Généralités
Master's Degree Title: Applied Mathematics
Semester: S2
Teaching Unit Title: Fundamental Unit 2.1
Subject Title: Discrete Dynamic Systems
Credits: 6
Coefficients: 3
Teaching Objectives
Teach students to master one-dimensional or two-dimensional recurrences that can be associated with differential systems. Definition of singularities such as nth-order cycles, critical points and the theory of bifurcations.
Recommended prior knowledge:
Real analysis. Linear algebra. Difference equations.
Subject content:
Chapter 1: General concepts of dynamic systems theory
1) Introduction
2) Concept of orbit of a system (Fixed points, cycles, critical points, ....)
3) Topological equivalence of systems.
4) Graphical study of dynamic systems.
a) Discrete dynamical systems of dimension 1.
b) Dynamical systems of dimension 2. Phase portraits.
Chapter 2: Discrete dynamical systems of dimension 1
1) Linear discrete dynamical systems of dimension 1 (fixed points, stability)
2) Nonlinear discrete dynamical systems of dimension 1 (fixed points, cycles, critical points.)
3) Stability
4) Non-uniqueness of the inverse, endomorphism.
Chapter 3: Discrete dynamical systems of dimension 2
1) Linear discrete dynamical systems of dimension 2 (fixed points, stability)
2) Nonlinear discrete dynamical systems of dimension 2 (fixed points, cycles, critical points.)
3) Stability
1) Definitions
2) Bifurcation Node-collar, Period doubling, Neimark.
Chapter 4: Introduction to chaos theory.
References:
[1] Abraham. R, Mira. C, Gardini. L, Chaos in discrete dynamical systems, Springer Science-Business Media, New York: Telos, 1997.
[2] Devaney. R. L, An introduction to chaotic dynamical dystems, CRC Press, third edition, Boca Raton, 2022.[3] Elaydi. S, An Introduction to difference equations, 3 edition, Sprigner, San Antonio, 2005.[4] Elaydi. S, Discrete chaos:With Applications in Science and Engineering, Second Edition,CRC Press, San Antonio, 2007[5] Hirsch. M. W, Smale. S, Devaney. R. L, Differential equations, dynamical systems, and an introduction to chaos, Elsevier, Waltham, 2013.[6] Galor. O, Discrete dynamical systems, Springer Science-Business Media, first edition, Berlin, 2007.