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Difference Equations and Applications

Code 14793
Year 3
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.
Teaching Methodologies and Assessment Criteria All classes will be both Theoretical and Practical. The teacher introduces the concepts, presents the results, proving several among them, and discusses applications. The students can participate during classes, interacting with the teacher and sometimes solving problems. Autonomous work, consisting mainly in solving exercises, several of them concerning equations or systems of equation originated in the applications and complementing the ones presented in the classes will be promoted. Assessment undertaken throughout the teaching-assessment period will consists in two written tests, each rated 10 values. The final classification will be the sum of the two written tests scores. The student will also be able to a final exam quoted for 20 values.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2022-02-07

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