logistical application, the complex quadratic family and the family of Arnold. An important part of this subject will be of practical character.

• The subject within the syllabus as a whole

The course Chaotic Dynamical Systems is a subject optativa of five crédits inside the máster interuniversity of Ingenieria Computational and Mathematical. It treats of a subject of mathematics with a clear component applied. 

• Professional fields to which it applies

The study of the dynamic systems, in spite of treating of a Mathematical discipline, has had a big influence in other scientific areas as for example Physical, Chemical, Engineerings, Biology or Economic.

• Prior knowledge

It treats of a subject fundamentally autocontenida. The necessary knowledges for cursar this subject are the conócimientos basic in Mathematics that can achieve in the mayoria of scientific Degrees. Also it is necessary to know basic technicians of programming.

• Information prior to enrolment

• Professor Coordinator:  Dr. Antonio Garijo Real (antonio.garijo[at]urv.cat)
  
• Credits: 5
• Description:  In this subject will enter the basic concepts of the dynamic system generated by the iteration of a function.  The study of these dynamic systems will enter us in the world of the groups fractales, conjoint autosimilares, chaotic groups, groups of Cantor and groups of Julia and Fatou. Two fundamental examples in the study of the chaotic dynamic systems will be the logistical application and the complex quadratic family. An important part of this subject will be of practical character.
• Requirements:  Capacity to read scientific texts in English. Basic knowledges of mathematics (level degree or engineering).

• Learning objectives and results

The objectivos  of this course are that the/the student/to purchase the basic knowledges of the theory of discreet dynamic systems.

The aims of learning are the following. Comprise the concept of dynamic system. Know some examples of dynamic systems. Comprise the concept of hiperbolicidad. Know the topological conjugation. Know and know apply the Theorem of Sarkovskii. Purchase the basic properties of the logistical application. Know the concept of Chaos. Know and know study the main types of bifurcations. Know the concept of conjoint Julia and Fatou of a quadratic polynomial. Know the group of Mandelbrot. Know the concept of number of rotation of a homeomorphism of the circle. Comprise the family of Arnold.

The competitions  in which it will deepen in in the course of systems dináimicos chaotic are those that to continuation enumerana. Dominate in an intermediate level a foreign tongue, preferably the English. Use of way advanced the tecnologias of information and the communication. Resolve complex problems of effective form in the field of the ingenieria mathematical and  computational. Work of autonomous form with responsibility and initiative. That the students know to apply the knowledges purchased and his capacity of resolution of problems in new surroundings or little known inside contexts wider (or multidisciplinary) related with his area of study. 

• Content

The course is divided in four thematic blocks.

The first block of the course devotes to the definción of the concept of dynamic system and the presentation of some examples. In particular it presents   the method of Newton-Raphson from the point of view an of dynamic system.

The second block is allocated to the study of the dynamic systems defined in the real interval I=[0,1]. The fundamental example of this type of dynamic systems is the Logistical application. So that the concepts that will present in this block will exemplify in this application. It initiates this block entering the concept of fixed point and periodic orbit. As well as the study of the hiperbolicidad of said objects. Also they present the most common bifurcations in this type of dynamic systems. Also it presents the topological and different conjugation models of dynamic systems conjugados. To continuation will study the concept of Chaos, also showing some examples in the Logistical application.

The third block is devoted to the dynamic systems defined in the complex plane. In this case the family that will use to present these examples is the quadratic family. This block initiates with the concept of succession of normal functions. This concept will allow us define the dicotomia fundamental that present these types of dynamic systems. The group of Fatou and of Julia. We will present different types of groups of Julia and Fatou. Finally we will present the group of Mandelbrot and some of his basic properties.

The last block is devoted to the dynamic systems defined in the circle unit. In this case we will present the contents using the family of functions estandar of Arnold. This family of functions serves like model of these systems. We will present the concept of number of rotation of a homeomorphism and will see his fundamental properties.

• Guidelines on assessment at the UOC

Assessment at the UOC is, in general, online, structured around the continuous assessment activities, the final assessment tests and exams, and the programme's final project.

Assessment activities and tests can be written texts and/or video recordings, use random questions, and synchronous or asynchronous oral tests, etc., as decided by each teaching team. The final project marks the end of the learning process and consists of an original and tutored piece of work to demonstrate that students have acquired the competencies worked on during the programme.

To verify students' identity and authorship in the assessment tests, the UOC reserves the right to use identity recognition and plagiarism detection systems. For these purposes, the UOC may make video recordings or use supervision methods or techniques while students carry out any of their academic activities.

The UOC may also require students to use electronic devices (microphones, webcams or other tools) or specific software during assessments. It is the student's responsibility to ensure that these devices work properly.

The assessment process is based on students' individual efforts, and the assumption that the student is the author of the work submitted for academic activities and that this work is original. The UOC's website on academic integrity and plagiarism has more information on this.

Submitting work that is not one's own or not original for assessment tests; copying or plagiarism; impersonation; accepting or obtaining any assignments, whether for compensation or otherwise; collaboration, cover-up or encouragement to copy; and using materials, software or devices not authorized in the course plan or instructions for the activity, including artificial intelligence and machine translation, among others, are examples of misconduct in assessments that may have serious academic and disciplinary consequences.

If students are found to be engaging in any such misconduct, they may receive a Fail (D/0) for the graded activities in the course plan (including final tests) or for the final grade for the course. This could be because they have used unauthorized materials, software or devices (such as artificial intelligence when it is not permitted, social media or internet search engines) during the tests; copied fragments of text from an external source (the internet, notes, books, articles, other students' work or tests, etc.) without the corresponding citation; purchased or sold assignments, or undertaken any other form of misconduct.

Likewise and in accordance with the UOC's academic regulations, misconduct during assessment may also be grounds for disciplinary proceedings and, where appropriate, the corresponding disciplinary measures, as established in the regulations governing the UOC community (Normativa de convivència).

In its assessment process, the UOC reserves the right to:

• Ask students to provide proof of their identity as established in the UOC's academic regulations.
• Ask students to prove the authorship of their work throughout the assessment process, in both continuous and final assessments, through a synchronous oral interview, of which a video recording or any other type of recording established by the UOC may be made. These methods seek to ensure verification of the student's identity, and their knowledge and competencies. If it is not possible to ensure the student's authorship, they may receive a D grade in the case of continuous assessment or a Fail grade in the case of the final assessment.

Artificial intelligence in assessments

The UOC understands the value and potential of artificial intelligence (AI) in education, but it also understands the risks involved if it is not used ethically, critically and responsibly. So, in each assessment activity, students will be told which AI tools and resources can be used and under what conditions. In turn, students must agree to follow the guidelines set by the UOC when it comes to completing the assessment activities and citing the tools used. Specifically, they must identify any texts or images generated by AI systems and they must not present them as their own work.

In terms of using AI, or not, to complete an activity, the instructions for assessment activities indicate the restrictions on the use of these tools. Bear in mind that using them inappropriately, such as using them in activities where they are not allowed or not citing them in activities where they are, may be considered misconduct. If in doubt, we recommend getting in touch with the course instructor and asking them before you submit your work.

• View the assessment model