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Difference Equations and Applications

Code 15372
Year 2
Semester S1
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements NA
Learning outcomes (i) To understand some concepts and fundamental results from the theory of difference equations;
(ii) To use concepts and results from the theory of difference equations to analyse some specific difference equation or system of difference equations;
(iii) To recognize some examples of application of difference equations in the modelling of some phenomena in the exact sciences and social sciences;
(iv) To analyse and understand mathematical proofs;
(v) To communicate using mathematical language, written and orally.
Syllabus 1. 1st order difference equations
1.1 Linear Equations;
1.2 Spider web diagrams and stability of equilibrium points;
1.3 Stability of periodic orbits;
1.4 Bifurcation and Sharkovsky's Theorem;
1.5 Basin of attraction and global stability.

2. Scalar difference equations
2.1 Shift, difference and anti-difference operators;
2.2 Homogeneous and non-homogeneous linear equations;
2.3 Asymptotic behavior.

3. Systems of difference equations
3.1 The discrete Putzer algorithm;
3.2 Fundamental matrix and variation of the constants formula;
3.3 Autonomous systems: Jordan's canonical form;
3.4 Periodic linear systems: Floquet exponents.

4. Stability of difference equation systems
4.1 Stability of linear systems;
4.2 Stability through linearization;
4.3 Lyapunov stability theorem and LaSalle's invariance principle.

5. Z Transform
5.1 Definition and properties;
5.2 Inverse transform;
5.3 Convolution-type equations.
Main Bibliography - Elaydi, S. (2005). An Introduction to Difference Equations. (3ª edição). Springer.
- Elaydi, S. (2008). Discrete Caos. (2ª edição). Chapman & Hall/CRC.
- Kelley, W.G. & Peterson, A.C. (2000). Difference Equations - An Introduction With Applications. Academic Press.
- Goldberg, S. (1986). Introduction to Difference Equation. New York: Dover.
- Agarwal, R.P. (1992). Difference Equations and Inequalities. New York: Marcel Dekker.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2023-01-24

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