On bifurcations in lifts of regular uniform coupled cell networks

Moreira, Catarina

doi:10.1098/rspa.2014.0241

Cited by 5 publications

(6 citation statements)

References 13 publications

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“…Proof. Note that for every positive integer t, the (tq)-ring is a lift of the q-ring [13]. Among these, the direct lifts are the ( pq)-rings, with p prime.…”

Section: Ringsmentioning

confidence: 99%

“…The result of lemma 4.1 can be extended to valency-1 regular networks. Indeed, these networks have a unique nontrivial strongly connected component, which is a unique q-cycle, with q 1 [13]. Therefore, they have only two types of connected direct lifts: those that result from splitting the q-cycle into a ( pq)-cycle, with p prime, and those that result from splitting exactly one cell into two cells (which is equivalent to adding a unique vertex with 0 outdegree).…”

Section: Ringsmentioning

confidence: 99%

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Direct lifts of coupled cell networks

Dias

Moreira

2018

Nonlinearity

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In networks of dynamical systems, there are spaces defined in terms of equalities of cell coordinates which are flow-invariant under any dynamical system that has a form consistent with the given underlying network structure-the network synchrony subspaces. Given a network and one of its synchrony subspaces, any system with a form consistent with the network, restricted to the synchrony subspace, defines a new system which is consistent with a smaller network, called the quotient network of the original network by the synchrony subspace. Moreover, any system associated with the quotient can be interpreted as the restriction to the synchrony subspace of a system associated with the original network. We call the larger network a lift of the smaller network, and a lift can be interpreted as a result of the cellular splitting of the smaller network. In this paper, we address the question of the uniqueness in this lifting process in terms of the networks' topologies. A lift G of a given network Q is said to be direct when there are no intermediate lifts of Q between them. We provide necessary and sufficient conditions for a lift of a general network to be direct. Our results characterize direct lifts using the subnetworks of all splitting cells of Q and of all split cells of G. We show that G is a direct lift of Q if and only if either the split subnetwork is a direct lift or consists of two copies of the splitting subnetwork. These results are then applied to the class of regular uniform networks and to the special classes of ring networks and acyclic networks. We also illustrate that one of the applications of our results is to the lifting bifurcation problem.

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“…Proof. Note that for every positive integer t, the (tq)-ring is a lift of the q-ring [13]. Among these, the direct lifts are the ( pq)-rings, with p prime.…”

Section: Ringsmentioning

confidence: 99%

Section: Ringsmentioning

confidence: 99%

Direct lifts of coupled cell networks

Dias

Moreira

2018

Nonlinearity

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“…Ring networks have been studied, for example, in Ganbat [19] and Moreira [31]. Definition 2.12 A network for which there is a cell c such that, for any other cell d, there is exactly one directed path from c to d is called a directed rooted tree.…”

Section: Definition 29 Letmentioning

confidence: 99%

Networks with asymmetric inputs: lattice of synchrony subspaces

Aguiar

2018

Nonlinearity

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We consider coupled cell networks with asymmetric inputs and study their lattice of synchrony subspaces. For the particular case of 1-input regular coupled cell networks we describe the join-irreducible synchrony subspaces for their lattice of synchrony subspaces, first in terms of the eigenvectors and generalized eigenvectors that generate them, and then by giving a characterization of the possible patterns of the associated balanced colourings. The set of the join-irreducible synchrony subspaces is join-dense for the lattice, that is, the lattice can be obtained by sums of those join-irreducible elements (M. Aguiar, P. Ashwin, A. Dias, and M. Field. Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation, J. Nonlinear Sci. 21 (2) (2011) 271-323), and we conclude about the possible patterns of balanced colourings associated to the synchrony subspaces in the lattice. We also consider the disjoint union of two regular coupled cell networks with the same cell-type and the same edge-type. We show how to obtain the lattice of synchrony subspaces for the network union from the lattice of synchrony subspaces for the component networks. The lattice of synchrony subspaces for a homogeneous coupled cell network is given by the intersection of the lattice of synchrony subspaces for its identical-edge subnetworks per each edge-type (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). This, together with the results in this paper, on the lattice of synchrony subspaces for 1-input regular networks and on the lattice of synchrony * Partial support by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020 subspaces for the disjoint union of networks, define a procedure to obtain the lattice of synchrony subspaces for homogeneous coupled cell networks with asymmetric inputs. 2010 Mathematics Subject Classification: 34C15 37C10 06B23 15A18 Coupled cell systems have been for long a focus of interest to the scientific community, including biologists, physicists and mathematicians, since these systems are used as models in a wide range of real-world applications. See, for example, Albert and Barabási [9], Newman [32], Boccaletti et al. [13], Arenas et al. [10], and references therein. The structure of a coupled cell system can be abstracted by a coupled cell network -each cell represents an individual dynamical system and the connections represent the mutual interactions between those individual dynamics. See, for example, the formalism of Golubitsky and Stewart [41], [27], [24] which is more algebraic and the formalism of Field [16] which is more combinatorial.Coupled cell networks can be represented by directed graphs where the vertices are the cells and the edges represent the connections between them. Edges of the same type indicate the same k...

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“…We denote by rings the cycles in the network involving only one edge type, and by the depth the maximal distance of any cell to a ring. Ring networks have been studied, for example, in Ganbat [5] and Moreira [8]. We start by looking at networks having a group structure ( § 7).…”

Section: Introductionmentioning

confidence: 99%

“…They showed that there exists an intrinsic relation between coupled cell systems and coupled cells networks, proving in particular, that robust patterns of synchrony of cells are in oneto-one correspondence to balanced colorings of cells in the network -see [12, theorem 6.5]. Coupled cell networks and coupled cell systems have been addressed, for example, from the bifurcation point of view, [1,5,7,8].…”

Section: Introductionmentioning

confidence: 99%

Characterization of fundamental networks

Aguiar¹,

Dias²,

Soares³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspaces, each network admissible system restricted to that subspace is a dynamical systems consistent with a smaller network. The original network is then said to be a lift of the smaller network. We characterize networks such that: its fundamental network is a lift of the network; the network is a subnetwork of its fundamental network, and the network is a fundamental network. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.

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On bifurcations in lifts of regular uniform coupled cell networks

Cited by 5 publications

References 13 publications

Direct lifts of coupled cell networks

Direct lifts of coupled cell networks

Networks with asymmetric inputs: lattice of synchrony subspaces

Characterization of fundamental networks

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