MATHS 761 : Dynamical Systems
Science
2025 Semester One (1253) (15 POINTS)
Course Prescription
Course Overview
Capabilities Developed in this Course
Capability 3: | Knowledge and Practice |
Capability 4: | Critical Thinking |
Capability 5: | Solution Seeking |
Capability 6: | Communication |
Capability 7: | Collaboration |
Learning Outcomes
- Use appropriate analytic, qualitative and numerical methods to construct phase portraits, bifurcation diagrams and bifurcation sets for systems of ODEs. (Capability 3, 4, 5 and 6)
- Use phase portraits, bifurcation diagrams and bifurcation sets for systems of ODEs to draw conclusions about the qualitative behaviour of solutions. (Capability 3, 4, 5 and 6)
- Explain and illustrate an understanding of dynamical systems through calculation, computation, description and discussion of dynamical phenomena. (Capability 3, 4, 5, 6 and 7)
Assessments
Assessment Type | Percentage | Classification |
---|---|---|
Exam | 50% | Individual Examination |
Assignments | 30% | Individual Coursework |
Team activities | 20% | Group & Individual Coursework |
3 types | 100% |
Assessment Type | Learning Outcome Addressed | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | ||||||||
Exam | ||||||||||
Assignments | ||||||||||
Team activities |
Key Topics
- Linear and nonlinear systems; topological equivalence; Hartman–Grobman theorem.
- Phase-space analysis; sketching phase portraits in 2D.
- Stable and unstable manifolds; stable manifold theorem.
- Periodic orbits; Poincaré maps; Poincaré-Bendixson theorem; Dulac's criterion; index theory.
- Dynamics in maps; stability of fixed point for maps.
- Non-hyperbolicity and structural stability; centre manifolds.
- Local bifurcations and normal forms (saddle-node, transcritical, pitchfork, Hopf).
- Global bifurcations (homoclinic, heteroclininc).
- Bifurcations for maps (period-doubling, route to chaos).
Special Requirements
Tuākana
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html
Whanaungatanga and manaakitanga are fundamental principles of the Mathematics Tuākana programme which provides support for Māori and Pasifika students who are taking mathematics courses. The Tuākana Mathematics programme consists of workshops and drop-in times, and provides a space where you are able to work alongside our Tuākana tutors and other Māori and Pasifika students who are studying mathematics.
Workload Expectations
It is expected that you spend 10 hours per week working on this course. The normal pattern of student study is expected to be (on average, each week):
- 2 hours lectures;
- 2-hour lab, including preparation;
- 3 hours lecture preparation and review;
- 3 hours assignments and exam preparation.
Students are expected to attend all class meetings. After each class you should review the material from the class and try any recommended examples. You are expected to preview the material provided for each lecture before you come to class.
Delivery Mode
Campus Experience
- Attendance is required at scheduled activities, including team activities, to receive credit for components of the course.
- The activities for the course are scheduled as a standard weekly timetable.
- Lectures will not normally be available as recordings.
Learning Resources
Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).
Please remember that the recording of any class on a personal device requires the permission of the instructor.
- Stability, Instability and Chaos by Paul Glendinning, Cambridge University Press (1994).
Student Feedback
During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.
At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.
Your feedback helps to improve the course and its delivery for all students.
Student feedback is considered on an on-going basis throughout the year.
Academic Integrity
The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework, tests and examinations as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. A student's assessed work may be reviewed against electronic source material using computerised detection mechanisms. Upon reasonable request, students may be required to provide an electronic version of their work for computerised review.
Class Representatives
Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.
Copyright
The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.
You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.
Inclusive Learning
All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.
Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz
Special Circumstances
If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.
If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.
This should be done as soon as possible and no later than seven days after the affected test or exam date.
Learning Continuity
In the event of an unexpected disruption we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and your assessment is fair, and not compromised. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator, and if disruption occurs you should refer to the University Website for information about how to proceed.
The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.
Student Charter and Responsibilities
The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.
Disclaimer
Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.
In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.