Ana Paula S. Dias homepage - Publications

Research on Coupled Cell Systems

Coupled cell systems can be seen as dynamical systems formed by interacting smaller dynamicals systems and are used to model in science a huge type of real world applications. The dynamics of such type of dynamical systems is determined in some aspects by the underlying coupling structure. Notably, we can see robust dynamical features occurring in coupled cell systems that are suprising, in case one was forgetting that they are not just dynamical systems, but they are coupled cell systems. When studying different problems in this field, many techniques can pop up to be useful, from the areas of Linear Algebra, Graph Theory, Geometry, Number Theory, Analysis, and so on.

Citing part of page 3 of the book by Golubistky and Stewart, Dynamics and Bifurcations in Networks, published by SIAM, on 2023, "One reason for studying networks of coupled dynamical systems is simple: they are fascinating ... they reveal new mathematical phenomena, inspire new techniques, and pose new questions."

An example is the paper by D. and Stewart on Linear Equivalence and ODE-equivalence for Coupled Cell Networks, Nonlinearity 18 (2005) 1003-1020, where it is shown that two networks are ODE-equivalent if and only if they determine the same space of linear vector fields; moreover, the variable associated with each node may be assumed one-dimensional for that purpose. That is, choosing different networks for one real world application may not result mathematically in different types of behaviour in case the linear coupling interaction structures basically generate the same linear space.

Theses

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

Papers

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 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

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 150 (2020) (1) 453-474 DOI: 10.1017/prm.2018.71 pdf

M.Aguiar, A.Dias and P.Soares. Towards a classification of networks with asymmetric inputs. Nonlinearity 34 (2021) 5630-5661 DOI: 10.1088/1361-6544/ac0b2e pdf

M.Aguiar and A.Dias. Synchrony and Anti-synchrony in Weighted Networks. SIAM J. Appl. Dyn. Syst. 20 (2021) (3) 1382-1420 DOI: 10.1137/20M1348248 pdf

M.Aguiar, A.Dias and H.Ruan. Synchrony Patterns in Gene Regulatory Networks. Physica D: Nonlinear Phenomena 429 (2022) 133065 DOI: 10.1016/j.physd.2021.133065 pdf

M.Aguiar, C.Bick and A.Dias. Network dynamics with higher-order interactions: coupled cell hypernetworks for identical cells and synchrony. Nonlinearity 36 (2023) (9) 4641 DOI: 10.1088/1361-6544/ace39f pdf

Preprints

M.Aguiar, A.Dias and P.Soares. Towards a classification of steady-state bifurcations for networks with asymmetric inputs. Submitted (2023) pdf

M.Aguiar, A.Dias and I.Stewart. Classification of 2-node Excitatory-Inhibitory Networks. Submitted (2024) arxiv:2403.02869

In preparation

M.Aguiar, A.Dias and I.Stewart. Classification of 3-node excitatory-inhibitory networks. In preparation (2024)

M.Aguiar, A.Dias and E.Nijholt. Homogeneous networks with partial inputs. In preparation (2024)

M.Aguiar, A.Dias and H.Ruan. Heteroclinic cycles in gene regulatory networks. In preparation (2024)

M.Aguiar, A.Dias and F.Mokhtari. Hopf-zero networks. In preparation (2024)

Book

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)

Publication in conference proceedings (with referee)

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

Publications in conference proceedings (without referee)

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

Other publication

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

Abstract of communication

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.

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