Learning Outcomes:
On completion of this module students should be able to:
1. Continuous time: First-Order Differential Equations
- understand geometry of flows direction fields, phase lines, equilibrium, stability
- identify bifurcation points.
- classify and describe different types of bifurcations.
2. Second-order differential equations: constant coefficient equations.
- construct fundamental solutions
3. Discrete time: First-Order Difference Equations
- understand dynamic stability to stability
- use graphical approach, phase plane, phase line.
- identify bifurcations
4. Discrete time: Second-Order Difference Equations
- solve constant coefficient equations
5. Systems of Differential and Difference Equations
- solve simultaneous dynamical equations
- construct two-variable phase diagrams
- carry out a stability analysis
Indicative Module Content:
Topics will be taken from the list below:
1. Continuous time: First-Order Differential Equations
a. first-order linear differential equations with constant coefficients.
- application: Dynamics of market price
- geometry of flows direction fields, phase lines, equilibrium, stability.
b. variable coefficient first order differential equations
- method of integrating factors.
c. nonlinear differential equations eg Bernoulli equation.
- application: Solow growth model, linear Cobb-Douglas function
d. elementary bifurcations – dependence on parameters.
- pitchfork, transcritical, saddle-node.
- application: population growth models
2. Second-order differential equations: constant coefficient equations.
a. fundamental solutions
b. applications: market model with price expectations, interaction of inflation and unemployment
3. Discrete time: First-Order Difference Equations
a. solving a first-order difference equation
b. dynamic stability to stability
c. cobweb models
d. application: market model with inventory
e. nonlinear difference equations – graphical approach, phase plane, phase line.
f. [application: market model with ceiling price]
g. bifurcations and chaos: dependence on parameters
h. application: logistic equation for population dynamics
4. Discrete time: Second-Order Difference Equations
a. constant coefficient equations
b. application: interaction models, inflation and unemployment
5. Systems of Differential and Difference Equations
a. solving simultaneous dynamical equations
b. application: input-output models, inflation-unemployment model
c. two-variable phase diagrams
d. linearisation of a nonlinear dynamical system
e. stability analysis