Differential Equations with Symmetry

Code:

M522

Acronym:

M522

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 2S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	2	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

Basic tools for studying differential equations with symmetry.

Learning outcomes and competences

To know how to apply the basic tools for studying differential equations with symmetry.

Working method

Presencial

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

Basic knowledge on Linear Algebra, Several Variables Calculus, Group Theory, Rings and Modules, and Ordinary Differential Equations.

Program

Group theory: Linear representations and actions of compact Lie groups.
Irreducibility.
Commuting linear mappings and absolute irreducibility.
Invariant theory: Invariant functions; Equivariant (nonlinear) mappings; Hilbert series and Molien Theorem; Hilbert series and hidden symmetries.
Symmetry-breaking steady-state bifurcation: Orbits and isotropy subgroups; Fixed-point subspaces; Equivariant Branching Lemma; Orbital asymptotic stability.
Example: Dn – standard action on C; Symmetric coupled cell systems.
Symmetry-breaking Hopf bifurcation: Hopf bifurcation; Equivariant Hopf Theorem; Example.
Coupled cell networks: Symmetric coupled cell networks; Coupled cell networks and coupled cell systems; ODE and linear equivalence of coupled cell networks; Quotient coupled cell networks.

Mandatory literature

Golubitsky Martin; The symmetry perspective. ISBN: 3-7643-6609-5
Golubitsky Martin; Singularities and groups in bifurcation theory. ISBN: 0-387-90999-0 (Vol. I)
M. Golubitsky, I.N. Stewart, and D.G. Schaeffer.; Singularities and Groups in Bifurcation Theory: Vol. 2, Applied Mathematical Sciences 69, Springer-Verlag, New York, 1988

Comments from the literature

Secondary bibliography:

M.A.D. Aguiar, A.P.S. Dias and F. Ferreira. Patterns of synchrony for feed-forward and auto-regulation feed-forward neural networks. Chaos 27 (2017) 013103.
M.A.D. Aguiar, A.P.S. Dias, M. Golubitsky and M.C.A. Leite. Homogeneous coupled cell networks with S3-symmetric quotient. Discrete and Continuous Dynamical Systems Supplement 1-9., 2007.
M.A.D. Aguiar, A.P.S. Dias, M. Golubitsky and M.C.A. Leite. Bifurcations from Regular Quotient Networks: A First Insight. Physica D 238 (2009) 137-155.
A.P.S. Dias and I. Stewart. Linear Equivalence and ODE-equivalence for Coupled Cell Networks. Nonlinearity 18 (2005) 1003-1020.
M.C.A. Leite and M. Golubitsky. Homogeneous three-cell networks. Nonlinearity 19 (2006) 2313-2363.
I. Stewart and A.P.S. Dias. Hilbert series for Equivariant Mappings Restricted to Invariant Hyperplanes. Journal of Pure and Applied Algebra 151 (2000) 89-106. 
I. Stewart and M. Golubitsky. Synchrony-breaking bifurcation at a simple real eigenvalue for regular networks 1: 1-dimensional cells. SIAM J. Appl. Dynam. Sys. 10 (4)  (2011) 1404-1442.

Teaching methods and learning activities

The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts. There are also practical exercises proposed at the lectures and discussed in the following lectures.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	50,00
Trabalho escrito	50,00
Total:	100,00

Calculation formula of final grade

The student assessment consists into two components: it considers one written test to be done at 2nd of May with classification out of 10 (ten); and an oral and written presentation of an individual project with classification out of 10 (ten). The oral presentations will occur at the last two weeks of the semester. The final mark is the sum of the marks obtained at the two components.

Observations

The lectures are given at the University of Coimbra.