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MATH3902 – Nonlinear Dynamics and Chaos

2022 – Session 1, In person-scheduled-weekday, North Ryde

General Information

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Unit convenor and teaching staff Unit convenor and teaching staff Unit Convenor and Lecturer
Catherine Penington
catherine.penington@mq.edu.au
Contact via Email
12 Wally's Walk, Office 717
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Unit Convenor and Lecturer
Elena Vynogradova
elena.vynogradova@mq.edu.au
Contact via Email
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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

LATE SUBMISSION OF WORK: All assessment tasks must be submitted by the official due date and time. Should these assessments be missed due to illness or misadventure, students should apply for Special Consideration. Assessments not submitted by the due date will receive a mark of zero.

Assessment Tasks

Name Weighting Hurdle Due
Assignment 1 15% No Week 4
Assignment 2 15% No Week 9
Assignment 3 10% No Week 12
Final Examination 60% No Final Exam Period

Assignment 1

Assessment Type 1: Problem set
Indicative Time on Task 2: 9 hours
Due: Week 4
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 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 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.

Final Examination

Assessment Type 1: Examination
Indicative Time on Task 2: 18 hours
Due: Final Exam 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.

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

Lectures: Concepts are introduced, explained and illustrated. There will be two formal contact hours per week, consisting of two lectures.

Small group teaching activity: Led by an SGTA instructor, students will discuss problems related to the previous week's lecture content, and work through similar problems.

Off-shore students must email the convenor as soon as possible to discuss study options

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 Assignment 1 due
5 Continuous Dynamical Systems: Poincare-Bendixson Theorem

 
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  

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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.

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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 ask.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. 

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Student Enquiries

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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.

Unit information based on version 2022.02 of the Handbook