Skip to main content
Unit of study_

MATH4063: Dynamical Systems and Applications

The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods. The theory has many applications and stimulates new developments in almost all areas of mathematics. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, linearisation, and analysis of asymptotic behaviour. The applications in this unit will be drawn from predator-prey systems, population models, chemical reactions, and other equations and systems from mathematical biology. You will learn how to use ordinary differential equations to model biological, chemical, physical and/or economic systems and how to use different methods from dynamical systems theory and the theory of nonlinear ordinary differential equations to find the qualitative outcome of the models. By doing this unit you will develop skills in using and analysing nonlinear differential equations which will prepare you for further studies in mathematics, systems biology or physics or for careers in mathematical modelling.

Code MATH4063
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
(A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 or MATH3066 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)]
MATH3063 or MATH3963
Assumed knowledge:
Linear ODEs (for example, MATH2921), eigenvalues and eigenvectors of a matrix, determinant and inverse of a matrix and linear coordinate transformations (for example, MATH2922), Cauchy sequence, completeness and uniform convergence (for example, MATH2923)

At the completion of this unit, you should be able to:

  • LO1. explain the principle of linear approximations to nonlinear systems and use this to analyse system behaviour close to steady state
  • LO2. synthesise graphical information from nullclines and flow to construct qualitative phase plane solutions to problems in nonlinear systems
  • LO3. demonstrate knowledge of the theory of existence and uniqueness of solutions of ordinary differential equations
  • LO4. interpret model results and evaluate and explain the limitations of models in representing real systems
  • LO5. Uunderstand the role of basic bifurcations in nonlinear systems, by synthesising graphical, symbolic and computational information, and evaluating the effect of parameter variation on observed model behaviour
  • LO6. explain and apply the concept of flow invariance
  • LO7. construct rigorous proofs and apply theory in novel and diverse applications