MATH3063: Semester 1, 2026

Units MATH3063 Semester 1 2026 [Normal day]

Unit outline_

MATH3063: Nonlinear ODEs with Applications

Semester 1, 2026 [Normal day] - Camperdown/Darlington, Sydney

Overview

This unit of study is an introduction to the theory of systems of ordinary differential equations. Such systems model many types of phenomena in engineering, biology and the physical sciences. The emphasis will not be on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behaviour. The aim is to develop a good geometrical intuition into the behaviour of solutions to such systems. Some background in linear algebra, and familiarity with concepts such as limits and continuity, will be assumed. The applications in this unit are from biology, physics, chemistry, and engineering, including population dynamics, epidemics, chemical reactions, and simple mechanical systems.

Unit details and rules

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	12 credit points of MATH2XXX units of study
Corequisites ?	None
Prohibitions ?	MATH3963 or MATH4063
Assumed knowledge ?	MATH2061 or MATH2961 or [MATH2X21 and MATH2X22]
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Holger Dullin, holger.dullin@sydney.edu.au

Assessment

The census date for this unit availability is 31 March 2026

Details
Criteria

Type	Description	Weight	Due	Length	Use of AI
Written exam	Final Exam Written exam	60%	Formal exam period	2 hours	AI prohibited
Outcomes assessed:
Written work	Assignment 1 Written assignment	10%	Week 04 Due date: 22 Mar 2026 at 23:59 Closing date: 01 Apr 2026	Approx two weeks to complete	AI allowed
Outcomes assessed:
In-person written or creative task	Class test 1 Written test	10%	Week 08 Due date: 13 Apr 2026 at 13:00 Closing date: 13 Apr 2026	45 minutes	AI prohibited
Outcomes assessed:
Written work	Assignment 2 Written assignment	10%	Week 10 Due date: 10 May 2026 at 13:59 Closing date: 20 May 2026	Approx two weeks to complete	AI allowed
Outcomes assessed:
In-person written or creative task	Class test 2 Written test	10%	Week 12 Due date: 18 May 2026 at 13:00 Closing date: 18 May 2026	45 minutes	AI prohibited
Outcomes assessed:
= hurdle task ?

Assessment summary

Detailed information for each assessment can be found on Canvas.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name	Mark range	Description
High distinction	85 - 100	Representing complete or close to complete mastery of the material.
Distinction	75 - 84	Representing excellence, but substantially less than complete mastery.
Credit	65 - 74	Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.
Pass	50 - 64	Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.
Fail	0 - 49	When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see guide to grades.

Use of generative artificial intelligence (AI)

You can use generative AI tools for open assessments. Restrictions on AI use apply to secure, supervised assessments used to confirm if students have met specific learning outcomes.

Refer to the assessment table above to see if AI is allowed, for assessments in this unit and check Canvas for full instructions on assessment tasks and AI use.

If you use AI, you must always acknowledge it. Misusing AI may lead to a breach of the Academic Integrity Policy.

Visit the Current Students website for more information on AI in assessments, including details on how to acknowledge its use.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

Deduction of 5% of the maximum mark for each calendar day after the due date.
After ten calendar days late, a mark of zero will be awarded.

Academic integrity

The University expects students to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

Our website provides information on academic integrity and the resources available to all students. This includes advice on how to avoid common breaches of academic integrity. Ensure that you have completed the Academic Honesty Education Module (AHEM) which is mandatory for all commencing coursework students

Penalties for serious breaches can significantly impact your studies and your career after graduation. It is important that you speak with your unit coordinator if you need help with completing assessments.

Visit the Current Students website for more information on AI in assessments, including details on how to acknowledge its use.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Support for students

The Support for Students Policy reflects the University’s commitment to supporting students in their academic journey and making the University safe for students. It is important that you read and understand this policy so that you are familiar with the range of support services available to you and understand how to engage with them.

The University uses email as its primary source of communication with students who need support under the Support for Students Policy. Make sure you check your University email regularly and respond to any communications received from the University.

Learning resources and detailed information about weekly assessment and learning activities can be accessed via Canvas. It is essential that you visit your unit of study Canvas site to ensure you are up to date with all of your tasks.

If you are having difficulties completing your studies, or are feeling unsure about your progress, we are here to help. You can access the support services offered by the University at any time:

Support and Services (including health and wellbeing services, financial support and learning support)
Course planning and administration
Meet with an Academic Adviser

Weekly schedule

WK	Topic	Learning activity
Week 01	Introduction to ODEs and modelling. Flows on the line; the phase line, equilibria, stability, linear stability, existence and uniqueness. Examples.	Lecture (3 hr)
Week 02	Bifurcation of flows on the line; saddle-node, transcritical, and pitchfork bifurcation. Bifurcation diagrams, hysteresis. Dimensional analysis, scaling. Examples.	Lecture (3 hr)
Week 02	Bifurcation of flows on the line; saddle-node, transcritical, and pitchfork bifurcation. Bifurcation diagrams, hysteresis. Dimensional analysis, scaling. Examples.	Tutorial (1 hr)
Week 03	Bifurcations continued. Qualitative bifurcation analysis. Examples. Introduction to phase plane analysis.	Lecture (3 hr)
Week 03	Bifurcations continued. Qualitative bifurcation analysis. Examples. Introduction to phase plane analysis.	Tutorial (1 hr)
Week 04	Linear systems, eigenvalues and eigenvectors. Classification of linear systems (nodes, focuses, saddles, centres, etc). Phase planes of linear systems. The Jacobian matrix; classifying behaviour using its trace and determinant.	Lecture (3 hr)
Week 04	Linear systems, eigenvalues and eigenvectors. Classification of linear systems (nodes, focuses, saddles, centres, etc). Phase planes of linear systems. The Jacobian matrix; classifying behaviour using its trace and determinant.	Tutorial (1 hr)
Week 05	Phase plane analysis of non-linear systems. Linearisation, linear stability analysis. Existence and uniqueness of solutions. Nullclines, index theorems.	Lecture (3 hr)
Week 05	Phase plane analysis of non-linear systems. Linearisation, linear stability analysis. Existence and uniqueness of solutions. Nullclines, index theorems.	Tutorial (1 hr)
Week 06	Examples: Lotka-Volterra equations, harvesting, predator-prey systems. Other predator-prey systems. Chemical reactions.	Lecture (3 hr)
Week 06	Examples: Lotka-Volterra equations, harvesting, predator-prey systems. Other predator-prey systems. Chemical reactions.	Tutorial (1 hr)
Week 07	Examples: Models for the spread of disease. Basic SIR model, SIS, SIRS and others.	Lecture (3 hr)
Week 07	Examples: Models for the spread of disease. Basic SIR model, SIS, SIRS and others.	Tutorial (1 hr)
Week 08	Conservative and reversible systems, first integrals, Hamiltonian systems and gradient systems. Examples.	Lecture (3 hr)
Week 08	Conservative and reversible systems, first integrals, Hamiltonian systems and gradient systems. Examples.	Tutorial (1 hr)
Week 09	Lyapunov stability. Finding and using Lyapunov functions. Sketch of Lyapunov theorems. Examples.	Lecture (3 hr)
Week 09	Lyapunov stability. Finding and using Lyapunov functions. Sketch of Lyapunov theorems. Examples.	Tutorial (1 hr)
Week 10	Limit cycles: definition, stability analysis, phase portraits, Poincare-Bendixson. Relaxation oscillations. Examples.	Lecture (3 hr)
Week 10	Limit cycles: definition, stability analysis, phase portraits, Poincare-Bendixson. Relaxation oscillations. Examples.	Tutorial (1 hr)
Week 11	Bifurcations in systems of two first order ODEs. Hopf bifurcation. Creation of limit cycles. Examples, e.g. Brusselator.	Lecture (3 hr)
Week 11	Bifurcations in systems of two first order ODEs. Hopf bifurcation. Creation of limit cycles. Examples, e.g. Brusselator.	Tutorial (1 hr)
Week 12	Chaos. Lorenz model, Poincare map.	Lecture (3 hr)
Week 12	Chaos. Lorenz model, Poincare map.	Tutorial (1 hr)
Week 13	Revision	Lecture (3 hr)
Week 13	Revision	Tutorial (1 hr)

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Required readings

All readings for this unit can be accessed through the Library eReserve, available on Canvas.

- Strogatz, S. H. (2018). Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering (2nd ed.). Westview Press.

Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

Outcomes
Graduate qualities

At the completion of this unit, you should be able to:

LO1. explain the principle of linear approximations to nonlinear systems and use this to analyse system behaviour close to steady states
LO2. synthesise graphical information from nullclines and flow to construct qualitative phase plane solutions to problems in nonlinear systems
LO3. demonstrate knowledge of the theory of existence and uniqueness, and the determination of stability of solutions of ordinary differential equations, including special cases such as Hamiltonian and gradient systems.
LO4. interpret model results and evaluate and explain the limitations of models in representing real systems
LO5. demonstrate a broad understanding the role of basic bifurcations in nonlinear systems by synthesising graphical, symbolic and computational information and evaluate the effect of parameter variation on observed model behaviour
LO6. apply mathematical theory in novel and diverse applications.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1	Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.
GQ2	Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.
GQ3	Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.
GQ4	Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.
GQ5	Inventiveness Generating novel ideas and solutions.
GQ6	Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.
GQ7	Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.
GQ8	Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.
GQ9	Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered.

Additional information

Site visit guidelines

There are no site visit guidelines for this unit.

Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General Laboratory Safety Rules

No eating or drinking is allowed in any laboratory under any circumstances
A laboratory coat and closed-toe shoes are mandatory
Follow safety instructions in your manual and posted in laboratories
In case of fire, follow instructions posted outside the laboratory door
First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

Disclaimer

Important: the University of Sydney regularly reviews units of study and reserves the right to change the units of study available annually. To stay up to date on available study options, including unit of study details and availability, refer to the relevant handbook.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Use of generative artificial intelligence (AI)

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Study commitment

Required readings

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Site visit guidelines

Work, health and safety

Disclaimer

On this page

Useful links