| Code |
15372
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| Year |
2
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| Semester |
S1
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
NA
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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.
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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.
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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.
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Teaching Methodologies and Assessment Criteria |
During the teaching-learning period, the assessment will consist of two written tests. Each test will be marked out of 10 points. A student will pass if the sum of the test scores is equal to or greater than 9.5 points. Whenever the score exceeds 16 points, the student will take an oral exam to defend their grade. Regardless of the student’s attendance or the outcome of the oral exam, a minimum grade of 16 points is guaranteed. All students are eligible to take the exam.
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Language |
Portuguese. Tutorial support is available in English.
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