APP MTH 7045 - Applied Mathematics Topic B

North Terrace Campus - Semester 1 - 2014

Please contact the School of Mathematical Sciences for further details, or view course information on the School of Mathematical Sciences web site at http://www.maths.adelaide.edu.au/students/honours

  • General Course Information
    Course Details
    Course Code APP MTH 7045
    Course Applied Mathematics Topic B
    Coordinating Unit Applied Mathematics
    Term Semester 1
    Level Postgraduate Coursework
    Location/s North Terrace Campus
    Units 3
    Course Description Please contact the School of Mathematical Sciences for further details, or view course information on the School of Mathematical Sciences web site at http://www.maths.adelaide.edu.au/students/honours
    Course Staff

    Course Coordinator: Dr Luke Bennetts

    Course Timetable

    The full timetable of all activities for this course can be accessed from Course Planner.

  • Learning Outcomes
    Course Learning Outcomes

    In 2014, the topic of this course is DYNAMICAL SYSTEMS.

    Syllabus

    This course provides an introduction to the theory of continuous and discrete dynamical systems with a particular emphasis on bifurcations and routes to chaotic behaviour. We begin by studying examples of how low-dimensional dynamical systems arise from approximations to complex physical systems. We observe that these systems can exhibit changes in behaviour as governing parameters are varied, including chaotic dynamics (extreme sensitivity to initial conditions). To understand this behaviour we
    study the stability and bifurcations of periodic structures, bringing together results from linear algebra, multivariable calculus, differential equations, topology and group theory.

    Assumed knowledge for the course: basic linear algebra (especially eigenvalues and eigenvectors); multivariable calculus (especially differentiability of vector fields and Taylor series); and differential equations (especially solution techniques for systems of ordinary differential equations).

    Learning Outcomes

    On successful completion of this course, students will be able to:

    1. understand the terminology surrounding both continuous and discrete dynamical systems;
    2. formulate mathematical models and reduce them to autonomous dynamical systems;
    3. integrate systems of ordinary differential equations numerically to high accuracy;
    4. construct phase portraits for pairs of autonomous ordinary differential equations;
    5. analyse stability and bifurcations of periodic structures and their implications for dynamics;
    6. understand the use of Poincare sections in reducing dimensionality of dynamical systems;
    7. understand the Poincare-Bendixson theorem and its various implications;
    8. understand the precise definition of chaos in dynamical systems;
    9. understand Sarkovskii's theorem and period doubling cascades as routes to chaos;
    10. understand the role of topology in continuous two-dimensional maps.

    University Graduate Attributes

    This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

    University Graduate Attribute Course Learning Outcome(s)
    Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all
    The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. all
    An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. all
    A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all
  • Learning Resources
    Required Resources
    None.
    Recommended Resources
    Books:

    1.  Differential equations, dynamical systems and an introduction to chaos by Hirsch, Smale and Devaney.

    2.  An introduction to chaotic dynamical systems by Devaney.

    Online Learning
    MyUni will be used to provide electronic resources, such as lecture notes, assignment papers, sample solutions, discussion boards, etc. It is recommended that the students make appropriate use of these resources.

    See https://myuni.adelaide.edu.au/webapps/login/
  • Learning & Teaching Activities
    Learning & Teaching Modes
    The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.
    Workload

    The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

    Lectures (x30) = 90 workload hours
    Assignments (x6) = 54 workload hours
    Test = 10 workload hours

    Total = 154 workload hours

    As a very rough estimate, the optional miniproject can be expected to add around 15 hours.
    Learning Activities Summary
    1.  Introduction to dynamical systems.
    2.  One-dimensional flows.
    3.  Two-dimensional flows.
    4.  Advanced material on flows.
    5.  From flows to maps.
    6.  One-dimensional maps.
    7.  Two-dimensional maps.
    8.  Advanced material on maps.
  • Assessment

    The University's policy on Assessment for Coursework Programs is based on the following four principles:

    1. Assessment must encourage and reinforce learning.
    2. Assessment must enable robust and fair judgements about student performance.
    3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
    4. Assessment must maintain academic standards.

    Assessment Summary
    Assessment: Assignments
    Due: Even weeks
    Weighting: 10-20%
    Learning outcomes: All

    Assessment: Test
    Due: Mid-semester
    Weighting: 10-20%
    Learning outcomes: 1,2

    Assessment: Examination
    Due: Examination period
    Weighting: 50-70%
    Learning outcomes: All

    Assessment: Mini-project
    Due: End of teaching
    Weighting: 0-15%
    Learning outcomes: 1
    Assessment Related Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    There will be a total of 6 homework assignments, distributed during each odd week of the semester and due at the end of the following week. Each will cover material from the lectures, and in addition, will sometimes go beyond that so that students may have to undertake some additional research. The best 5 of the 6 assignments will be used towards the final grade, totalling between 10 and 20% of it. Weightings for each of the assessment items specified in the table presented in the Assessment Summary will be applied individually to maximise each student's final mark.
    Submission
    Homework assignments must either be given to the lecturer in person or left in the box outside the lecturer's office by the given due time. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.  The last day on which a miniproject may be submitted is the last teaching day of the semester.
    Course Grading

    Grades for your performance in this course will be awarded in accordance with the following scheme:

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    FNS   Fail No Submission
    F 1-49 Fail
    P 50-64 Pass
    C 65-74 Credit
    D 75-84 Distinction
    HD 85-100 High Distinction
    CN   Continuing
    NFE   No Formal Examination
    RP   Result Pending

    Further details of the grades/results can be obtained from Examinations.

    Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

    Final results for this course will be made available through Access Adelaide.

  • Student Feedback

    The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

    SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

  • Student Support
  • Policies & Guidelines
  • Fraud Awareness

    Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's student’s disciplinary procedures.

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