PURE MTH 7023 - Pure Mathematics Topic D

North Terrace Campus - Semester 2 - 2023

This course is available for students taking a Masters degree in Mathematical Sciences. The course will cover an advanced topic in pure mathematics. For details of the topic offered this year please refer to the Course Outline.

  • General Course Information
    Course Details
    Course Code PURE MTH 7023
    Course Pure Mathematics Topic D
    Coordinating Unit Mathematical Sciences
    Term Semester 2
    Level Postgraduate Coursework
    Location/s North Terrace Campus
    Units 3
    Available for Study Abroad and Exchange Y
    Assessment Ongoing assessment, exam
    Course Staff

    Course Coordinator: Associate Professor Sanjeeva Balasuriya

    Course Timetable

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

  • Learning Outcomes
    Course Learning Outcomes
    This year, the topic of this course is Nonlinear Analysis of Dynamical Systems.


    The course is at the interface of applied and pure mathematics, and thus  earns either a pure or an applied contribution towards your honours or  higher degree.  Nonlinear dynamics aspects will be considered from both  the perspective of theory (establishing conditions on  existence/uniqueness/bifurcations/chaos which appear in the form of  theorems), and applications and intuition (will consider applied problems which link to such theory).  These two aspects respectively are usually interpreted as pure and applied mathematics, but this course  will emphasise the connections between them, demonstrating that the division is somewhat artificial.  The theory will include the implicit function theorem; Gronwall's inequality; Lyapunov functions; Lasalle's invariance principal; stable/unstable manifolds; Hartman-Grobman theorem; Poincare-Bendixson theorem; saddle-node, transcritical, pitchfork, period-doubling and Hopf bifurcations; symbolic dynamics; Smale horseshoe map; Smale-Birkhoff theorem; and Melnikov theory.  Applications will be in areas such as vibrations, fluid mechanics, invasion waves, Hamiltonian dynamics and mathematical modelling; however, the emphasis will be on applying methods to an application, rather than focussing principally on the application itself.

    The course will be run mainly on active learning principles, with the delivery designed to be student-centred.  There will be no lectures in
    the traditional form, and the class meetings will be all in workshop format.  The workshops will be based on a range of activities: worksheets which students will work on in small groups for discovery-based learning, discussions based on assigned recordings or readings, and student presentations on assigned topics or problems.

    Learning Outcomes
    On successful completion of the course, students will be able to:
    1.  Appreciate the connections between the pure and applied aspects of nonlinear dynamical systems;
    2. Apply relevant theorems to identify and predict the presence of entities (such as stable and unstable manifolds) whose characteristics
    govern global flow patterns;
    3.  Be cognisant of computational tools for detection of the above entities to empower an understanding of long-term behaviour of nonlinear dynamical systems;
    4.  Explain the mathematical concept of chaos, and be able to apply both analytical and computational tools to investigate and detect chaotic behaviour;
    5.  Employ a suite of mathematical and computational tools for analysing dynamical systems emerging from a variety of applications.
    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)

    Attribute 1: Deep discipline knowledge and intellectual breadth

    Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.


    Attribute 2: Creative and critical thinking, and problem solving

    Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

  • Learning Resources
    Required Resources
    Video recordings, short writeups, research publications and other ancillary material will all be provided via the course's MyUni site.
    Recommended Resources
    Online Learning
    This course will have an active MyUni website.
  • Learning & Teaching Activities
    Learning & Teaching Modes
    The learning in this course will be governed by modern pedagogical techniques, with no traditional lectures.  A combination of the concepts of flipped classrooms, active learning, discovery-based learning, peer evaluations, and assessments for learning will be employed. 

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

    Activity Quantity Workload Hours
    Workshops (includes presentations, and prior work) 24 90
    Problem sets 4 36
    Final project 1 30
    Total 156
    Learning Activities Summary
    The course material will be associated with the following sections, each of which will be of roughly two weeks duration:

    1. Theoretical preliminaries: existence, uniqueness, continuity in initial conditions
    2.  Phase space: invariant sets, Lasalle invariance, Hamiltonian systems, Lyapunov functions, alpha and omega limits sets
    3.  Critical points and local behaviour: stability, Hartman-Grobman theorem, stable and unstable manifolds, centre manifolds
    4.  Poincare maps: critical points for maps, Poincare maps, Poincare-Bendixson theorem, van der Pol oscillator
    5.  Local bifurcations: saddle-node, transcritical, pitchfork, Hopf, period-doubling, establishment via the implicit function theorem
    6.  Chaos: Smale horseshoe map, symbolic dynamics, Smale-Birkhoff theorem, Melnikov methods

    The delivery of these sections will be via the two workshop sessions per week, run in student-centred mode, coupledwith prior readings/viewings.
  • 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 Task Type Weighting Learning Outcomes
    Problem sets (4) Formative and Summative 44% All
    Presentations (instructor-reviewed) Formative and Summative 16% All
    Presentations (peer-reviewed)    Formative and Summative 6% All
    Active participation (instructor+peer-reviewed) Formative and Summative 10% All
    Final project Summative 24% All
    Assessment Detail

    No information currently available.


    No information currently available.

    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
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