MATH3063: Semester 1, 2020

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Units MATH3063 Semester 1 2020 [Normal day]

Unit of study_

MATH3063: Nonlinear ODEs with Applications

Semester 1, 2020 [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 will be drawn from predator-prey systems, transmission of diseases, chemical reactions and other equations and systems from mathematical biology.

Unit details and rules

Unit code	MATH3063
Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prohibitions ?	MATH3963 or MATH4063
Prerequisites ?	12 credit points of MATH2XXX units of study
Corequisites ?	None
Assumed knowledge ?	MATH2061 or MATH2961 or [MATH2X21 and MATH2X22]
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Mary Myerscough, mary.myerscough@sydney.edu.au
Lecturer(s)	Ian Lizarraga, ian.lizarraga@sydney.edu.au
Lecturer(s)	Mary Myerscough, mary.myerscough@sydney.edu.au

Type	Description	Weight	Due	Length
Final exam	Examination Written exam with calculations and graphs	70%	Formal exam period	2 hours
Outcomes assessed:
Assignment	Assignment 1 Written assignment	5%	Week 04	Approx two weeks to complete
Outcomes assessed:
Small test	Class test 1 Written quiz completed on paper	10%	Week 07	45 minutes
Outcomes assessed:
Assignment	Assignment 2 Written assignment	10%	Week 10	Approx two weeks to complete
Outcomes assessed:
Small test	Class test 2 Written test completed on paper	5%	Week 12	30 minutes
Outcomes assessed:

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 sydney.edu.au/students/guide-to-grades.

For more information see guide to grades.

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 Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

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.

Weekly schedule

WK	Topic	Learning activity
Week 01	First order equations: Mathematical models–exponential and logistic growth; definition of linear and nonliner, autonomous and non-autonomous. Phase portraits, equilibria, stability (using phase portraits), linear stability, linearisation for single first order equations	Lecture and tutorial (4 hr)
Week 02	Bifurcation in single, first order equations, including the expential and logistic models. Harvesting, both constant rate and constant effort. Population catastrophes. Bifurcation diagrams. Hysteresis	Lecture and tutorial (4 hr)
Week 03	Introduction to predator-prey models, specifically the Lotka-Volterra model. Nonlinear models cannot be solved explicitly, hence the need for new mathematical tools. Introduction of the phase plane; nullclines, flows, sketching solutions. Phase plane of the Lotka-Volterra equations motivates the need for more information	Lecture and tutorial (4 hr)
Week 04	Nonlinear systems and linearisation. Solving linear systems. Classification of the behaviour of linear systems (nodes, focuses, saddles, centres, etc). Phase planes of linear systems. The Jacobian matrix and classifying behaviour using its trace and determinan	Lecture and tutorial (4 hr)
Week 05	Phase portraits and linear stability analysis of a variety of nonlinear systems (lots of examples). Existence and uniqueness of solutions. Different types of stability	Lecture and tutorial (4 hr)
Week 06	Lotka-Volterra equations using linear analysis and phase planes. Harvesting the Lotka-Volterra equations. Structural instability. Other predator-prey systems. Models for ecological competition, mutualism	Lecture and tutorial (4 hr)
Week 07	Models for the spread of disease. Definition of an epidemic. Basic SIR model. Critical population sizes. Vaccination effects. What happens as t → ∞. SIS and SIRS models, crisscross infections and STDs	Lecture and tutorial (4 hr)
Week 08	Lyapunov stability. Finding and using Lyapunov functions. Sketch of Lyapunov theorems.	Lecture and tutorial (4 hr)
Week 09	First integrals, Hamiltonian systems and gradient systems. Definition of first integral. The Lotka-Volterra equations as an example of a Hamiltonian system. Conservative systems. nonlinear pendulum, Duffing equation, the Van der Pol oscillator	Lecture and tutorial (4 hr)
Week 10	Limit cycles: definition, stability analysis, phase protraits. Biological examples (mainly computational)	Lecture and tutorial (4 hr)
Week 11	Bifurcation in systems of two first order ODEs. Statement of the Hopf bifurcation theorem. Creation of limit cycles. The Brusselator model. Predator-prey and epidemiological models with Hopf bifurcations	Lecture and tutorial (4 hr)
Week 12	Fitzhugh-Nagumo equations, relaxation oscillations and excitable media	Lecture and tutorial (4 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.

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 of the role of basic bifurcations in nonlinear systems 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

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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

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

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

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

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

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