Registry

Module Specifications

Current Academic Year 2012 - 2013
Please note that this information is subject to change.

Module Title	Dynamical Systems
Module Code	MS413
School	School of Mathematics
Online Module Resources
NFQ level	8	Credit Rating	7.5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

Description

The course gives an introduction to the qualitative theory of nonlinear ordinary differential equations, and applying the theory to case studies in economics and population dynamics. The course is aimed at upper level undergraduate and first year taught masters students in mathematics. The module complements other modules that students of this module cover in the same academic year, which cover topics such as financial economics, stochastic finance, time series and econometrics. In each of these modules, dynamic economic models with randomness, but without extensive nonlinearities, are studied.

Learning Outcomes

1. Solve simple scalar differential equations
2. Establish qualitative properties, such as boundedness, monotonicity, and asymptotic stability for scalar autonomous differential equations
3. Model problems from economics and population biology as first order equations, analyse the equation, and interpret and critique the model.
4. Determine the normal form for systems of linear differential equations, and prove results concerning the structure of solutions of linear differential equations
5. Model problems from economics and finance using systems of linear differential equations, to analyse the mathematical problem, and to interpret and critique the results of the model
6. Identify conservative dynamical systems, their first integrals, and sketch their phase portraits
7. Analyse nonlinear differential equations with hyperbolic equilibria, and to use linearisation theory and global methods to sketch phase portraits
8. Model problems from economics using systems of differential equations, analyse the resulting mathematical model, and to interpret and critique the results of this analysis

Workload	Full-time hours per semester
*Type*	*Hours*	*Description*
Lecture	36	Lecture
Tutorial	12	Whole group tutorial
Independent learning time	140	Self study
Total Workload: 188

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Review (1 week).
Review of scalar separable, homogeneous and linear differential equations; defintions.

First order differential equations (2 weeks).
Existence and uniqueness; equilibria and stability; monotonicity, boundedness and convexity of trajectories; flow and the flow property. Applications: Solow-Swan, Haavelmo growth model, market saturation and tipping points, harvesting in biology.

Systems of linear differential equations (5 weeks).
Structure of solutions; principle of superposition; normal forms; sketching phase portraits. Applications to economics: Keynesian IS-LM model, stability of general equilibrium, Leontief model, Hicks' model of the business cycle.

Conservative systems (2 weeks).
First integrals; level sets. Nonlinear oscillator and potential; separatrices; sketch of phase portraits. Applications: Predator/prey model, Goodwin's model of the class struggle.

Nonlinear systems with hyperbolic equilibria (2 weeks).
Linearisation, sketching phase portraits. Applications: Competiting species, general equilibrim with production, Samuelson's model of increasing returns, epidemiological models.

Assessment Breakdown
Continuous Assessment	25%	Examination Weight	75%

Course Work Breakdown
Type	Description	% of total	Assessment Date
Group project	Group project	25%	Week 28

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
This module is category 3

Indicative Reading List

Hale, J. & Kocak, H.: 1991, Dynamics and bifurcation, SpringerVerlag,
Arrowsmith, D.K. and Place, C.M.: 1992, Dynamical systems, Chapman and Hall,
M. W. Hirsch, S. Smale & R. Devaney: 2004, Differential Equations and Dynamical Systems & An Introduction to Chaos, 2nd edition, Elsevier,
Alligood, K.T., T.D. Sauer, J.A. Yorke: 1997, Chaos: an introduction to dynamical systems, SpringerVerlag,

Other Resources

None

Array

Programme or List of Programmes

APM

B.Sc. Applicable Mathematics

BSSA

Study Abroad (DCU Business School)

BSSAO

Study Abroad (DCU Business School)

ECSA

Study Abroad (Engineering & Computing)