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12786 Finite Difference Equations - Five-year degree in Mathematics

Center

Faculty of Mathematics

Departament

Applied Mathematics

Lecturers in charge

Sin datos cargados

Met. Docent

Met. Avaluació

Final examination - -

Bibliografia

1 ) Berry, J. Introduction to Non-Linear Systems. Modular Mathematics Series. Ed. Arnold, 1996.

( 2 ) Collet P., Eckmann J. P. Iterated Maps of the Interval as Dynamical Systems. Birkhõuser, 1980.

( 3 ) De Melo, W, M., Van Strien, S. One-Dimensional Dynamics. Springer-Verlag, 1994

( 4 ) Devaney R. L. An Introduction to Chaotic Dynamical Systems Addison-Wesley, 1989.

( 5 ) Elaydi, S. N. An introduction to Difference Equations. Springer. 1995.

( 6 ) Easton, R. W. Geometric Methods for Discrete Dynamical Systems. Oxford Engineering Science Series nº 50. Oxford University Press, 1988.

( 7 ) Hildebrand F. B. Finite-Difference Equations and Simulations. Prentice-Hall, Inc. 1968.

( 8 ) Holmgren, R. A. A first Course in Discrete Dynamical Systems. Springer - Verlag. 19

Continguts

Objectives
This is a first approximation to the dynamic systems in the first cycle of Mathematics. In second place it covers the mathematical bases that are necessary for other subjects, particularly numerical calculus, differential equations and mathematics for finance. Finally the general training process of the mathematics students must be complemented, increasing their capacity for abstract reasoning and geometric intuition.

Theory Program

1.- Introduction and Basic Concepts.

2.- Equations in differences.

3.- Solving an equation in differences.
Definition, autonomous and non-autonomous, lineal equationsà

4.- Orbits.
Fixed points and periodic orbits, limit sets, invariant sets and functions...

5.- Diagrams of orbits.
Fixed points, graphic determination, logistic equation and quadratic applicationsà.

6.- Fixed points.
Stability of fixed points, hyperbolicity, permanence of fixed points...

7.- Periodic Orbits.
Stability, orbits of period three and Sarkovski theorem, developing continuous fractions...

8.- Introduction.
Systems on a plane, matrices, resolution according to own values, extension to the non-dimensional caseà

9.- Lineal differential equations.
Definitions, independence of solutions, homogeneous and non homogeneous equations,
method of indeterminate coefficients, variation of constants methodà

10.- Rotations on a circumference.
Transformations and rotations on S1