Unit outline_

# MATH4063: Dynamical Systems and Applications

## Overview

The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods. The theory has many applications and stimulates new developments in almost all areas of mathematics. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, linearisation, and analysis of asymptotic behaviour. The applications in this unit will be drawn from predator-prey systems, population models, chemical reactions, and other equations and systems from mathematical biology. You will learn how to use ordinary differential equations to model biological, chemical, physical and/or economic systems and how to use different methods from dynamical systems theory and the theory of nonlinear ordinary differential equations to find the qualitative outcome of the models. By doing this unit you will develop skills in using and analysing nonlinear differential equations which will prepare you for further studies in mathematics, systems biology or physics or for careers in mathematical modelling.

### Unit details and rules

Academic unit Mathematics and Statistics Academic Operations 6 (A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 or MATH3066 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)] None MATH3063 or MATH3963 Linear ODEs (for example, MATH2921), eigenvalues and eigenvectors of a matrix, determinant and inverse of a matrix and linear coordinate transformations (for example, MATH2922), Cauchy sequence, completeness and uniform convergence (for example, MATH2923) Yes

### Teaching staff

Coordinator Robert Marangell, robert.marangell@sydney.edu.au

## Assessment

Type Description Weight Due Length
Final exam (Take-home short release) Final exam
Written exam, 50% to pass
55% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
10% Week 04 n/a
Outcomes assessed:
Written midterm
25% Week 07 45 minutes
Outcomes assessed:
Assignment Assignment 2
10% Week 12 n/a
Outcomes assessed:
= Type D final exam

### Assessment summary

• Assignments: These assignments will require you to integrate information from lectures and tutorial's to solve mathematical problems. The use of mathematical computer software is recommended. Assignments must be typed and a pdf copy must be submitted via the turnitin. Hand written assignments or photographs thereof will not be accepted.
• Midterm quiz: This in-class midterm quiz will test your understanding of material covered in the first 5 weeks of this course (lectures and tutorials).
• Final exam: The exam will cover all material in the unit from both lectures and tutorial classes.

To pass the course you need to (1) achieve at least 50% of the total assessment mark and (2) achieve at least 50% of the exam mark. If you miss the mid semester exam (due to a granted special consideration), then the exam mark weight will be adjusted accordingly.

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.

### 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.

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 Learning outcomes
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)
Week 13 Revision 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.

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

• Meiss, J. D. (2007). Differential Dynamical Systems . Philadelphia: Society for Industrial and Applied Mathematics.

## 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.

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 state
• 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 of solutions of ordinary differential equations
• LO4. interpret model results and evaluate and explain the limitations of models in representing real systems
• LO5. Uunderstand the role of basic bifurcations in nonlinear systems, by synthesising graphical, symbolic and computational information, and evaluating the effect of parameter variation on observed model behaviour
• LO6. explain and apply the concept of flow invariance
• LO7. construct rigorous proofs and apply theory in novel and diverse applications

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

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## 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.