- Symmetric Dynamical Systems -- Bifurcation theory with symmetry

Interest at the role of the symmetry in the study of the steady-state and Hopf bifurcations occurring in bifurcations problems symmetric under compact groups. - Coupled Cell Systems

Interest at the dynamics and bifurcations of sets of cells (systems of ordinary differential equations) that are coupled together exploring the role of the coupled cell network representing the cells and the interactions between them, and the internal cell dynamics.

A.P.S.Dias, 1993. Caos Instantâneo, MSc dissertation, Universidade do Porto, Portugal.

A.P.S.Dias, 1997. Bifurcations with wreath product symmetry, PhD Thesis, Math. Institute, University of Warwick, UK. pdf

I.S.Labouriau and A.P.S.Dias. Instant chaos is chaos in slow motion, Journal of Mathematical Analysis and Applications 199 (1996) 138-48.

A.P.S.Dias. Hopf bifurcation for wreath products, Nonlinearity 11 (1998) 247-264. pdf

A.P.S.Dias and I.Stewart. Hopf bifurcation on a simple cubic lattice, Dynamics and Stability of Systems 14 (1999) 3-55. pdf

A.P.S.Dias and I.Stewart. Symmetry-breaking Bifurcations of Wreath Product Systems, Journal of Nonlinear Science 9 (1999) 671-695. pdf

A.P.S.Dias and I.Stewart. Invariant Theory for Wreath Product Groups, Journal of Pure and Applied Algebra 150 (2000) 61-84. pdf

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. pdf

I.Stewart and A.P.S.Dias. Toric Geometry and Equivariant Bifurcations, Physica D 143 (2000) 235-261. pdf

A.P.S.Dias, B.Dionne and I.Stewart. Heteroclinic Cycles and Wreath Product Symmetries, Dynamics and Stability of Sytems 15 (2000) 353-385. pdf

A.P.S.Dias and I.Stewart. Secondary bifurcations in systems with All-to-All coupling, Proceedings of the Royal Society of London Ser. A 459 (2003) 1-18. pdf

A.P.S.Dias and I.Stewart. Symmetry Groupoids and Admissible Vector Fields for Coupled Cell Networks, Journal of the London Mathematical Society 69 (2004) 707-736. pdf

A.P.S.Dias and I.Stewart. Linear Equivalence and
ODE-equivalence for Coupled Cell Networks, Nonlinearity **18** (2005)
1003-1020.
pdf

F.Antoneli, A.P.S.Dias, M.Golubitsky and Y.Wang.
Patterns of Synchrony in Lattice Dynamical Systems,
Nonlinearity **18** (2005) 2193-2209.
pdf

S.M.C.Abreu and A.P.S.Dias. Hopf Bifurcation on Hemispheres. Nonlinearity **19** (2006) 553-574.
pdf

A.P.S.Dias and R.C.Paiva. Hopf bifurcation with D_n-symmetry. Glasgow Mathematical Journal
**48** (2006) 41-51. pdf

A.P.S.Dias and R.C.Paiva. Hopf bifurcation with S_3-symmetry, PortugaliÆ Mathematica
**63** (2) (2006) 127-155. pdf

A.P.S.Dias and J.S.W.Lamb.
Local bifurcation in symmetric coupled cell networks: linear theory. Physica D **223** (2006) 93-108.
pdf

A.P.S.Dias and A.Rodrigues.
Secondary Bifurcations in Systems with All-to-All Coupling. Part II. Dynamical Systems **21** (2006)
439-463.
pdf

M.A.D.Aguiar and A.P.S.Dias. Minimal Coupled Cell Networks. Nonlinearity **20** (2007) 193-219.
pdf

M.A.D.Aguiar, A.P.S.Dias, M.Golubitsky and M.C.A.Leite.
Homogeneous coupled cell networks with S_3-symmetric quotient. Discrete and Continuous Dynamical Systems
**Supplement** (2007) 1-9. pdf

F. Antoneli, A.P.S.Dias and R.C. Paiva.
Hopf Bifurcation in Coupled Cell Networks with Interior Symmetries.
SIAM Journal on Applied Dynamical Systems **7** (1) (2008) 220-248.pdf

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** (2008) 477-512.pdf

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: Nonlinear Phenomena **238** (2009) 137-155.
pdf

F.Antoneli, P.H. Baptistelli, A.P.S.Dias and M.Manoel.
Invariant Theory and Reversible-Equivariant Vector Fields.
Journal of Pure and Applied Algebra
**213** (2009) 649-663.pdf pdf2

A.P.S.Dias and A.Rodrigues.
Hopf bifurcation with S_n symmetry. Nonlinearity ** 22 ** (2009) 627-666.
pdf

A.P.S.Dias and E.M.Pinho.
Spatially Periodic Patterns of Synchrony in Lattice Networks. SIAM Journal on Applied Dynamical
Systems ** 8 ** (2) (2009) 641-675.
pdf

A.P.S.Dias, P.C.Matthews and A.Rodrigues.
Generating Functions for Hopf Bifurcation with S_n-Symmetry. Discrete and Continuous
Dynamical Systems - Series A ** 25 ** (3) (2009) 823-842.
pdf

A.P.S.Dias and E.M.Pinho.
On the enumeration of periodic patterns of synchrony via finite bidirectional
networks.
Proceedings A of the Royal Society of London ** 466 ** (2010) 891-910. pdf

F.Antoneli, A.P.S.Dias and C.M.A.Pinto.
Quasi-periodic States in Coupled Rings of Cells. Communications in Nonlinear Science and Numerical Simulations ** 15 ** (4) (2010) 1048-1062.
pdf

F.Antoneli, A.P.S.Dias and R.C.Paiva.
Coupled Cell Networks: Hopf bifurcation and Interior Symmetry. Discrete and Continuous Dynamical Systems
**Supplement** (2011) 71-78. pdf

A.P.S.Dias and C.S.Moreira.
Spectrum of the elimination of loops and multiple arrows in coupled cell networks.
Nonlinearity ** 25 ** (2012) 3139-3154, DOI:10.1088/0951-7715/25/11/3139
pdf

M.A.D.Aguiar and A.P.S.Dias.The Lattice of Synchrony Subspaces of
a Coupled Cell Network: Characterization and Computation Algorithm.
Journal of Nonlinear Science ** 24 ** (6) (2014) 949-996, DOI: 10.1007/s00332-014-9209-6 pdf

M.A.D.Aguiar and A.P.S.Dias. Regular Synchrony Lattices for
Product Coupled Cell Networks.
Chaos
** 25 ** (2015) 013108, DOI: 10.1063/1.4905703 pdf

M.A.D.Aguiar, A.P.S.Dias and H.Ruan. Synchrony and Elementary Operations
on Coupled Cell Networks.
SIAM J. Appl. Dyn. Syst.
** 15 ** (1) (2016) 322-337, DOI:10.1137/140980119 pdf

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: An Interdisciplinary Journal of Nonlinear Science ** 27 ** (2017) 013103,
DOI: 10.1063/1.4973234 pdf

M.A.D.Aguiar and A.P.S.Dias. Heteroclinic network dynamics on
joining coupled cell networks.
Dynamical Systems An International
Journal ** 32 ** (1) (2017) 4-22,
DOI: 10.1080/14689367.2016.1197889 pdf

M.A.D.Aguiar, A.P.S.Dias, and P. Soares. Characterization of fundamental networks. Proceedings of the Royal Society of Edinburgh Section A: Mathematics (2019) DOI: 10.1017/prm.2018.71 pdf

M.A.D.Aguiar and A.P.S.Dias. (2018) An Overview of Synchrony in Coupled Cell Networks. In: A. Pinto, D. Zilberman (eds) Modeling, Dynamics, Optimization and Bioeconomics III. DGS 2016, BIOECONOMY 2015. Springer Proceedings in Mathematics & Statistics, vol 224. Springer, Cham, pp 25-48, DOI: 10.1007/978-3-319-74086-7_2 pdf

A.P.S.Dias and C.Moreira. Direct lifts of coupled cell
networks. Nonlinearity
**31** (2018) (4) 1299-1312, DOI: 10.1088/1361-6544/aa9e5b
pdf

M.A.D.Aguiar and A.P.S.Dias. Synchronization and Equitable
Partitions in Weighted Networks.
Chaos: An Interdisciplinary Journal of Nonlinear Science **28** (2018) 073105, DOI: 10.1063/1.4997385 pdf

M.A.D.Aguiar, A.P.S.Dias and M. Field. Feedforward networks:
adaptation, feedback, and synchrony. Journal of Nonlinear Science **29** (2019) (3)
1129-1164 DOI:10.1007/s00332-018-9513-7 pdf

M.A.D.Aguiar, A.P.S.Dias and P. Soares. The Steady-state Lifting
Bifurcation Problem Associated with the Valency on Networks. Physica D: Nonlinear Phenomena **390**
(2019) 36-46 DOI: 10.1016/j.physd.2018.10.006
pdf

M.A.D.Aguiar, A.P.S.Dias and M. Manoel. Gradient and Hamiltonian
coupled systems on undirected networks. Mathematical Biosciences and Engineering **16**
(2019) (5) 4622-4644 DOI: 10.3934/mbe.2019232 pdf

J.Buescu, S.B.S.D.Castro, A.P.S.Dias and I.S.Labouriau (eds). Bifurcations, symmetry and patterns. A conference in honour of M.Golubitsky and I.Stewart, Birkhauser Verlag, Basel, Switzerland, 2003. (ISBN 3-7643-7020-3)

F.Antoneli, A.P.S.Dias and C.M.A.Pinto. Rich phenomena in a network of two rings coupled through a `buffer' cell. In: Proceedings 2nd Conference on Nonlinear Science and Complexity, ISEP, July 28-31 2008, Porto, Portugal To appear. pdf

F.Antoneli, A.P.S.Dias, M.Golubitsky and Y.Wang. Flow Invariant
Subspaces for Lattice Dynamical Systems. In:
*Workshop on Bifurcation Theory and Spatio-Temporal
Pattern Formation in PDE.*
(W. Nagata and N.S. Namachchivaya, eds.)
Fields Institute Communications, 2006, 1-8.
pdf

F.Antoneli, A.P.S.Dias, M.Golubitsky and Y.Wang.
Synchrony in Lattice Differential Equations. In: Some topics in industrial and
applied mathematics. (R. Jeltsch, T. Li and I.H. Sloan, eds.) Contemporary Applied Mathematics Series ** 8 ** World Scientific Publ. Co., 2007,
43-56. pdf

M.A.D.Aguiar and A.P.S.Dias. Coupled cell networks: minimality. PAMM Proc. Appl. Math. Mech. ** 7 ** , 1030501-1030502 (2007).
pdf

A.P.S.Dias and R.C. Paiva. Hopf bifurcation in coupled cell networks with abelian symmetry. In: Actas do Encontro Nacional da SPM Leiria 2010. Boletim da Sociedade Portuguesa de Matemática, 2011, 110-115. pdf

A.P.S.Dias and E.M.Pinho. Regras de boa vizinhança. Gazeta de Matemática ** 162 ** (2010) 49-56. pdf

A.P.S.Dias, B.Dionne and I.Stewart. Heteroclinic cycles and wreath product symmetries, p.53-57. In: D.Bambusi, G.Gaeta and M.Cadoni, 2001. Proceedings of the International Conference on Symmetry and Perturbation Theory, Sardinia, Italy, 2001, World Scientific, Singapore.