Complements of Ordinary Differential Equations
Code
11631
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
José Maria Nunes de Almeida Gonçalves Gomes, Oleksiy Karlovych
Weekly hours
4
Teaching language
Português
Objectives
At the end of this course the student will have acquired knowledge, skills and competences to:
- Understand issues involving: equations and systems of ODE’s linear and non-linear, existence and uniqueness of solution, geometry related to ODE''s, local and global stability, and bifurcation problems.
- Be able to solve problems involving the previous issues.
- Know examples and applications of the theory of ODE''s.
Prerequisites
Students should have attended and low achieved an undergraduate course on Ordinary Differential Equations.
Subject matter
Although in each year some adjustments can be made, depending on student’s interests, the program focuses on the following topics:
Existence, uniqueness and extension solutions. Geometry of ODE’s. Flow. Stability and linearization. Method of Lyapunov. Periodic solutions. Stability of linear and non-linear systems. Gronwall inequality. Principle of superposition.
Floquet theory. Relations with partial differential equations. Invariant manifolds. Hartman-Grobman theorem. Perturbations. Forced systems. Homoclinic orbits. Melnikov method. Local and global bifurcations. Classical applications.
According to the interests of the students the course can also be directed to some specific type of applications, non-autonomous systems, invariant manifolds, stability, strange attractors, or other directions.
Bibliography
C. Chicone, Ordinary Differential Equations with Applications, Springer, 2006.
J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983.
M. Hirsch, S. Smale, R. Devaney, Differential Equations, Dynamical Systems & an Introduction to Chaos, Elsevier, 2004.
L. Perko, Differential Equations and Dynamical Systems, Springer, 1991.
S. Wiggins, Intoduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 2009.