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Complements of Ordinary Differential Equations

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.

General characterization

Code

12974

Credits

9.0

Responsible teacher

Available soon

Hours

Weekly - Available soon

Total - 43

Teaching language

Português

Prerequisites

Students should have attended and low achieved an undergraduate course on Ordinary Differential Equations.

Bibliography

Differential Equations: A Dynamical Systems Approach to Theory and Practice, Marcelo Viana, José M. Espinar, Graduate Studies in Mathematics, vol. 212, Editora American Mathematical Society, 2021.

M. Hirsch, S. Smale, R. Devaney, Differential Equations, Dynamical Systems & an Introduction to Chaos, Elsevier, 2004.

Ordinary Differential Equations (An Introduction to Nonlinear Analysis), Herbert Amann, Gruyter Studies in Mathematics, vol. 13, 1990

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983.

L. Perko, Differential Equations and Dynamical Systems, Springer, 1991.

S. Wiggins, Intoduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 2009.

Teaching method

There will be a special emphasis on geometry and intuitive side of each topic so that the mathematical structure stay motivated and be grasped in a solid and profound way.

Classes are theoretical-practical, alternating between more expository sessions and other parts where concrete problems will be worked out.

The evaluation is based on problem solving and observation of the student’s progress throughout the semester. Alternatively it may be considered the possibility of carrying out an exam at the end of the semester.

Evaluation method

The evaluation is based on problem solving, observation of the student’s progress throughout the semester and two oral presentations on the contents taught.

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.