Chaotic Dynamical Systems Code:  M0.534    :  6
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This is the course plan for the second semester of the academic year 2023/2024. To check whether the course is being run this semester, go to the Virtual Campus section More UOC / The University / Programmes of study section on Campus. Once teaching starts, you'll be able to find it in the classroom. The course plan may be subject to change.

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.

 

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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. 

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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.

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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.

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  • 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).

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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. 

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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.

 

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The assessment process is based on the student's personal work and presupposes authenticity of authorship and originality of the exercises completed.

Lack of authenticity of authorship or originality of assessment tests, copying or plagiarism, the fraudulent attempt to obtain a better academic result, collusion to copy or concealing or abetting copying, use of unauthorized material or devices during assessment, inter alia, are offences that may lead to serious academic or other sanctions.

Firstly, you will fail the course (D/0) if you commit any of these offences when completing activities defined as assessable in the course plan, including the final tests. Offences considered to be misconduct include, among others, the use of unauthorized material or devices during the tests, such as social media or internet search engines, or the copying of text from external sources (internet, class notes, books, articles, other students' essays or tests, etc.) without including the corresponding reference.

And secondly, the UOC's academic regulations state that any misconduct during assessment, in addition to leading to the student failing the course, may also lead to disciplinary procedures and sanctions.

The UOC reserves the right to request that students identify themselves and/or provide evidence of the authorship of their work, throughout the assessment process, and by the means the UOC specifies (synchronous or asynchronous). For this purpose, the UOC may require students to use a microphone, webcam or other devices during the assessment process, and to make sure that they are working correctly.

The checking of students' knowledge to verify authorship of their work will under no circumstances constitute a second assessment.

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You can only pass the course if you participate in and pass the continuous assessment. Your final mark for the course will be the mark you received in the continuous assessment.

 

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