Differential Equations with Symmetry

Code:

M522

Acronym:

M522

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2009/2010 - 2S

Active?	Yes
Responsible unit:	Departamento de Matemática Pura
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	8	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

Aims of the course:
- Use of basic concepts as for example, symmetry group, ring, module, ideal, vector space, basis and the notions of integral at some applications in Algebra, with emphasis to classic results of invariant theory, as for example the Hilbert-Weyl and Molien Theorems.

- Understand that in symmetric systems of ordinary differential autonomous equations, depending on parameters and presenting singular points, the way symmetry acts affects the types, the multiplicities and the stability of the solutions of the equations.

- Learn the basic results in the area of Bifurcation Theory with Symmetry.

Program

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
5.3 Hilbert series
5.4 Molien Theorem
5.5 Hilbert series and hidden symmetries

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 Example: Dn – standard action on C

8 Symmetry-breaking Hopf bifurcation
8.1 Hopf bifurcation
8.2 Equivariant Hopf Theorem
8.3 Example

9 Coupled cell networks
9.1 Symmetric coupled cell networks
9.2 Coupled cell networks and coupled cell systems
9.3 ODE and linear equivalence of coupled cell networks
9.4 Quotient coupled cell networks

Mandatory literature

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
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

Complementary Bibliography

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, 137-155., 2009
F. Antoneli, A.P.S. Dias and P.C. Matthews; Invariants, Equivariants and Characters in Symmetric Bifurcation Theory, Proceedings of the Royal Society of Edinburgh 138A, 477-512, 2008
A.K. Bajaj and P.R. Sethna; Bifurcations in three-dimensional motions of articulated tubes, Trans. ASME 49, 606-618., 1982
J.M. Ball and D.G. Schaeffer; Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions, Math. Proc. Camb. Phil. Soc. 94, 315-339, 1983
N. Bourbaki; Groupes et Algebras de Lie, Ch. I, Act. Sc. et Ind. 1285, Ch.IV, V, VI Hermann, Paris, 1960,1968
A.P.S. Dias and J.S.W. Lamb; Local bifurcation in symmetric coupled cell networks: linear theory, Physica D 223, 93-108, 2006
A.P.S. Dias and I. Stewart; Linear Equivalence and ODE-equivalence for Coupled Cell Networks, Nonlinearity 18, 1003-1020, 2005
M. Field; Combinatorial dynamics, Dynamical Systems 19, 217-243, 2004
M. Forger; Invariant polynomials and Molien functions, J. Math. Phys. 39, 1107-1141, 1998
M. Golubitsky, J.E. Marsden and D.G. Schaeffer; Bifurcation problems with hidden symmetries. In: Partial Differential Equations and Dynamical Systems (W.E. Fitzgibbon III, Ed.), Research Notes in Math. 101 Pitman, San Francisco, pp 181-210, 1984
M. Golubitsky, I. Stewart and A. Torok ; Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows, SIAM J. Appl. Dynam. Sys. 4 (1), 78–100, 2005
000045751. ISBN: 0-12-349550
G. Hochschild; The Structure of Lie Groups, Holden-Day, San Francisco, 1965
T. Molien; Uber die Invarianten der Linearen Substitutionsgruppe, Sitzungsber. K¨onig. Preuss. Akad. Wiss. 1152-1156., 1897
000049916. ISBN: 3-540-09715-5
G.W. Schwarz; Smooth functions invariant under the action of a compact Lie group, Topology 14, 63-68, 1975
I. Stewart and A.P.S. Dias; Hilbert Series for Equivariant Mappings Restricted to Invariant Hyperplanes, Journal of Pure and Applied Algebra 151, 89-106, 2000
I. Stewart, M. Golubitsky and M. Pivato; Symmetry groupoids and patterns of synchrony in coupled cell networks, IAM J. Appl. Dynam. Sys. 2, 609-646, 2003
000050635. ISBN: 3-211-82445-6

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

Eligibility for exams

Evaluation
The student assessment considers three written tests to be done during the semester. If T1, T2, T3 are the classifications obtained by the student at Test 1, Test 2, Test 3, respectively, each in the scale 0 to 20, then the final mark of the student (in the scale 0 to 20) is the average of the two best marks out of T1, T2, T3.

The dates of the three tests:
March 12, 2010; time: 9hrs; room: M042;
April 12, 2010; time:13h30m; room: M042;
May 21, 2010,; time: 9hrs; room: M042.

Calculation formula of final grade

Evaluation
The student assessment considers three written tests to be done during the semester. If T1, T2, T3 are the classifications obtained by the student at Test 1, Test 2, Test 3, respectively, each in the scale 0 to 20, then the final mark of the student (in the scale 0 to 20) is the average of the two best marks out of T1, T2, T3.