Advanced Topics in Dynamics and Geometry

Code:

M6002

Acronym:

M6002

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 1S

Active?	Yes
Web Page:	http://cmup.fc.up.pt/cmup/islabour/Contents/DS2016.html
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	4	Official Study Plan	1	-	6	56	162

Teaching language

English

Objectives

Provide the students with basic tools in Geometry of Manifolds and Dynamical Systems.

Learning outcomes and competences

Acquisition of basic tools in Geometry of Manifolds and Dynamical Systems.

Working method

Presencial

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

Pre-requisits: notions of analysis in R^n

Program

First Module: Manifolds (Domenico Catalano, U. Aveiro)

Euclidean spaces as metric spaces. Metric spaces as topological spaces.
Continuous functions and homeomorphisms. Topological manifolds as locally euclidean spaces. Topological obstructions: connected spaces, connected components, invariance of the number of connected components under continuous functions.

Differentiable Manifolds: Definition and examples. Diffeomorphisms. Bump functions.
The tangent space: tangent vectors as derivations. Differential maps. The tangent bundle as a manifold.

Vector fields as derivations. Lie bracket. Integral curves.
The flow of a vector field. One forms. Submanifolds, immersions and submersions.
Tensors and tensor fields.

Second Module: Dynamics (Isabel Labouriau, U. Porto)

Abstract definition of a dynamical system, ordinary differential equations, difference equations. Vector fields and ordinary differential equations: flow; phase portrait; equilibrium points. Difference equations, relation with differential equation: discretisation; first return map.

Hyperbolic linear difference equation, stable and unstable subspaces. For non hyperbolic equations, centre subspace.

Linear differential equations. Exponential of a linear map, relation to the flow of a linear differential equation. Methods for computing the exponential. Liapunov stability, asymptotic stability of an equilibrium point. Unstable equilibrium. Hyperbolic linear differential equations, stable and unstable subspaces. For non hyperbolic equations, centre subspace.

Phase portraits for all linear differential equations in the plane, stability of equilibria, invariant subspaces.

Proof that the set of hyperbolic linear isomorphisms is open and dense in the set of invertible linear maps of R^n. Proof that the set of hyperbolic linear vector fields is open and dense in the set of linear maps of R^n.

Equivalence relations for classification of dynamical systems: differentiable conjugacy, topological conjugacy, topological equivalence. Flow-box theorem. The dimension of the stable subspace as a complete topological invariant for hyperbolic linear vector fields. Alpha- and omega-limit sets. Local section of a vector field, Poincaré-Bendixson theorem.

The C^r metric on the space C^r(M,R^s) for M a compact smooth manifold. and brief discussion of the C^infinity topology. Proof that the set of C^r diffeomorphisms is an open subset of C^r(M,M).

Transversality of vector subspaces, of a function to a submanifold, of two submanifolds, of two maps between manifolds, of two submanifolds, of a map and a submanifold, relation to codimension. Proof that the set of maps from a compact manifold that are transverse to a closed submanifold is open. The weak Thom transversality theorem and structural stability.

Stability of equilibria of vector fields, first and second theorems of Liapunov. Analogous results for difference equations.

Local structural stability of vector fields under topological equivalence: simple equilibria, if a vector field has an equilibrium that is not simple, then it is not locally structurally stable. The set of vector fields on a manifold M with only simple equilibria is residual, and if M is compact then it is open and dense. Example of a simple vector field that is not structurally stable. The set of vector fields on a manifold M with only hyperbolic equilibria is residual, and if M is compact then it is open and dense. Grobman-Hartman theorem, and structural stability. Analogous results for difference equations.

Stable and unstable manifolds, existence, invariance under topological equivalence. In structurally stable vector fields, invariant manifolds intersect transverselly. Statement of Peixoto Theorem.

Smale's horseshoe, symbolic dynamics, existence of infinitely many periodic orbits of arbitrary periods, existence of a dense orbit.

Mandatory literature

Palis Jr. Jacob; Geometric theory of dynamical systems. ISBN: 0-387-90668-1
Arnold Vladimir igorevich; Geometrical methods in the theory of ordinary differential equations. ISBN: 0-387-96649-8 (for the second module)
Brickell F.; Differentiable manifolds. ISBN: 442-01048-6 (for the first module)

Complementary Bibliography

Carmo Manfredo Perdigão do; Geometria Riemanniana (for the first module)
O.Neil Barrett; Semi-Riemannian geometry with applications to relativity. ISBN: 0-12-526740-1 (for the first module)
Arrowsmith D. K.; Ordinary differential equations (for the second module)
Hirsch Morris W.; Differential equations, dynamical systems, and linear algebra. ISBN: 0-12-349550 (for the second module)

Teaching methods and learning activities

Presentation of the subject in class. Resolution by the students, outside classes, of proposed exercises.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

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

Calculation formula of final grade

Average of the marks obtained in the tests of the two modules.