| 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. First-order difference equations: Solutions, orbits, and linear equations; Stability of equilibrium points and periodic orbits; Basin of attraction and global stability; Period doubling, bifurcation, and chaos.
2. Higher-order difference equations: Difference calculus; Linear scalar equations of order ??
3. n; Method of undetermined coefficients; Asymptotic behavior of solutions; Poincaré and Perron theorems.
4. Systems of linear difference equations: Putzer's algorithm; Fundamental matrix and general solution; Jordan canonical form; Periodic linear systems: Floquet theory.
Stability of difference equation systems: Notions of stability; Stability of linear systems; Stability of periodic linear systems; Lyapunov's direct method; Stability through linearization.
5. Z-transform: Z-transform; Inverse Z-transform; Volterra systems.
|
|
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.
|