Advanced Topics in Dynamics and Geometry

Code:

M6002

Acronym:

M6002

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Applied Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
PDMATAPL	0	Official Study Plan	1	-	6	56	162

Teaching language

Portuguese and english

Objectives

General knowledge and uniformisation of skills in Differential Equations, Dynamical Systems and Differential Geometry.

Learning outcomes and competences

General knowledge and uniformisation of skills in Differential Equations, Dynamical Systems and Differential Geometry.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Basic knowledge on Linear Algebra, Analytic Geometry, Vectorial Analysis and Diferential Equations.

Program

Elementary geometry of submanifolds of R^n:
Parametrisations (or charts), tangent bundle, differentiable functions, submanifolds, transversality. Differential forms, Stokes theorem on manifolds.

 

Basic concepts of dynamics in R^n (or in submanifolds of R^n):
Differential equations, stability of equilibria and of periodic solutions, hyperbolicity, stable and unstable manifolds, Poincaré map. Structural stability and bifurcations. Liapuvov-Schmidt Reduction. Symmetric differential equations and coupled cell systems. 

 

Mandatory literature

Victor Guillemin; Differential topology. ISBN: 0-13-212605-2
V. I. Arnold; Ordinary differential equations. ISBN: 0-262-01037-2
Morris W. Hirsch; Differential equations, dynamical systems, and linear algebra. ISBN: 0-12-349550
Martin Golubitsky; Singularities and groups in bifurcation theory. ISBN: 0-387-90999-0 (Vol. I)
Martin Golubitsky; The symmetry perspective. ISBN: 3-7643-6609-5
Martin Golubitsky; Dynamics and bifurcation in networks. ISBN: 978-1-61197-732-5

Complementary Bibliography

L., ed. lit. Lerman; Methods of qualitative theory of differential equations and related topics. ISBN: 0-8218-2663-8
Clark Robinson; Dynamical systems. ISBN: 0-8493-8493-1
Anatole Katok; Introduction to the modern theory of dynamical systems. ISBN: 0-521-57557-5
John M. Lee; Introduction to smooth manifolds. ISBN: 0-387-95448-1

Teaching methods and learning activities

The syllabus consists of a minimum list of basic concepts in Differential

keywords

Physical sciences > Mathematics > Chaos theory
Physical sciences > Mathematics > Mathematical analysis > Differential equations
Physical sciences > Mathematics > Mathematical analysis > Functions
Physical sciences > Mathematics > Geometry > Algebraic geometry

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Frequência das aulas	56,00
Estudo autónomo	106,00
Total:	162,00

Eligibility for exams

Absence from lectures will not be registered.

Calculation formula of final grade

Assessment with two components. One, G, on geometry, the other, D, on dynamics, marked in a scale from 0 to 20.
The final mark will be the the average G/2+D/2.
The component G will be the result of a written test.
The component D will be either the result of a written test or partly the result of an oral test, to be agreed with the students.
A minimum of 6 points in each component is required for aproval.
In the resit period there will be a written exam with two parts.
Each part may replace the marks in one of the components.

Special assessment (TE, DA, ...)

Written exam with two parts corresponding to the two  components.

Classification improvement

Written exam with two parts in the resit period.
Each part may replace the marks in one of the components.