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MAT3103 Mathematical Modelling and Dynamical Systems

Semester 2, 2022 Toowoomba On-campus
Units : 1
Faculty or Section : Faculty of Health, Engineering and Sciences
School or Department : School of Mathematics, Physics & Computing
Grading basis : Graded
Course fee schedule : /current-students/administration/fees/fee-schedules

Staffing

Examiner:

Requisites

Pre-requisite: MAT2100 or MAT2500 or ENM2600

Overview

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 practical 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.

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. The oncampus offering of this course is normally available only in even-numbered years. The external offering of this course is available yearly.

Course learning outcomes

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 of solutions
  3. illustrate solutions by sketching phase portraits; deduce qualitative conclusions
  4. apply mathematical equations, modelling processes and principles to a range of authentic and real-life problems

Topics

Description Weighting(%)
1. Systems of differential equations: solution of linear ODE's, the conversion of higher-order linear ODE's to first-order systems; fixed points and phase portraits for second order ODEs, qualitative solution of nonlinear ODE in the vicinity of critical points. 15.00
2. Potentials, bifurcations, catastrophes. 15.00
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. 10.00
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

Introductory Book (current year), Course MAT3103, Mathematical Modelling for Dynamics, USQ Distance and e-Learning Centre, Toowoomba.
(Available on course 精东传媒appDesk.)
精东传媒app Book (current year), Course MAT3103, Mathematical Modelling for Dynamics, USQ Distance and e-Learning Centre, Toowoomba.
(Available on course 精东传媒appDesk.)
Access to any graphical package to visualise solutions.
Access to computer or internet facilities for mathematical typesetting.

Student workload expectations

To do well in this subject, students are expected to commit approximately 10 hours per week including class contact hours, independent study, and all assessment tasks. If you are undertaking additional activities, which may include placements and residential schools, the weekly workload hours may vary.

Assessment details

Approach Type Description Group
Assessment
Weighting (%) Course learning outcomes
Assignments Written Problem Solving 1 No 30 1
Assignments Written Problem Solving 2 No 30 2,3
Assignments Written Report No 40 4
Date printed 10 February 2023