Registry
Module Specifications
Current Academic Year 2012 - 2013
Please note that this information is subject to change.
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| Description | |||||||||||||||||||||||||||||||||||||
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The course gives an introduction to the qualitative theory of nonlinear ordinary differential equations, and applying the theory to case studies in economics, finance, population dynamics and epidemiology. The course is aimed first year taught masters students in financial and industrial mathematics. The module complements other modules taken by these students in financial economics, stochastic finance, and differential equations. In these complementary modules, dynamic economic models with randomness, but without extensive nonlinearities, or deterministic differential equations which can be solved, are studied. The emphasis in this module is to analyse the long term behaviour of systems when explicit solutions cannot be found, and applying this theory to model and analyse real-world dynamic phenomena | |||||||||||||||||||||||||||||||||||||
| Learning Outcomes | |||||||||||||||||||||||||||||||||||||
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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 | |||||||||||||||||||||||||||||||||||||
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 |
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| Indicative Content and Learning Activities | |||||||||||||||||||||||||||||||||||||
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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: Competing species, general equilibrim with production, Samuelson's model of increasing returns. Case studies from economics and biology. Students in MS413M will cover 2 of the following as case studies: Holling-Tanner model, Solow model with human capital, Tobin macrodynamic general equilibrium model, Cournot duopoly model, SIRS model. | |||||||||||||||||||||||||||||||||||||
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