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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 equations1.1 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 equations2.1 Shift, difference and anti-difference operators;2.2 Homogeneous and non-homogeneous linear equations;2.3 Asymptotic behavior.3. Systems of difference equations3.1 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 systems4.1 Stability of linear systems;4.2 Stability through linearization;4.3 Lyapunov stability theorem and LaSalle's invariance principle.5. Z Transform5.1 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.

### Instructors César Augusto Teixeira Marques da Silva

### Course

Mathematics and Applications
Last updated on: 2023-01-24

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