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MATH3902 – Nonlinear Dynamics and Chaos

2024 – Session 1, In person-scheduled-weekday, North Ryde

General Information

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Unit convenor and teaching staff Unit convenor and teaching staff
Catherine Penington
catherine.penington@mq.edu.au
Elena Vynogradova
elena.vynogradova@mq.edu.au
Credit points Credit points
10
Prerequisites Prerequisites
(MATH2010 or MATH235) and (MATH2020 or MATH2110 or MATH232 or MATH236)
Corequisites Corequisites
Co-badged status Co-badged status
Unit description Unit description

The remarkable fact that determinism does not guarantee regular or predictable behaviour is having a major impact on many fields of science and engineering, as well as mathematics. The discovery of chaos, or of chaotic motions, in simple dynamical systems changed our understanding of the foundations of physics and has found many practical applications. Dynamical systems involve the study of maps and systems of differential equations. In this unit, the diversity of nonlinear phenomena is explored through the study of second-order differential equations and second-order systems, in which nonlinearity is usually ignored in simpler treatments.

Important Academic Dates

Information about important academic dates including deadlines for withdrawing from units are available at https://www.mq.edu.au/study/calendar-of-dates

Learning Outcomes

On successful completion of this unit, you will be able to:

  • ULO1: Explain the principles and basic concepts of Nonlinear Dynamical Systems, both of discrete systems and continuous ones through Differential Equations. In particular, gain an appreciation of the characteristics of ‘chaotic’ behaviour.
  • ULO2: Competently use modern computing software to model a range of phenomena in science and engineering, displaying the complexity that can occur with nonlinear systems.
  • ULO3: Demonstrate an understanding of the breadth of the theory of Nonlinear Systems, and how the distinction between periodic and non-periodic orbits is related to the very numbers used to model or describe the state of a system.
  • ULO4: Construct logical, clearly presented and justified mathematical arguments incorporating deductive reasoning especially in the context of the Dynamical Systems, and to produce appropriate computer graphics to aptly illustrate the phenomena involved.

General Assessment Information

REQUIREMENTS TO PASS THIS UNIT

To pass this unit you must: Achieve a total mark equal to or greater than 50%.

LATE SUBMISSION OF WORK:

Unless a Special Consideration request has been submitted and approved, a 5% penalty (of the total possible mark) will be applied each day a written assessment is not submitted, up until the 7th day (including weekends). After the 7th day, a grade of 0 will be awarded even if the assessment is submitted. Submission time for all written assessments is set at 11:55 pm. A 1-hour grace period is provided to students who experience a technical concern.

For any late submission of time-sensitive tasks, such as scheduled exams, students need to submit an application for Special Consideration.

Assessments where Late Submissions will be accepted:

  • Assignments 1, 2, 3 – YES, Standard Late Penalty applies
  • Final Exam – NO, unless Special Consideration is Granted

SPECIAL CONSIDERATION

The Special Consideration Policy aims to support students who have been impacted by short- term circumstances or events that are serious, unavoidable and significantly disruptive, and which may affect their performance in assessment. If you experience circumstances or events that affect your ability to complete the written assessments in this unit on time, please inform the convenor and submit a Special Consideration request through ask.mq.edu.au.

Assessment Tasks

Name Weighting Hurdle Due
Final Examination 60% No Examination period
Assignment 3 10% No Week 12
Assignment 2 15% No Week 9
Assignment 1 15% No Week 5

Final Examination

Assessment Type 1: Examination
Indicative Time on Task 2: 18 hours
Due: Examination period
Weighting: 60%

 

This will be an invigilated exam, held during the final exam period. It will test the ability of students to utilise concepts and techniques learnt in lectures. The final examination is a hurdle requirement. To satisfy the hurdle requirement students must score at least 50% on the final examimnation.

 
On successful completion you will be able to:
  • Explain the principles and basic concepts of Nonlinear Dynamical Systems, both of discrete systems and continuous ones through Differential Equations. In particular, gain an appreciation of the characteristics of ‘chaotic’ behaviour.
  • Competently use modern computing software to model a range of phenomena in science and engineering, displaying the complexity that can occur with nonlinear systems.
  • Demonstrate an understanding of the breadth of the theory of Nonlinear Systems, and how the distinction between periodic and non-periodic orbits is related to the very numbers used to model or describe the state of a system.
  • Construct logical, clearly presented and justified mathematical arguments incorporating deductive reasoning especially in the context of the Dynamical Systems, and to produce appropriate computer graphics to aptly illustrate the phenomena involved.

Assignment 3

Assessment Type 1: Problem set
Indicative Time on Task 2: 9 hours
Due: Week 12
Weighting: 10%

 

The assignment will test the ability of the students to develop and analyse mathematical problems using concepts and techniques learnt in lectures.

 
On successful completion you will be able to:
  • Explain the principles and basic concepts of Nonlinear Dynamical Systems, both of discrete systems and continuous ones through Differential Equations. In particular, gain an appreciation of the characteristics of ‘chaotic’ behaviour.
  • Competently use modern computing software to model a range of phenomena in science and engineering, displaying the complexity that can occur with nonlinear systems.
  • Demonstrate an understanding of the breadth of the theory of Nonlinear Systems, and how the distinction between periodic and non-periodic orbits is related to the very numbers used to model or describe the state of a system.
  • Construct logical, clearly presented and justified mathematical arguments incorporating deductive reasoning especially in the context of the Dynamical Systems, and to produce appropriate computer graphics to aptly illustrate the phenomena involved.

Assignment 2

Assessment Type 1: Problem set
Indicative Time on Task 2: 9 hours
Due: Week 9
Weighting: 15%

 

The assignment will test the ability of the students to develop and analyse mathematical problems using concepts and techniques learnt in lectures.

 
On successful completion you will be able to:
  • Explain the principles and basic concepts of Nonlinear Dynamical Systems, both of discrete systems and continuous ones through Differential Equations. In particular, gain an appreciation of the characteristics of ‘chaotic’ behaviour.
  • Competently use modern computing software to model a range of phenomena in science and engineering, displaying the complexity that can occur with nonlinear systems.
  • Demonstrate an understanding of the breadth of the theory of Nonlinear Systems, and how the distinction between periodic and non-periodic orbits is related to the very numbers used to model or describe the state of a system.
  • Construct logical, clearly presented and justified mathematical arguments incorporating deductive reasoning especially in the context of the Dynamical Systems, and to produce appropriate computer graphics to aptly illustrate the phenomena involved.

Assignment 1

Assessment Type 1: Problem set
Indicative Time on Task 2: 9 hours
Due: Week 5
Weighting: 15%

 

The assignment will test the ability of the students to develop and analyse mathematical problems using concepts and techniques learnt in lectures.

 
On successful completion you will be able to:
  • Explain the principles and basic concepts of Nonlinear Dynamical Systems, both of discrete systems and continuous ones through Differential Equations. In particular, gain an appreciation of the characteristics of ‘chaotic’ behaviour.
  • Competently use modern computing software to model a range of phenomena in science and engineering, displaying the complexity that can occur with nonlinear systems.
  • Demonstrate an understanding of the breadth of the theory of Nonlinear Systems, and how the distinction between periodic and non-periodic orbits is related to the very numbers used to model or describe the state of a system.
  • Construct logical, clearly presented and justified mathematical arguments incorporating deductive reasoning especially in the context of the Dynamical Systems, and to produce appropriate computer graphics to aptly illustrate the phenomena involved.

1 If you need help with your assignment, please contact:

  • the academic teaching staff in your unit for guidance in understanding or completing this type of assessment
  • the Writing Centre for academic skills support.

2 Indicative time-on-task is an estimate of the time required for completion of the assessment task and is subject to individual variation

Delivery and Resources

CLASSES

Lectures (beginning in Week 1): There are two one-hour lectures each week.

SGTA classes (beginning in Week 2): there is one one-hour SGTA each week. Led by an SGTA instructor, students will discuss problems related to the previous week's lecture content, and work through similar problems.

TEXTBOOKS:

There is no set textbook for MATH3902. The following texts provide useful references for various sections of the course:

  • Hirsch, M.W., Smale S. & Devaney, R.L., Differential Equations, Dynamical Systems & An Introduction to Chaos, Elsevier Academic Press. (Available online via the library.)
  • Drazin, P.G., Nonlinear Systems, Cambridge University Press.
  • Strogatz. S. Nonlinear dynamics with chaos. Westview Press. (Available online via the library.)
  • Salinelli, E. Discrete dynamical models. Springer. (Available online via the library.)

COMMUNICATION

We will communicate with you via your university email or through announcements on iLearn. Queries to convenor can either be placed on the iLearn discussion board or sent to your lecturers from your university email address.

COVID Information

For the latest information on the University’s response to COVID-19, please refer to the Coronavirus infection page on the Macquarie website: https://www.mq.edu.au/about/coronavirusfaqs. Remember to check this page regularly in case the information and requirements change during semester. If there are any changes to this unit in relation to COVID, these will be communicated via iLearn.

Unit Schedule

Week

Material

Assessment Due

1

Continuous Dynamical Systems: Introduction, Autonomous Systems

  

2

Continuous Dynamical Systems: Autonomous Systems

  

3

Continuous Dynamical Systems: Energy

  

4

Continuous Dynamical Systems: Energy, Poincare-Bendixson Theorem

  

5

Continuous Dynamical Systems: Poincare-Bendixson Theorem

Assignment 1 due

6

Continuous Dynamical Systems: Bifurcations

  

7

Discrete Dynamical Systems: Introduction

  

8

​​​​​​​Discrete Dynamical Systems: Equilibrium points & Stability

  

9

Discrete Dynamical Systems: Equilibrium points & Stability, continued

Assignment 2 due

10

Discrete Dynamical Systems: Periodic orbits, Sharkovskii's Theorem

  

11

Discrete Dynamical Systems: Bifurcations, Period doubling, and Stability

  

12

Discrete Dynamical Systems: Logistic map, Chaos

Assignment 3 due

13

Revision

  

Policies and Procedures

Macquarie University policies and procedures are accessible from Policy Central (https://policies.mq.edu.au). Students should be aware of the following policies in particular with regard to Learning and Teaching:

Students seeking more policy resources can visit Student Policies (https://students.mq.edu.au/support/study/policies). It is your one-stop-shop for the key policies you need to know about throughout your undergraduate student journey.

To find other policies relating to Teaching and Learning, visit Policy Central (https://policies.mq.edu.au) and use the search tool.

Student Code of Conduct

Macquarie University students have a responsibility to be familiar with the Student Code of Conduct: https://students.mq.edu.au/admin/other-resources/student-conduct

Results

Results published on platform other than eStudent, (eg. iLearn, Coursera etc.) or released directly by your Unit Convenor, are not confirmed as they are subject to final approval by the University. Once approved, final results will be sent to your student email address and will be made available in eStudent. For more information visit connect.mq.edu.au or if you are a Global MBA student contact globalmba.support@mq.edu.au

Academic Integrity

At Macquarie, we believe academic integrity – honesty, respect, trust, responsibility, fairness and courage – is at the core of learning, teaching and research. We recognise that meeting the expectations required to complete your assessments can be challenging. So, we offer you a range of resources and services to help you reach your potential, including free online writing and maths support, academic skills development and wellbeing consultations.

Student Support

Macquarie University provides a range of support services for students. For details, visit http://students.mq.edu.au/support/

The Writing Centre

The Writing Centre provides resources to develop your English language proficiency, academic writing, and communication skills.

The Library provides online and face to face support to help you find and use relevant information resources. 

Student Services and Support

Macquarie University offers a range of Student Support Services including:

Student Enquiries

Got a question? Ask us via the Service Connect Portal, or contact Service Connect.

IT Help

For help with University computer systems and technology, visit http://www.mq.edu.au/about_us/offices_and_units/information_technology/help/. 

When using the University's IT, you must adhere to the Acceptable Use of IT Resources Policy. The policy applies to all who connect to the MQ network including students.

Changes from Previous Offering

None

Unit information based on version 2024.01R of the Handbook