PURE MTH 7002 - Pure Mathematics Topic B
North Terrace Campus - Semester 1 - 2024
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General Course Information
Course Details
Course Code PURE MTH 7002 Course Pure Mathematics Topic B Coordinating Unit Mathematical Sciences Term Semester 1 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.
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Learning Outcomes
Course Learning Outcomes
his year, the topic of this course in Nonlinear Analysis of Dynamical Systems.
Synopsis
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 this 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.
all 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.
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Learning Resources
Required Resources
Video recordings, short writeups, research publications and other ancillary material will all be provided via the course's MyUni site.Online Learning
The 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.Workload
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 36 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:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Assessment Summary
Assessment task Type Weighting Learning Outcomes Problem sets (4) Formative and Summative 44% All Presentations/active participation (instructor-reviewed) Formative and Summative 16% All Presentations/active participation (peer-reviewed) Formative and Summative 16% All Final project Summative 24% All Assessment Related Requirements
An aggregate score of 50% is required to pass the course.Assessment Detail
There will be five homework assignments distributed approximately once every two weeks across the semester. There will also be a final
examination.Submission
Homework assignments must be submitted on MyUni as pdf files. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.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.
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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.
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Student Support
- Academic Integrity for Students
- Academic Support with Maths
- Academic Support with writing and study skills
- Careers Services
- International Student Support
- Library Services for Students
- LinkedIn Learning
- Student Life Counselling Support - Personal counselling for issues affecting study
- Students with a Disability - Alternative academic arrangements
- YouX Student Care - Advocacy, confidential counselling, welfare support and advice
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Policies & Guidelines
This section contains links to relevant assessment-related policies and guidelines - all university policies.
- Academic Credit Arrangements Policy
- Academic Integrity Policy
- Academic Progress by Coursework Students Policy
- Assessment for Coursework Programs Policy
- Copyright Compliance Policy
- Coursework Academic Programs Policy
- Elder Conservatorium of Music Noise Management Plan
- Intellectual Property Policy
- IT Acceptable Use and Security Policy
- Modified Arrangements for Coursework Assessment Policy
- Reasonable Adjustments to Learning, Teaching & Assessment for Students with a Disability Policy
- Student Experience of Learning and Teaching Policy
- Student Grievance Resolution Process
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