Heteroclinic network dynamics on joining coupled cell networks

Aguiar, Manuela A. D.; Dias, Ana Paula S.

doi:10.1080/14689367.2016.1197889

Cited by 4 publications

(4 citation statements)

References 25 publications

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“…Moreover, the knowledge of the set of all synchrony subspaces of a coupled cell graph and the associated quotients can be used, for example, to investigate the possibility of the associated coupled cell systems to support heteroclinic behavior. See for example [1,3].…”

Section: Application To Coupled Cell Systemsmentioning

confidence: 99%

Quotients and lifts of symmetric directed graphs

Aguiar¹,

Dias²,

Manoel³

2018

Preprint

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Given a directed graph, an equivalence relation on the graph vertex set is said to be balanced if, for every two vertices in the same equivalence class, the number of directed edges from vertices of each equivalence class directed to each of the two vertices is the same. In this paper we describe the quotient and lift graphs of symmetric directed graphs associated with balanced equivalence relations on the associated vertex sets. In particular, we characterize the quotients and lifts which are also symmetric. We end with an application of these results to gradient and Hamiltonian coupled cell systems, in the context of the coupled cell network formalism of Golubitsky, Stewart and Török (Patterns of synchrony in coupled cell networks with multiple arrows. SIAM Journal of Applied Dynamical Systems 4 (1) (2005) 78-100).

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Section: Application To Coupled Cell Systemsmentioning

confidence: 99%

Quotients and lifts of symmetric directed graphs

Aguiar¹,

Dias²,

Manoel³

2018

Preprint

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“…In the fully symmetric case, Ashwin et al [39,51] have examples of complex patterns of synchronization and desynchronization in phase oscillator networks that are related to heteroclinic networks. Examples of heteroclinic cycles and networks are given in [6,5] for coupled cell networks with no global symmetry group.…”

Section: Semilinear Feedback Systems and Coupled Cell Systemsmentioning

confidence: 99%

“…Analogously, in the equivariant case, given a cubic truncation, dynamics is uniquely defined on the (spherical) simplex ∆ k−1 = S k−1 ∩ O k using the phase vector field (see [25,Chapters 4,5] and section 2.1). Dynamics on the simplex and spherical simplex are topologically conjugate by the transformation x ↔ x 2 and a time rescaling by a factor of 2.…”

Section: Examples 23 (1) a (Generalized) Lotka-volterra System Is Dmentioning

confidence: 99%

Patterns of desynchronization and resynchronization in heteroclinic networks

Field

2016

Nonlinearity

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We prove results that enable the efficient and natural realization of a large class of robust heteroclinic networks in coupled identical cell systems. We also propose some general conjectures that relate a natural and large class of robust heteroclinic networks that occur in networks modelled by equations of Lotka-Volterra type, and certain networks of symmetric systems, to robust heteroclinic networks in coupled cell networks.

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“…The characterization of the set of synchrony subspaces for a network is important because the existence of these flow-invariant subspaces can have a large impact on the dynamics, and favour the existence of non-generic dynamical behaviour like robust heteroclinic cycles and networks, and bifurcation phenomena. See, for example, [3,5,11,17,18] and [15,21,22,40].…”

Section: Introductionmentioning

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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Heteroclinic network dynamics on joining coupled cell networks

Cited by 4 publications

References 25 publications

Quotients and lifts of symmetric directed graphs

Quotients and lifts of symmetric directed graphs

Patterns of desynchronization and resynchronization in heteroclinic networks

Networks with asymmetric inputs: lattice of synchrony subspaces

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