Course specification
The current and official versions of the course specifications are available on the web at http://www.usq.edu.au/course/specification/current.
Please consult the web for updates that may occur during the year.

# MAT3103 Mathematical Modelling and Dynamical Systems

 Semester 2, 2012 External Toowoomba Units : 1 Faculty or Section : Faculty of Sciences School or Department : Maths and Computing Version produced : 22 May 2013

## Staffing

Examiner: Yury Stepanyants

## Requisites

Pre-requisite: MAT2100 or MAT2500

## Rationale

Mathematical modelling is a process of fundamental importance to the practising researcher. Differential equations and an understanding of their qualitative behaviour provide a structure for the analysis of a wide variety of problems. This course uses mathematical tools developed so far and introduces dimensional analysis, the phase-plane concept, elements of bifurcation theory and theory of catastrophe, the calculus of variations and other contemporary methods to explore many problems of practical applications.

## Synopsis

The course uses mathematical tools introduced in pre-requisite studies to model a variety of realistic phenomena surrounding us in everyday life and introduces calculus of variations for optimisation problems. The course emphasises the importance of the dimensional analysis and demonstrates the close connection between phase-plane concept and qualitative analysis of solutions of ODE. The basics of technical communication in the mathematical sciences are developed throughout the course. This course is offered only in even-numbered years.

## Objectives

On completion of this course students will be able to:

1. solve systems of linear differential equations;
2. analyse the dynamics of systems of differential equations to determine stability, sketch phase portraits, and draw qualitative conclusions;
3. demonstrate the ability to apply the modelling process to real-life problems;
4. demonstrate an understanding of the principles of mathematical modelling applied to a range of problems and using mathematical content from previous studies;
5. demonstrate the ability to solve applied problems found in mechanics, physics, engineering and many other areas;
6. apply the Euler-Lagrange equations to find optimal solutions for various optimisation problems;
7. demonstrate the ability to prepare, structure and deliver documents and presentations of technical material.

## Topics

Description Weighting(%)
1. Systems of differential equations: the solution of linear DE's, the conversion of higher-order linear DE's to first-order systems; fixed points and phase portraits, especially in 2-D; qualitative solution of nonlinear, first-order DE's, especially in the region of fixed points. 15.00
2. Potentials, bifurcations, catastrophes. 10.00
3. Dimensions, scaling, dimensional analysis. 10.00
4. Growth and relaxation: exponential growth and decay, autoregulation. 10.00
5. Vibrations in complex systems: free vibrations, mechanical vibrations, nonlinear oscillations, forced vibrations, linear response, resonance, nonlinear response; coupled oscillators. 25.00
6. Dynamic and chaotic oscillations and waves. Simple and strange attractors. Auto-oscillations and auto-waves. 15.00
7. Calculus of variations: challenge problems and functionals; Euler-Lagrange equation, comparison functions, fundamental lemma; special cases; straight lines minimise arc length; geodesics; brachistochrone; the Lagrangian of dynamical systems. 15.00

## Text and materials required to be purchased or accessed

ALL textbooks and materials available to be purchased can be sourced from USQ's Online Bookshop (unless otherwise stated). (https://bookshop.usq.edu.au/bookweb/subject.cgi?year=2012&sem=02&subject1=MAT3103)

• Introductory Book 2012, Course MAT3103, Mathematical Modelling for Dynamics, USQ Distance and e-Learning Centre, Toowoomba.
• Study Book 2012, Course MAT3103, Mathematical Modelling for Dynamics, USQ Distance and e-Learning Centre, Toowoomba.
• Svobodny, Thomas 1998, Mathematical Modeling for Industry and Engineering, Prentice Hall, Upper Saddle River, NJ.

## Reference materials

Reference materials are materials that, if accessed by students, may improve their knowledge and understanding of the material in the course and enrich their learning experience.
• Department of Maths & Computing, University of Southern Queensland 2006, Mathematics and Computing CD-Rom Set,
<http://www.sci.usq.edu.au/courses/mat3103>.
(Some electronic resources for this course may be available via its home page.)
• Fabrikant, A.L., Stepanyants, Yu.A 1998, Propagation of Waves in Shear Flows, World Scientific, Singapore,
<http://www.worldscibooks.com/books/chaos/2557.html>.
• Highham, NJ 1998, Handbook of writing for the mathematical sciences, 2nd edn, SIAM, Philadelphia.
• Kreyszig, E 2006, Advanced Engineering Mathematics, 9th edn, Wiley, New York.
• Weinstock, R 1974, Calculus of variations: with applications to physics and engineering, Dover Publications, New York.

Activity Hours
Assessments 30.00
Directed Study 55.00
Examinations 2.00
Private Study 83.00

## Assessment details

Description Marks out of Wtg (%) Due Date Notes
ASSIGNMENT 1 200 20 09 Nov 2012 (see note 1)
ASSIGNMENT 2 200 20 09 Nov 2012 (see note 2)
ASSIGNMENT 3 200 20 09 Nov 2012 (see note 3)
HOMEWORK 100 10 09 Nov 2012 (see note 4)
EXAM 2HR RESTRICTED 100 30 End S2 (see note 5)

NOTES
1. Further details about the due dates are given in the study schedule of the Introductory Book.
2. Further details about the due dates are given in the study schedule of the Introductory Book.
3. Further details about the due dates are given in the study schedule of the Introductory Book.
4. Further details about the due dates are given in the study schedule of the Introductory Book.
5. Examination dates will be available during the Semester. Please refer to Examination timetable when published.

## Important assessment information

1. Attendance requirements:
There are no attendance requirements for this course. However, it is the students' responsibility to study all material provided to them or required to be accessed by them to maximise their chance of meeting the objectives of the course and to be informed of course-related activities and administration.

2. Requirements for students to complete each assessment item satisfactorily:
To complete each assessment item satisfactorily, students must obtain at least 50% of the marks available for each item.

3. Penalties for late submission of required work:
If students submit assignments after the due date without (prior) approval then a penalty of 5% of the total marks gained by the student for the assignment may apply for each working day late up to ten working days at which time a mark of zero may be recorded. No assignments will be accepted after model answers have been posted.

4. Requirements for student to be awarded a passing grade in the course:
To be assured of receiving a passing grade a student must achieve at least 50% of the total weighted marks available for the course.

5. Method used to combine assessment results to attain final grade:
The final grades for students will be assigned on the basis of the aggregate of the weighted marks /grades obtained for each of the summative assessment items in the course.

6. Examination information:
In a Restricted Examination, candidates are allowed access only to specific materials. The only materials that candidates may use in the restricted examination for this course are: writing materials (non-electronic and free from material which could give the student an unfair advantage in the examination); calculators which do not hold textual information (students must indicate on their examination paper the make and model of any calculator(s) they use during the examination). One A4 sheet of paper, written or typed on one or both sides with any material the student wishes to include. Students whose first language is not English, may take an appropriate unmarked non-electronic translation dictionary (but not technical dictionary) into the examination. Dictionaries with any handwritten notes will not be permitted. Translation dictionaries will be subject to perusal and may be removed from the candidate's possession until appropriate disciplinary action is completed if found to contain material that could give the candidate an unfair advantage.

7. Examination period when Deferred/Supplementary examinations will be held:
Any supplementary or deferred examinations for this course will be held during the next examination period.

8. University Student Policies:
Students should read the USQ policies: Definitions, Assessment and Student Academic Misconduct to avoid actions which might contravene University policies and practices. These policies can be found at http://policy.usq.edu.au/portal/custom/search/category/usq_document_policy_type/Student.1.html.

## Assessment notes

1. Assignments: The due date for an assignment is the date by which a student must despatch the assignment to the USQ. The onus is on the student to provide proof of the despatch date, if requested by the Examiner. Students must retain a copy of each item submitted for assessment. This must be despatched to USQ within 24 hours of receipt of a request from the Examiner to do so. In accordance with University Policy, the examiner may grant an extension of the due date of an assignment in extenuating circumstances.

2. It is strongly recommended that external students have regular reliable access to email and the Internet for submitting homework and discussing the course material with the lecturers.

3. Exam paper presentation: All exam papers should be presented in accurate and clear writing by blue or black pen. Pencil writing is not acceptable. Assignments can be presented using any word processor such as Word or Latex, or can be neatly written by blue or black pen (but not by pencil!).