Skip to Main Content

Content

Course Tabs

Overview

Course Prescription

Mathematical models of systems that change are frequently written in the form of nonlinear differential equations, but it is usually not possible to write down explicit solutions to these equations. This course covers analytical and numerical techniques that are useful for determining the qualitative properties of solutions to nonlinear differential equations.

Course Overview

Mathematical models of systems that change are frequently written in the form of nonlinear differential equations, but it is usually not possible to write down explicit solutions to these equations. This course covers analytical and numerical techniques that are useful for determining the qualitative properties of solutions to nonlinear differential equations. This is a core applied mathematics course for the BSc(Hons), BAdvSci and PGDipSci. It is also of interest to students majoring in Physics, Engineering Science, Computer Science or Statistics.

Key Topics

  1. Linear and nonlinear systems; topological equivalence; Hartman–Grobman theorem.
  2. Phase-space analysis; sketching phase portraits in 2D.
  3. Stable and unstable manifolds; stable manifold theorem.
  4. Periodic orbits; Poincaré maps; Poincaré-Bendixson theorem; Dulac's criterion; index theory.
  5. Dynamics in maps; stability of fixed point for maps.
  6. Non-hyperbolicity and structural stability; centre manifolds.
  7. Local bifurcations and normal forms (saddle-node, transcritical, pitchfork, Hopf).
  8. Global bifurcations (homoclinic, heteroclininc).
  9. Bifurcations for maps (period-doubling, route to chaos).

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.

Course Prerequisites, Corequisites and Restrictions

Prerequisite

Locations and Semesters Offered

LocationSemester
City

Teaching and Learning

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

Taught courses use a learning and collaboration tool called Canvas to provide students with learning materials including 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.

Additional Information on Learning Resources

Required Text:

  • Stability, Instability and Chaos by Paul Glendinning, Cambridge University Press (1994). 
Students are also expected to use and become proficient with a software package to study phase-space analysis and bifurcation diagrams numerically. We will use the software package CoCo which runs under MATLAB; you can download CoCo for free from https://sourceforge.net/p/cocotools/wiki/Home/.
An alternative stand-alone software package that is also very suitable is XPPAUT, which is freely downloadable from http://www.math.pitt.edu/~bard/xpp/xpp.html and particularly suitable for students interested in Mathematical Biology. 
If you wish to work with CoCo, you can run MATLAB remotely via https://www.auckland.ac.nz/flexit/ or download this software for home use by following the instructions here: https://www.software.auckland.ac.nz/en/matlab.html

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.

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 course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework 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. This requirement also applies to sources on the internet. A student's assessed work may be reviewed for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

Similarly, research students must meet the University’s expectations of good research practice. This requires:

  • Honesty - in all aspects of research work
  • Accountability - in the conduct of research
  • Professional courtesy and fairness – in working with others
  • Good stewardship – on behalf of others
  • Transparency – of research process and presentation of results
  • Clarity - communication to be understandable, explainable and accessible

For more information on the University’s expectations of academic integrity, please see the Academic Conduct section of the University policy hub.

Disclaimer

Elements of this outline may be subject to change. The latest information about taught courses is made available to enrolled students in Canvas.

Students may be asked to submit assessments digitally. The University reserves the right to conduct scheduled tests and examinations online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.


Assessment and Learning Outcomes

Special Requirements

The course makes use of the software package CoCo which runs under MATLAB (see the Learning Resources). You will either need a home computer to run CoCo and MATLAB on, or you will need to be regularly on campus to make use of the University of Auckland's computer labs to work on any assignment tasks that use CoCo. 
MATLAB is available in the University of Auckland's computer labs, or you can use it remotely via https://www.auckland.ac.nz/flexit/. You can also download MATLAB and install it on your own computer or laptop  by following the instructions here: https://www.software.auckland.ac.nz/en/matlab.html
You can find details about our computer labs here: https://www.library.auckland.ac.nz/services/it-essentials/computer-facilities

Course Learning Outcomes

CLO #OutcomeProgramme Capability Link
1
2
3

Assessments

Assessment TypeAssessment PercentageAssessment Classification

Assessment to CLO Mapping

Assessment Type123

Student Feedback, Support and Charter

Student Feedback

Feedback on taught courses is gathered from students at the end of each semester through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback and respond with summaries and actions. Your feedback helps teachers to improve the course and its delivery for future students. In addition, class Representatives in each class can take feedback to the department and faculty staff-student consultative committees.

Additional Information on Student Feedback

Student feedback is considered on an on-going basis throughout the year.

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.

Tuākana

Tuākana is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at Tuakana Learning Communities.

Additional Information on Tuākana

Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at
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.

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.

Wellbeing

We all go through tough times during the semester, or see our friends struggling. There is lots of help out there - please see the Support Services page for information on support services in the University and the wider community.

Special Circumstances

If your ability to complete assessed work 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. This should be done as soon as possible and no later than seven days after the affected test or exam date.

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.

Student Academic Complaints and Disputes

Students with concerns about teaching including how a course is delivered, the resources provided, or supervision arrangements, have the right to express their concerns and seek resolution. The university encourages informal resolution where possible, as this is quicker and less stressful. For information on the informal and formal complaints processes, please refer to the Student Academic Complaints Statute in the Student Policies and Guidelines section of the Policy Hub.