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İzmir Ekonomi Üniversitesi

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    GRADUATE SCHOOL

    Ph.D. In Electrical-Electronics Engineering

    EEE 612 | Course Introduction and Application Information

    Course Name
    Chaos and Fractals
    Code
    Semester
    Theory
    (hour/week)
    Application/Lab
    (hour/week)
    Local Credits
    ECTS
    EEE 612
    Fall/Spring
    3
    0
    3
    7.5

    Prerequisites
    None
    Course Language
    English
    Course Type
    Elective
    Course Level
    Third Cycle
    Mode of Delivery -
    Teaching Methods and Techniques of the Course -
    National Occupation Classification -
    Course Coordinator
    Course Lecturer(s) -
    Assistant(s) -
    Course Objectives The course aims the students: i) to know basic concepts on chaotic systems and fractal sets and ii) to use this knowledge in engineering applications.
    Learning Outcomes

    The students who succeeded in this course;

    • Know the features of a chaotic behavior distinguishing it from other dynamical behaviors,
    • Know the features of fractal sets distinguishing them from other sets,
    • Calculate numerically the Liapunov exponents of chaotic systems, and obtain and analyze bifurcation diagrams,
    • Construct fractal sets by iterated function systems, Use chaos and fractals in engineering applications.
    Course Description A unification of chaotic dynamics and fractal sets in a dynamical system and set theory background. Sensitive dependency and topological transitivity in invariant sets. From stable fixed points to period doubling, and entrance to chaos. Symbolic dynamics and examples for strange attractors. From cantor set to classical fractals. Self-similarity and fractal dimension. Image encoding by iterated function systems. Randomness in fractal construction. Chaotification. Engineering applications of chaos and fractals.

     



    Course Category

    		Core Courses
    Major Area Courses
    X
    Supportive Courses
    Media and Management Skills Courses
    Transferable Skill Courses

     

    WEEKLY SUBJECTS AND RELATED PREPARATION STUDIES

    Week Subjects Related Preparation Learning Outcome
    1 Historical remarks. First examples of chaotic systems and fractal sets in nature and engineering. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    2 Nonlinear dynamics: From equilibrium to chaos. Multiple equilibria. Limit cycle. Quasi periodic trajectories. Almost periodicity. Period doubling. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    3 Definition of chaos. Invariant set. Sensitive dependence on initial conditions. Denseness of periodic and aperiodic trajectories. Topological transitivity. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    4 Liapunov exponents. Identification of chaos by numerical methods. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    5 Analysis of discrete time chaotic systems: Logistic map. Fixed point. Contraction mapping. Bifurcation. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    6 Analysis of continuous time chaotic systems: Lorenz system. Circuit realizations of Lorenz system. Rössler system. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    7 Chua’s chaotic circuit and its hardware realizations Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    8 Bifurcation: Pitchfork, saddle-node, Hopf, homoclinic and heteroclinic bifurcations. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    9 1. Midterm
    10 Fractal set examples from nature. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    11 Definition of fractal set. Self-similarity. Fractal dimension. Box counting dimension. Hausdorff dimension. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    12 Construction of fractal sets by Iterated Function Systems. Contraction mapping. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    13 2. Midterm
    14 Engineering applications of chaotic systems. Secure communication by synchronized chaotic systems. Chaotification. Chaotic liquid mixing. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    15 Engineering applications of fractal sets. Fractal image compression. Chaos and Fractals: New Frontiers of Science, Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, 2nd Edition, Springer, ISBN-10: 0387202293, ISBN-13: 978-0387202297. Lecture Notes.
    16 Review of the Semester  

     

    Course Notes/Textbooks The textbook referenced above and lecture notes
    Suggested Readings/Materials Related Books and Research Papers

     

    EVALUATION SYSTEM

    Semester Activities Number Weigthing
    Participation
    Laboratory / Application
    Field Work
    Quizzes / Studio Critiques
    Portfolio
    Homework / Assignments
    2
    10
    Presentation / Jury
    Project
    2
    30
    Seminar / Workshop
    Oral Exams
    Midterm
    2
    30
    Final Exam
    1
    30
    Total

    Weighting of Semester Activities on the Final Grade
    70
    Weighting of End-of-Semester Activities on the Final Grade
    30
    Total

    ECTS / WORKLOAD TABLE

    Semester Activities Number Duration (Hours) Workload
    Theoretical Course Hours
    (Including exam week: 16 x total hours)
    16
    3
    48
    Laboratory / Application Hours
    (Including exam week: '.16.' x total hours)
    16
    0
    Study Hours Out of Class
    15
    3
    45
    Field Work
    0
    Quizzes / Studio Critiques
    0
    Portfolio
    0
    Homework / Assignments
    2
    15
    30
    Presentation / Jury
    0
    Project
    2
    20
    40
    Seminar / Workshop
    0
    Oral Exam
    0
    Midterms
    2
    15
    30
    Final Exam
    1
    32
    32
        Total
    225

     

    COURSE LEARNING OUTCOMES AND PROGRAM QUALIFICATIONS RELATIONSHIP

    #
    		 PC Sub Program Competencies/Outcomes
    * Contribution Level
    1
    2
    3
    4
    5
    1 Accesses information in breadth and depth by conducting scientific research in Electrical and Electronics Engineering; evaluates, interprets and applies information.
    -
    -
    -
    -
    X
    2 Is well-informed about contemporary techniques and methods used in Electrical and Electronics Engineering and their limitations.
    -
    -
    -
    -
    X
    3 Uses scientific methods to complete and apply information from uncertain, limited or incomplete data; can combine and use information from different disciplines. Knows and applies the research methods in studies of the area with a high level of skill.
    -
    -
    X
    -
    -
    4 Is informed about new and upcoming applications in the field and learns them whenever necessary.
    -
    -
    -
    X
    -
    5 Defines and formulates problems related to Electrical and Electronics Engineering, develops methods to solve them and uses progressive methods in solutions. Can independently realize novel studies that bring innovation to the field, or methods, or design, or known methods.
    -
    -
    X
    -
    -
    6 Develops novel and/or original methods, designs complex systems or processes and develops progressive/alternative solutions in designs.
    -
    -
    -
    -
    X
    7 Designs and implements studies based on theory, experiments and modeling; analyses and resolves the complex problems that arise in this process. Performs critical analysis, synthesis and evaluation of new and complex ideas.
    -
    -
    -
    X
    -
    8 Can work effectively in interdisciplinary teams as well as teams of the same discipline, can lead such teams and can develop approaches for resolving complex situations; can work independently and takes responsibility.
    -
    X
    -
    -
    -
    9 Engages in written and oral communication at least in Level C1 of the European Language Portfolio Global Scale.
    -
    -
    X
    -
    -
    10 Communicates the process and the results of his/her studies in national and international venues systematically, clearly and in written or oral form.
    -
    -
    X
    -
    -
    11 Evaluates the results of scientific, technological and engineering research and development activities in terms of the social, environmental, health, safety and legal aspects. Examines social relations and norms related to the field, and develops and makes attempts to change them if necessary. Knows their project management and business applications, and is aware of their limitations in Electrical and Electronics Engineering applications.
    -
    -
    X
    -
    -
    12 Highly regards scientific and ethical values in data collection, interpretation, communication and in every professional activity. Adheres to the principles of research and publication ethics.
    -
    X
    -
    -
    -

    *1 Lowest, 2 Low, 3 Average, 4 High, 5 Highest

     

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