Differential Equations with Symmetry

Code:

M522

Acronym:

M522

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 2S

Active?	Yes
Web Page:	http://elearning2.fc.up.pt/aulasweb0910/course/view.php?id=1915
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	4	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

- Study of differential equations using basic concepts such as symmetry group, ring, module, ideal, vector space, with emphasis ion classic results of invariant theory.

- Study of how symmetry affects types, multiplicity, stability and genericity of solutions of ordinary and partial differential equations.

- Study of basic results in Bifurcation with Symmetry.

Program

Introduction to the study of symmetric systems of nonlinear ordinary or partial differential equations. Specifically, it will be shown how the symmetries of symmetric systems may be used, in a systematic way, to analyze, predict, and understand many general mechanisms of pattern formation and dynamic behaviour. Results from invariant theory, theory of differential equations and singularity theory will be used.

1 Introduction

2 Group theory
2.1 Brief reference to Lie groups
2.2 Representations and actions
2.3 Invariant integration

3 Irreducibility

4 Commuting linear mappings and absolute irreducibility

5 Invariant theory
5.1 Invariant functions
5.2 Equivariant (nonlinear) mappings

6 Symmetry-breaking steady-state bifurcation
6.1 Orbits and isotropy subgroups
6.2 Fixed-point subspaces
6.3 Equivariant Branching Lemma
6.4 Orbital asymptotic stability

7 Spatio-temporal symmetries
7.1 Hopf bifurcation, Liapunov-Schmidt reduction
7.2 Equivariant Hopf Theorem
7.3 Spatio-temporal symmetries, subgroups of the cross-product with S^1


8 Examples
8.1 Bifurcations with symmetry Z_n
8.2 Bifurcations with symmetry D_n in its standard action on C
8.4 Bifurcations with symmetry Z_2+Z_2
8.5 Models for gaits in bipedal locomotion
8.6 Models for gaits in quadrupedal locomotion.

Mandatory literature

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
000054353. ISBN: 981-02-3828-2
000054352. ISBN: 0-582-30346-X
000044936. ISBN: 0-387-90999-0 (Vol. I)
000054916. ISBN: 3-7643-6609-5

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	56,00
	Total:	-	0,00