# Current students

Unit of study_

## Overview

In applied mathematics, dynamical systems are systems whose state is changing with time. Examples include the motion of a pendulum, the change in the population of insects in a field or fluid flow in a river. These systems are typically represented mathematically by differential equations or difference equations. Dynamical systems theory reveals universal mechanisms behind disparate natural phenomena. This area of mathematics brings together sophisticated theory from many areas of pure and applied mathematics to create powerful methods that are used to understand and control the dynamical building blocks which make up physical, biological, chemical, engineered and even sociological systems. By doing this unit you will develop a broad knowledge of methods and techniques in dynamical systems, and know how to use these to analyse systems in nature and in technology. This will provide a strong foundation for using mathematics in a broad sweep of applications and for research or further study.

### Details

Academic unit Mathematics and Statistics Academic Operations MATH4414 Advanced Dynamical Systems Semester 2, 2020 Normal day Camperdown/Darlington, Sydney 6

### Enrolment rules

 Prohibitions ? None None None Assumed knowledge is vector calculus (e.g., MATH2X21), linear algebra (e.g., MATH2X22), dynamical systems and applications (e.g., MATH4063 or MATH3X63) or equivalent. Some familiarity with partial differential equations (e.g., MATH3978) and mathematical computing (e.g., MATH3976) is also assumed. Yes

### Teaching staff and contact details

Coordinator Geoffrey Vasil, geoffrey.vasil@sydney.edu.au

## Assessment

Type Description Weight Due Length
Assignment Final take-home exam
Combination analytical/computational work. Typed written report
50% Formal exam period
Due date: 03 Dec 2020 at 23:59
4 days
Outcomes assessed:
Assignment Assignment 1
Combination analytical/computational work. Type written report.
25% Week 06 4 weeks
Outcomes assessed:
Assignment Assignment 2
Combination analytical/computational work. Typed written report
25% Week 11 4 weeks
Outcomes assessed:
• Assignment 1: Will cover the fundamentals of finite maps. Among other things, this will involve the use of composition and transfer operators.

• Assignment 2: Will cover the dynamics of continuous systems. Among other things, this will involve studying chaos and sensitive dependance on initial conditions.
• Final Exam: This will involve synthesising concepts througout the semester.

In each case, the topics will be presented with much reference to soecific examples.

### Assessment criteria

The work will include a mixture of analytical and computer work. The computer results will be presented in the form of figures and text. Not code specifically. The work will be marked on the basis of correctness and quality of presentation.

### 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.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

No late assessments will be accepted.

### Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

The Current Student website provides information on academic honesty, academic dishonesty, 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 dishonesty or plagiarism seriously.

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

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Introduction to definitions and examples Block teaching (4 hr)
Week 02 Further explore foundational definitions and examples Block teaching (4 hr)
Week 03 Explore classic discrete models in depth. Block teaching (4 hr)
Week 04 Explore classic discrete models in depth Block teaching (4 hr)
Week 05 Explore classic discrete models in depth Block teaching (4 hr)
Week 06 Investigate classic continuous models in dynamical systems Block teaching (4 hr)
Week 07 Investigate classic continuous models in dynamical systems Block teaching (4 hr)
Week 08 Investigate classic continuous models in dynamical systems Block teaching (4 hr)
Week 09 Investigate classic continuous models in dynamical systems Block teaching (4 hr)
Week 10 Study chaotic dynamics via classic modest. E.g Double pendulum, Lorentz equations, Logistic map, and renormalisation group theory. Block teaching (4 hr)
Week 11 Study chaotic dynamics via classic modest. E.g Double pendulum, Lorentz equations, Logistic map, and renormalisation group theory. Block teaching (4 hr)
Week 12 Consider the stochastic dynamics and its relation to fast-slow chaotic dynamics. Block teaching (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.

None required. Students can consult the book “Nonlinear Dynamics and Chaos” by Steve Strogatz as a reference.

## 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. Understand the distinction between continuous, discrete, and finite dynamical systems.
• LO2. Understand the concept of transfer and composition operators.
• LO3. Explain how the single and double pendulum are important paradigms for much of general dynamical systems.
• LO4. Be able to gain intuition about a system by computing examples numerically.
• LO5. Understand bifurcations, period doubling, and transition to chaos,
• LO6. Understand the statistical nature of ensembles of deterministic and stochastic solutions to dynamical systems.

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.