# MATHS 761 : Dynamical Systems

## Science

### 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.

### Course Requirements

Prerequisite: B- in both MATHS 340 and 361

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity
Graduate Profile: Master of Science

### Learning Outcomes

By the end of this course, students will be able to:
1. Use appropriate analytic, qualitative and numerical methods to construct phase portraits, bifurcation diagrams and bifurcation sets for systems of ODEs. (Capability 1, 2, 3, 4 and 5)
2. Use phase portraits, bifurcation diagrams and bifurcation sets for systems of ODEs to draw conclusions about the qualitative behaviour of solutions. (Capability 1, 2, 3, 4 and 5)
3. Explain and illustrate an understanding of dynamical systems through calculation, computation, description and discussion of dynamical phenomena. (Capability 1, 2, 3, 4 and 5)

### Assessments

Assessment Type Percentage Classification
Exam 50% Individual Examination
Assignments 30% Individual Coursework
Team activities 20% Group & Individual Coursework
Assessment Type Learning Outcome Addressed
1 2 3
Exam
Assignments
Team activities

### 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.

For further information, please visit
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

### 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).

### Special Requirements

The course makes use of the software package MatCont which runs under MATLAB (see the Learning Resources). You will either need a home computer to run 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 MatCont.

You can find details about our computer labs here: https://www.library.auckland.ac.nz/services/it-essentials/computer-facilities

### 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.

### 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.

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 MatCont which runs under MATLAB; you can download it for free via https://sourceforge.net/projects/matcont/

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 MacCont, you can download MATLAB for home use by following the instructions here: https://www.software.auckland.ac.nz/en/matlab.html

### 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 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 against online source material using computerised detection mechanisms.

### 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

### 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 .

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 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.

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 .

### 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 students may be asked to submit 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. 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.

Published on 28/10/2022 11:28 a.m.