## Abstract

A *lift* of a given network is a network that admits the first network as quotient. Assuming that a bifurcation occurs for a coupled cell system consistent with the structure of a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type), it is well known that some lifts exhibit new bifurcating branches of solutions. In this work, we approach this problem restricting attention to uniform networks, that is, networks that have no loops and no multiple arrows. We show that, from the bifurcation point of view, rings and their lifts are special networks. We also prove that generically there are lifts that just exhibit the bifurcating branches determined by the quotient network and, moreover, we identify all generic situations where lifts exist that may exhibit bifurcating branches that do not appear in the quotient itself.

## 1. Introduction

Networks represent interactions between individual dynamical units and they are abundant in science. The most frequently cited examples include electrical power grids, food webs, the Internet, chemical reaction schemes, molecular interactions, neural circuits and individual relationships [1,2]. Scientists are trying to understand global properties of networks and, more specifically, the interplay between structure and dynamics. Mathematicians are aware of this general growing interest in networks and are also contributing to this study using their powerful mathematical tools. We highlight the theory of coupled cell networks formalized by Stewart, Golubitsky and co-workers [3,4].

Before introducing all the precise concepts and results necessary to understand the exact main achievements of this paper, we give the reader an idea of the key importance of this work. Architecturally, a *lift* of a given network is interpreted as a network that results from the former by a splitting of cells (nodes), satisfying a specific fundamental property of the splittings [5]. Assuming that a codimension-1 bifurcation occurs for a coupled cell system consistent with the structure of the network, Aguiar *et al.* [6] proved that some networks admit lifts that exhibit additional bifurcating branches of solutions and some other networks do not admit such lifts. We are then led to ask the following questions:

(1) Do all networks admit lifts with no additional bifurcating branches?

(2) How do we identify networks that admit lifts exhibiting new bifurcating branches?

In this paper, we answer these two questions for regular uniform networks, that is, for networks with no loops and no multiple arrows that have only one type of cell, one type of coupling and all cells receive the same number of inputs. We also show how to obtain lifts in each case.

### (a) Regular and uniform coupled cell networks

A *cell* is a system of ordinary differential equations and a *coupled cell system* is a finite collection of interacting cells. A coupled cell system can be associated with a *network*, a directed graph (abbreviated to *digraph*) whose nodes represent cells and whose arrows represent couplings between cells. In this paper, it is assumed that all networks have a connected underlying graph. The general theory allows for *loops* and *multiple arrows*. All couplings of the same type between two cells are represented by a single arrow with the number of couplings attached to it, unless this number is equal to 1, in which case it is simply omitted. The general theory associates with each network a class of admissible vector fields, consistent with the structure of the network.

In this paper, we restrict attention to *regular and uniform networks*, that is, networks with no loops and no multiple arrows (uniform) associated with coupled cell systems where all cells have the same differential equation, up to reordering of coordinates, and one kind of coupling (regular). In this case, the state spaces of the cells are all identical, say a euclidean space *k*≥1, and so if the network has *n* cells then the *total phase space* is *valency* of a regular network is the number of arrows that input to each cell. The *j*th coordinate of an admissible vector field of a regular network with valency *m* has the form
*k*≥1 and

A *polydiagonal* is a subspace of the total phase space that is defined by the equalities of certain cell coordinates. A *synchrony subspace* is a polydiagonal that is flow invariant for every admissible vector field.

Golubitsky *et al.* [4] proved that every coupled cell system associated with a network when restricted to a synchrony subspace corresponds to a coupled cell system associated with a smaller network, called the *quotient network*. If *Q* is a quotient network of a network *G*, then we say that *G* is a *lift* of *Q*.

### (b) Spectrum of regular networks

The *adjacency matrix* of a regular network is a square matrix *A*=[*a*_{ij}], where *a*_{ij} is the number of arrows that cell *i* receives from cell *j*. Note that each row sum of the adjacency matrix equals the valency of the network.

Recall that a *multi-set* is a collection of objects that are not necessarily distinct. Given an object *a* and a multi-set *A*, the multiplicity of *a* in *A* is the number of times that it occurs in *A*. The object *a* is an *element* of *A* when it occurs in *A* and this fact is denoted by *a*∈*A*. Commonly, the elements of a multi-set are written between square brackets. Given two multi-sets *A* and *B*, we define the difference *A*−*B* to be the multi-set consisting of all elements of *A* that have a greater multiplicity in *A* than in *B*; the multiplicity of these elements in *A*−*B* is the difference between the multiplicities in *A* and in *B*.

The *spectrum of a matrix* is the multi-set of its eigenvalues. The *spectrum of a regular network* *G* is the spectrum of its adjacency matrix and it is denoted by *S*_{G}.

Observe that the network adjacency matrix can be seen as the matrix of a linear admissible vector field. If Δ is a synchrony subspace of a regular network *G* then Δ is, in particular, left invariant by all linear admissible vector fields of *G* and so by its adjacency matrix. If *Q* is the corresponding quotient network, then the adjacency matrix of *Q* is similar to the restriction of the adjacency matrix of *G* to Δ. Thus, all eigenvalues of a (quotient) regular network are eigenvalues of any lift, including multiplicities.

### Definition 1.1

Consider a lift *G* of a regular network *Q*. The elements of *S*_{G}−*S*_{Q} are the *extra eigenvalues* (of *G* with respect to *Q*). Given an eigenvalue λ of *Q*, the lift *G* is λ-*preserving* (for *Q*) if the real part of any extra eigenvalue is distinct from the real part of λ. The lift *G* is *spectrum-preserving* (for *Q*) if *G* is λ-preserving, for every λ∈*S*_{Q}.

For example, in figure 2, we present a 3-cell network *A* and one of its 5-cell lifts, *B*. For these networks, *S*_{B}=[−1,−*i*,0,*i*,1] and *S*_{A}=[−1,0,1] and so the extra eigenvalues are *i* and −*i* and *B* is not 0-preserving (therefore, *B* is not spectrum-preserving).

### (c) Motivation

Given an *n*-cell regular network *G*, consider a one-parameter system of ordinary differential equations
*F* is a *G*-admissible vector field and λ is the bifurcation parameter. If *k* is the dimension of the internal dynamics, then

Suppose that there exists a synchronous equilibrium in the *fully synchronous subspace* {*x*_{1}=…=*x*_{n}}, which we assume, after a change of coordinates, to be the origin for λ=0, that is,
*steady-state bifurcations* and *Hopf bifurcations*. The first type occurs when the Jacobian *J*=(d*F*)_{0,0} has a zero eigenvalue and the second one occurs when the Jacobian *J* has a pair of non-zero purely imaginary eigenvalues. In this paper, we only consider these two bifurcation types and so, to simplify, whenever we write *bifurcation* it should be understood to mean *steady-state or Hopf bifurcation*. Moreover, each of these bifurcations divides into *synchrony-preserving* and *synchrony-breaking*, depending on whether the centre subspace is contained or not, respectively, in the fully synchronous subspace. Bifurcation theory has been applied to the theory of coupled cell networks in various ways (e.g. [6–9]).

There is an interesting issue concerning a comparative study of a bifurcation from a fully synchronous equilibrium in two different systems, one associated with a given (quotient) network and the other with a lift of that network. This issue was raised by Aguiar *et al.* [6,10], and examples are given where, assuming a bifurcation occurs for a coupled cell system restricted to a fixed (quotient) network, new branches of solutions occur in some of the lifts. More precisely: assume that *Q* is a quotient of *G* determined by a synchrony subspace Δ and impose a degeneracy condition on *F*, implying that the centre subspace of *J*|_{Δ} is non-trivial. We may now ask about the impact of that degeneracy condition on the bifurcation problem (1.1), assuming the same degeneracy condition on *F*. A necessary condition for new branches of bifurcating solutions for (1.1) to exist is the increasing dimension of the centre subspace of *J* comparative to *J*|_{Δ}.

The results in Leite & Golubitsky [7] and Golubitsky & Lauterbach [8] relate the eigenvalues of the Jacobian *J* to the eigenvalues of the adjacency matrix of the network. More specifically, if *μ*_{1},…,*μ*_{n} are the eigenvalues of the adjacency matrix of the network then the *kn* eigenvalues of the Jacobian *J* are the union of the eigenvalues of the *k*×*k* matrices
*α* is the *k*×*k* matrix of the linearized internal dynamics at the origin and *β* is the *k*×*k* matrix of the linearized coupling at the origin, both matrices being found by differentiating *F*.

For *k*=1, every eigenvalue of the Jacobian *J* has the form
*α*+*βμ* of the Jacobian *J* is on the imaginary axis if and only if the eigenvalue *μ* of the adjacency matrix is such that *α*+*Re*(*μ*)*β*=0. Therefore, in this case, if *α*+*Re*(*μ*)*β*=0 is the degeneracy condition that has been assumed for the one-parameter bifurcation that is being studied, then that bifurcation is also associated with all the eigenvalues *μ*_{i} of the adjacency matrix such that *α*+*Re*(*μ*)*β*=0=*α*+*Re*(*μ*_{i})*β*. Besides, as mentioned in §1*a*, it is assumed that all networks have a connected underlying graph and, thus, it is naturally assumed that the linearized coupling *β* at the origin is generically not zero. Thus, for *k*=1, generically, the bifurcation is associated with at least an eigenvalue of the adjacency matrix and all of them have the same real part. For *k*≥2, generically, the centre subspace at a codimension-1 synchrony-breaking bifurcation is isomorphic to the real part of a generalized eigenspace of the network adjacency matrix [8]. Thus, for a general dimension *k* of the internal dynamics, generically, the codimension-1 bifurcation is associated with at least an eigenvalue of the adjacency matrix and all of them have the same real part. In this paper, these eigenvalues are called *bifurcation eigenvalues*. In this sense, the issue of preserving the number of eigenvalues on the imaginary axis is translated, in terms of the adjacency matrix, into the preservation of the number of eigenvalues with a specific real part. Hence, if a lift of a regular network is spectrum-preserving, then generically no new branches of solutions can occur for the lift equations, assuming that a codimension-1 synchrony-breaking bifurcation occurs for the quotient equations.

In this work, we address the following questions concerning bifurcations of coupled cell networks: given a regular network and a bifurcation occurring in a coupled cell system consistent with its structure, is it always possible to find lifts that just exhibit the bifurcating branches determined by the quotient? Why do some networks admit lifts that exhibit bifurcating branches that do not appear in the quotient itself and some other networks do not admit such lifts? What is the role of the network architecture in this existence?

Dias & Moreira [5] studied the spectrum of regular uniform lifts when the initial network is regular but non-uniform. The main result shows that, when the initial network *Q* has loops or multiple arrows, it is not always possible to construct spectrum-preserving uniform lifts. Nevertheless, it has been proved that it is always possible to obtain a spectrum-preserving uniform lift of a network that is ODE-equivalent to *Q* (two networks are *ODE-equivalent* if they give rise to the same space of admissible vector fields, for a suitable choice of cell phase spaces [4,11]; thus, the set of bifurcations of the admissible vector fields for each of two ODE-equivalent networks is the same).

In this paper, we focus our attention on regular and uniform networks and we prove that, given a valency-*v* regular uniform network and an eigenvalue λ of the corresponding adjacency matrix, there are always non-trivial λ-preserving lifts. Moreover, we prove that there are lifts that are not λ-preserving if and only if λ=0 or *v*>1. As a consequence, assuming that a codimension-1 synchrony-breaking bifurcation occurs for a coupled cell system associated with a given general regular uniform network, we show that generically there are always non-trivial lifts that just exhibit the bifurcating branches determined by the quotient network. We also identify all generic situations where lifts exist that may exhibit bifurcating branches not predicted by the quotient.

### (d) Structure of the paper

In §2, we present all basic concepts and results that are necessary to understand this work. In §3, we study some architectural and spectral properties of rings and of their lifts. The main results here are lemma 3.3 and theorem 3.4. In §4, we prove that rings and their lifts play a special role from the bifurcation point of view, because they are used to construct lifts with specific spectral properties. To be precise, given a bifurcation occurring in a coupled cell system associated with a regular uniform network, we use rings to construct non-trivial lifts that just exhibit the bifurcating branches determined by the quotient (theorem 4.4 and corollary 4.5), as well as to describe all generic situations where lifts exist that may exhibit additional bifurcating branches (theorem 4.7 and corollary 4.9).

## 2. Basic concepts and results

In this section, we present all basic concepts and results from the theory of coupled cell networks and from the general theory of digraphs that are necessary to understand this work.

### (a) Digraphs

A regular network is a digraph and, so, all concepts and results in the theory of digraphs can be applied to regular networks. In particular, we point out the following concepts and results in the theory of digraphs that are important in our work:

(1) The

*sources*of a digraph are all vertices that have 0 indegree.(2) The

*condensation*of a digraph is a digraph with no loops in which each strongly connected component is replaced by a unique vertex, and all directed edges from one strongly connected component to another are replaced by a single directed edge.(3) A

*simple cycle*consists of a sequence of vertices starting and ending at the same vertex and with no other repetition of vertices, such that, for each two consecutive vertices, there exists an edge directed from the earlier vertex to the later one.(4) All eigenvalues of the adjacency matrix of a digraph are often called the

*eigenvalues of the digraph*itself.(5) The eigenvalues of a digraph are the eigenvalues of each of its strongly connected components, counting multiplicities.

For further details about this subject, see, for example, Brualdi & Ryser [12].

### (b) Lifts and colourings

Theorem 4.3 of Golubitsky *et al.* [4] describes a very easy method to identify lifts of a general network and the corresponding synchrony subspace in the following way: a network *G* is a lift of a *q*-cell network *Q* if and only if it is possible to colour all cells in *G* with *q* distinct colours in such a way that any pair of cells with colour *c* receives the same number of inputs from cells of colour *d*, for each *d* (for general non-regular networks it is assumed that all cells with the same colour are of the same type); moreover, in the affirmative case, the corresponding synchrony subspace is defined by all equalities of cell coordinates that have the same colour. For example, using this *colouring method*, it is easy to understand that the 11-cell network in figure 3 is a lift of the 4-cell in the same figure and that the corresponding synchrony subspace is

### (c) Lift as splitting of cells

A lift can be interpreted as resulting from the cellular splitting of the initial network. For example, all lifts with exactly one more cell than the initial network result from the splitting of exactly one of its cells into two cells. A cell that splits is called a *splitting cell* and every cell resulting from this splitting is called a *split cell*.

Consider a regular network and assume that some of its cells split. In order to obtain a lift after the decomposition, the cellular splitting has to satisfy the following property:

*Fundamental property of the splittings*[5]: assume that*i*is a cell that receives*k*arrows from cell*j*and that at least one of these two cells splits. There are three cases to be considered:(1)

: after the splitting, each split cell associated with cell*i*splits but*j*does not*i*receives*k*arrows from cell*j*.(2)

: after the splitting, cell*j*splits but*i*does not*i*receives*k*cells from the set of split cells associated with cell*j*.(3)

*both*: after the splitting, each split cell associated with cell*i*and*j*split*i*receives*k*cells from the set of split cells associated with cell*j*.

For example, in figure 4, we show a 3-cell regular network *Q* and a 6-cell lift *G*, obtained by splitting cell 1 and cell 2.

### (d) Lifts and extra eigenvalues

The following result is useful to study extra eigenvalues as it simplifies their calculation. Recall that given a network *N* and a subset *C* of cells, the *subnetwork* of *N* consisting of all cells in *C* is the digraph whose nodes are all cells in *C* and whose arrows are all existing arrows in the original network whose heads and tails are cells in *C*.

### Theorem 2.1 ([5])

*Given a lift G of a regular network Q, consider the subnetworks S and S′ of all splitting cells and of all split cells, respectively. The extra eigenvalues of G with respect to Q are precisely the extra eigenvalues of S′ with respect to S.*

For example, in figure 4, consider the 3-cell network *Q* and the 6-cell network *G*, which is a lift of *Q*. The characteristic polynomials of the corresponding adjacency matrices are *x*^{3}−*x*^{2}−2*x* and
*G* with respect to *Q* are the three complex roots of the cubic *x*^{3}+2*x*−1. In figure 5, we present the subnetworks *S* and *S*′ of splitting cells and of split cells, respectively. The characteristic polynomials of the corresponding adjacency matrices for *S* and *S*′ are, respectively, *x*^{2}−*x*−1 and
*S*′ with respect to *S* are also the three complex roots of the previous cubic.

## 3. Rings and their lifts

In this section, we study rings and their lifts. These networks are highlighted in this paper because, as we will see in the next section, their architectural and spectral properties play a special role from the bifurcation point of view.

### Definition 3.1

Let *q* and *s* be positive integers.

(1) The

*q*-*ring*is the*q*-cell regular network depicted in figure 6.(2) The (

*q*+*s*)-*chain with feedback*is the (*q*+*s*)-cell regular network depicted in figure 7.

### Remark 3.2

The adjacency matrix of the *q*-ring is a (*q*×*q*)-permutation matrix. Well-known results in the theory of matrices allow us to conclude that the eigenvalues of the *q*-ring are the *q*th complex roots of the unity, being simple eigenvalues and lying on the unit circle [13].

### (a) Chains with feedback

In this section, we show that chains with feedback are lifts of rings, using the cellular splitting point of view.

### Lemma 3.3

*For all positive integers* *q* *and* *s*, *the* (*q*+*s*)-*chain with feedback is a lift of the* *q*-*ring. Moreover*, 0 *is the unique extra eigenvalue*.

### Proof.

Let *q* and *s* be positive integers. Taking the synchrony subspace defined by the *s* equalities of the form *x*_{j}=*x*_{i}, with *q*<*j*≤*q*+*s* and 1≤*i*≤*q* satisfying *j*≡*i*(mod *q*), we have that the (*q*+*s*)-chain with feedback (figure 7) is a lift of the *q*-ring (figure 6). It is also easy to prove this fact using the colouring method described in §2*b* (figure 3 shows the procedure for the case *q*=4 and *s*=7).

To emphasize the cellular splitting perspective, we also prove this result by describing how the (*q*+*s*)-chain with feedback can be obtained from the *q*-ring using splittings of cells. When *q*=1, the (*q*+*s*)-chain with feedback is obtained by a unique splitting of the cell of the ring into exactly (*q*+*s*) cells. When *q*>1, the (*q*+*s*)-chain with feedback can be obtained from the *q*-ring by *s* sequential splittings. Let *r* be the smallest positive integer such that *s*≡*r*(mod *q*). The (*q*+*s*)-chain with feedback is constructed from the *q*-ring using the following steps (the scheme is shown in figure 8):

— The first step consists in splitting cell

*r*into cells*r*and (*q*+*s*), in such a way that cell (*q*+*s*) has no output (and so, in the lift, cell (*r*+1) depends on cell*r*). Note that, in the lift, cell (*r*−1) has two outputs because both split cells depend on the same cell (recall the fundamental property of the splittings).— In the

*l*th step, 2≤*l*≤*s*, let*c*be the cell on which cell (*q*+*s*−*l*+2) depends. It is clear that cell*c*has two outputs. Split cell*c*into cell*c*and cell (*q*+*s*−*l*+1) in such a way that, in the lift, both split cells send an arrow and cell (*q*+*s*−*l*+1) sends an arrow to cell (*q*+*s*−*l*+2).

In each step, the corresponding splitting cell splits into exactly two cells and thus the extra eigenvalue is 0, because the ring has no loops. Therefore, 0 is the unique extra eigenvalue. ▪

Using the example of rings and chains with feedback, we can understand that the type of splitting can have a huge impact from the spectral point of view, and, consequently, from the bifurcation point of view. Indeed, in figure 9, we present two lifts of the *q*-ring, namely the (*q*+*q*)-chain with feedback and the (2*q*)-ring, *q*≥1. Note that there is only a small difference in the architecture of these two lifts, namely one of the inputs of a unique cell (cell 1). In both cases, the lifts are obtained by splitting all cells of the ring, each of them into exactly two cells. However, the first lift is obtained with *s* sequential splittings, one after the other, and the second is obtained with a unique splitting where all cells are decomposed simultaneously. This big difference in the type of splitting is accomplished by a big difference from the spectral point of view: in the chain with feedback, all extra eigenvalues are zero; in the ring, all extra eigenvalues are simple and lie on the unit circle, as pointed out in remark 3.2. In particular, the first lift is spectrum-preserving if and only if *q*≢0 (mod 4) and the second lift is always spectrum-preserving, for all values of *q*.

### (b) General lifts

The following result describes all lifts of rings and their spectra.

### Theorem 3.4

*For every positive integer q, a network is a lift of the q-ring if and only if it is a valency-1 regular network having a unique non-trivial strongly connected component, which is a (sq)-ring, for some s≥1. In particular, the possible extra eigenvalues of lifts of the q-ring are zero and the complex solutions of equations of the form* (*x*^{tq}−1)/(*x*^{q}−1)=0, *with t>1, and in this last case the eigenvalues are simple.*

### Proof.

Let *q* be a positive integer and *Q* be the *q*-ring. Firstly, we consider a lift *G* of *Q* and prove that it is a valency-1 regular uniform network having a unique non-trivial strongly connected component, which is an (*sq*)-ring, for some *s*≥1. The lifting process preserves the regularity, uniformity and the valency of the quotient network and, so, *G* is a valency-1 regular uniform network. Moreover, the sources of the condensation are valency-1 non-trivial strongly connected components and so they are rings. As *v*=1, the network has a unique non-trivial strongly connected component. Hence, *G* has a unique non-trivial strongly connected component and this component is a ring, say, with *n* cells. It remains to be proved that *q*|*n*. If *n*=*q*, the result is obvious and so, in what follows, we assume *n*>*q*. As *Q* is a quotient of *G*, there is a colouring of the *n*-ring using *q* distinct colours, *c*_{1},…,*c*_{q}, in such a way that any pair of cells with colour *c* receives an input (unique) from a cell of colour *d*. So, if *c*_{1} is the colour of cell 1, then cell 2 is coloured with a colour *c*_{2} distinct from *c*_{1} because otherwise all cells would have the same colour, contradicting the fact that the corresponding quotient is a *q*-ring. Similarly, colour *c*_{3} of cell 3 is distinct from *c*_{2} and *c*_{3}=*c*_{1} if and only if *q*=2. Continuing this reasoning, we obtain that *c*_{i}≠*c*_{j} for 1≤*i*<*j*≤*q* and so the first *q* cells of the *n*-ring must be coloured using the *q* distinct colours; say, cell *k* has colour *c*_{k}, for 1≤*k*≤*q*. As there are only *q* available distinct colours, cells (*q*+1) until cell (2*q*) must be coloured using the same coloured periodic sequence. Generalizing for all 1≤*l*≤*n*, cell *l* has colour *c*_{r}, where 1≤*r*≤*q* and *l*≡*r*(mod *q*). Thus, because cell 1 depends on cell *n*, this cell has colour *c*_{q}, and, so, *q*|*n*.

We also use the colouring method to prove the other implication. Consider a valency-1 regular uniform network having a unique non-trivial strongly connected component, which is an (*sq*)-ring, for some *s*≥1. In what follows, we describe the colouring of this network that shows that it is a lift of *Q* (in figure 10, we present an illustration of this colouring as an example). Consider *q* distinct colours *c*_{1},…, *c*_{q} and, for each 1≤*i*≤*q*, colour cell *i* in the (*sq*)-ring with colour *c*_{i}. If *s*>1, proceed colouring all the other cells of this ring as follows: colour cell (*q*+1) in the (*sq*)-ring with colour *c*_{1}, cell (*q*+2) with colour *c*_{2} and, in general, cell *j* with colour *c*_{r}, where 1≤*r*≤*q* and *j*≡*r* (mod *q*). After the colouring of the (*sq*)-ring, colour all trivial strongly connected components as follows:

(1) First step: colour all cells that depend on a cell of the (

*qs*)-ring as follows: if*l*is a cell that depends on a cell of colour*c*_{α}of the (*qs*)-ring, then it is coloured with colour*c*_{α+1}, assuming*c*_{q+1}=*c*_{1}.(2) Second step: colour all cells that depend on a cell that was coloured in the previous step using the same argument, that is, if cell

*j*is a cell that depends on a cell of colour*c*_{α}that was coloured in the previous step, then it is coloured with colour*c*_{α+1}, assuming*c*_{q+1}=*c*_{1}.(3) Remaining steps: continue this process repeating the previous step until all cells of the network are coloured.

In this colouring, any pair of cells with colour *c* receives an input (unique) from a cell of colour *d*. Therefore, the regular network is a lift of the *q*-ring and the corresponding synchrony subspace is defined by the equality of cell coordinates that have the same colour. The extra eigenvalues are obtained taking into account that the eigenvalues of a regular network are the eigenvalues of each of its strongly connected components, including multiplicities, and that the eigenvalues of an *n*-ring are the *n*th complex (simple) roots of the unity. ▪

## 4. Application to bifurcation theory

Recall that a codimension-1 synchrony-breaking bifurcation (steady-state or Hopf) is generically associated with at least an eigenvalue of the adjacency matrix, all of them with the same real part (when the dimension of the internal dynamics of each cell is at least 2, the bifurcation is generically associated with a unique eigenvalue of the network adjacency matrix). In this paper, these eigenvalues are called *bifurcation eigenvalues*. The issue of preserving the number of eigenvalues on the imaginary axis is translated, in terms of the adjacency matrix, into the preservation of the number of eigenvalues with a specific real part. Hence, if a lift of a regular network is spectrum-preserving, then generically no new branches of solutions can occur for the lift equations, assuming that a synchrony-breaking bifurcation occurs for the quotient equations.

In [6], the following question is addressed: assuming that a codimension-1 synchrony-breaking bifurcation occurs in a coupled cell system consistent with the structure of a given regular network, and considering a lift of this network, how does the bifurcation lift to the overall space? It is proved in particular that every lift of the 2-ring is spectrum-preserving. The following corollaries of theorem 3.4 generalize this result.

### Corollary 4.1

*For every positive integer* *q*, *all lifts of the* *q*-*ring are spectrum-preserving if and only if* *q*≢0(mod 4).

### Proof.

Let *q* be a positive integer and *Q* be the *q*-ring. Consider a non-trivial lift *G* of *Q*. The set *S*_{1} of eigenvalues of *Q* consists of complex conjugate pairs on the unit circle. Owing to theorem 3.4, the set *S*_{2} of eigenvalues of *G* consists of complex conjugate pairs on the unit circle and possibly zero. *S*_{2} contains properly *S*_{1}. Taking into account that all eigenvalues of *S*_{2} on the unit circle are simple (theorem 3.4) and that *S*_{1} is closed under complex conjugation, we have that *G* is not spectrum-preserving if and only if
*Q* are spectrum-preserving if and only if ±*i*∉*S*_{1}, that is, if and only if *q*≢0(mod 4). ▪

### Corollary 4.2

*For every positive integer* *q*, *suppose that a codimension*-1 *synchrony-breaking bifurcation occurs in a coupled cell system associated with the* *q*-*ring. If* *q*≢0(mod 4) *then, generically, all lifts just exhibit the bifurcating branches determined by the quotient ring, imposing the same degeneracy condition at the coupled cell system as assumed for the* *q*-*ring*.

### Remark 4.3

The hypothesis of this result could be less restrictive, depending on the type of bifurcation, on the dimension *k* of the internal dynamics and on the bifurcation eigenvalues. In fact, if the codimension-1 synchrony-breaking bifurcation is

(1) a Hopf bifurcation, then the bifurcation eigenvalues are a pair of non-zero complex conjugate eigenvalues (of the

*q*-ring adjacency matrix).(a) If

*k*>1 or if*k*=1 and ±*i*are not bifurcation eigenvalues then, generically, all lifts just exhibit the bifurcating branches determined by the ring, because all lifts are spectrum-preserving.(b) If

*k*=1 and ±*i*are the bifurcation eigenvalues then, generically, only lifts that are also rings preserve the number of branches from the quotient. Recall that the addition of cells ‘outside’ the ring provides only 0 as an extra eigenvalue in the lift.

Note, however, that

*many generic codimension-1 Hopf bifurcations cannot occur unless the dimension of the internal dynamics is at least 2*[8], p. 53.(2) a steady-state bifurcation, then there is a unique bifurcation eigenvalue, either 1 or −1. As all lifts are ±1-preserving, generically, all lifts just exhibit the bifurcating branches determined by the quotient ring.

In [6], it is also shown that if the quotient network is the 2-ring or the 3-cell bidirectional ring, then it is possible to find lifts that just exhibit the bifurcating branches that appear in the quotient. We ask whether this fact can be generalized, that is, whether it is always possible to find, for a given regular uniform network, lifts that just exhibit the bifurcating branches determined by the quotient. Also, in that paper, it is proved that some lifts of the 3-cell bidirectional ring exhibit steady-state bifurcating branches in addition to the ones determined by the quotient for the same fixed degeneracy condition. Here, we ask for necessary and sufficient conditions to characterize networks that admit lifts exhibiting additional bifurcating branches. In the next two sections, we answer these questions.

### (a) Preserve the number of bifurcating branches

In this section, we prove that, given a codimension-1 synchrony-breaking bifurcation occurring in a coupled cell system associated with a regular uniform network, there are always lifts that just exhibit the bifurcating branches that appear in the quotient (for the same fixed degeneracy condition).

### Theorem 4.4

*Given a regular uniform network and an eigenvalue* λ *of the corresponding adjacency matrix, there are non-trivial* λ-*preserving lifts.*

### Proof.

If *Re*(λ)≠0, then it is possible to obtain lifts (with as many cells as we choose), having only 0 as an extra eigenvalue. In fact, it is possible to split a unique cell into any finite number of cells or, more generally, split sequentially more than one cell, one by one, and introduce only 0 as an extra eigenvalue, taking into account that no loops are allowed. Thus, in this case, there are always non-trivial λ-preserving lifts (with any finite number of additional cells).

If *Re*(λ)=0, consider the strongly connected components of the network that correspond to the sources of the corresponding condensation. These components are non-trivial because no loops are allowed. Fix one of them. In what follows, we prove that it is possible to split cells in this component and preserve the number of eigenvalues on the imaginary axis. Consider all simple cycles in this component with minimal length and fix one of them. The subnetwork defined by this cycle is an *n*-ring. As noted in remark 3.2, the eigenvalues of the *n*-ring are the *n*th complex roots of the unity. The splitting depends on *n* and we consider the following three distinct situations (see also figure 11):

(1) If 2

*n*≢0(mod 4) or*n*≡0(mod 4): split all cells of the*n*-ring and obtain the (2*n*)-ring. In the first case, no eigenvalue of the (2*n*)-ring is on the imaginary axis and so, in particular, no extra eigenvalue has zero real part. In the second case, ±*i*are eigenvalues of the*n*-ring and so, because the (2*n*)-ring is spectrum-preserving, no extra eigenvalue has zero real part.(2) If 2

*n*≡0(mod 4) and*n*≢0(mod 4): split all cells of the*n*-ring and obtain the (3*n*)-ring. As 3*n*≡*n*(mod 4)≢0(mod 4), no eigenvalue of the lift is on the imaginary axis and so, in particular, no extra eigenvalue has zero real part.

Hence, the lift of the regular uniform network is obtained by splitting the chosen *n*-ring into a (2*n*)-ring or a (3*n*)-ring, depending on which of the previous conditions is satisfied. Owing to theorem 2.1, no extra eigenvalue of this lift has zero real part. ▪

### Corollary 4.5

*Given a codimension*-1 *synchrony-breaking* (*steady-state or Hopf*) *bifurcation occurring in a coupled cell system associated with a regular uniform network, generically, there are non-trivial lifts that just exhibit the bifurcating branches determined by the quotient* (*associated with the same degeneracy condition determining the bifurcation for the quotient*).

### Example 4.6

In figure 12, we show a 10-cell network and an example of a 0-preserving lift.

Note that this result cannot be extended to non-uniform networks in the context of uniform lifts, as proved in Dias & Moreira [5].

### (b) Increase the number of bifurcating branches

If a codimension-1 synchrony-breaking bifurcation occurs in a coupled cell system associated with a regular uniform network, it is well known that it is not always possible to find lifts exhibiting bifurcating branches in addition to the ones determined by the quotient [6]. In this section, we describe all generic situations where this increase of bifurcating branches may occur.

### Theorem 4.7

*Given a valency-v regular uniform network and an eigenvalue* λ *of the corresponding adjacency matrix, there are lifts that are not* λ-*preserving if and only if v*>1 *or* λ *has zero real part.*

### Proof.

Firstly, we prove that if *v*=1 and *Re*(λ)≠0, then all lifts are λ-preserving. Indeed, it is straightforward that every valency-1 regular uniform network has a unique non-trivial strongly connected component and it is also clear that such a component is a ring. Consider the set *S*_{1} of eigenvalues of this network that lie outside the imaginary axis. The lifting process preserves the valency and the uniformity and, so, every lift also has a unique strongly connected component and this component is a ring. As the set *S*_{2} of eigenvalues of this lift lying outside the imaginary axis consists of simple eigenvalues on the unit circle, *S*_{1}⊂*S*_{2} and *S*_{1} is closed under complex conjugation, we obtain that the lift is λ-preserving, for all λ∈*S*_{1}.

To prove the other implication, we assume that *v*>1 or *Re*(λ)=0 and show that there are lifts that are not λ-preserving. In this case, we consider the following three distinct situations:

(1) If

*Re*(λ)=0, then it is possible to obtain lifts that are not 0-preserving (with as many cells as we choose) and having only 0 as an extra eigenvalue, as explained above.(2) If

*Re*(λ)=*Re*(*μ*), for some eigenvalue*μ*of a strongly connected component of the network that is not a source of the condensation, then we can split all cells of that component and produce a copy of it. Since the eigenvalues of a regular network are the eigenvalues, counting multiplicities, of its strongly connected components, the extra eigenvalues of the constructed lift are all eigenvalues of the repeated component and, thus,*μ*is an extra eigenvalue. Note that the copy receives arrows from another component and thus the lift is connected. (An example of this construction is given in figure 14.)(3) If none of the previous situations occur, then λ is an eigenvalue of a source

*C*of the condensation, with*Re*(λ)=0 and*v*>1. Thus,*C*is a valency-*v*regular uniform network and so it is not a ring. One way of obtaining a lift where the multiplicity of the eigenvalue λ increases is by splitting an arbitrary cell in*C*into exactly two cells in such a way that one of the split cells,*c*, does not send any arrow. Note that*C*is a strongly connected component of this lift and that*c*receives at least two inputs from*C*. After that, produce a copy*C*′ of*C*in such a way that*c*receives inputs from both*C*and*C*′, to guarantee the connectivity of the lift. (An example of this construction is given in figure 17; for specific architectures or values of λ, lifts can admit simpler constructions, as illustrated in figures 15 and 16.)

▪

### Remark 4.8

This theorem can be extended to non-uniform regular networks in the context of uniform lifts. If *v*=1 and *Re*(λ)≠0, then all uniform lifts are λ-preserving. Indeed, a valency-1 regular network has a unique non-trivial strongly connected component. Additionally, if the network is non-uniform, then this component consists of a unique cell with a loop. Therefore, this network has only two distinct eigenvalues, namely 1 (simple) and 0, implying that λ=1. Hence, the unique non-vanishing extra eigenvalues that can be introduced in uniform lifts of this network are the *q*th complex (simple) roots of the unity distinct from 1, for some *q*>1, obtained by splitting the cell with the loop into exactly *q* cells. So, all (uniform) lifts are 1-preserving. To prove the other implication, the construction is the same for each of the three situations considered in the proof of theorem 4.7. Note that, after this construction, it is possible to obtain a uniform lift of the initial network using the construction suggested in [5].

### Corollary 4.9

*Given a codimension*-1 *synchrony-breaking* (*steady-state or Hopf*) *bifurcation occurring in a coupled cell system associated with a valency*-*v* *regular uniform network, generically, there are lifts that may exhibit bifurcating branches not predicted by the quotient* (*associated with the same degeneracy condition determining the bifurcation for the quotient*) *if and only if* *v*>1 *or the bifurcation eigenvalues have zero real part*.

### Example 4.10

We present examples of networks and some of their lifts that are not λ-preserving, for some eigenvalues λ of the corresponding quotient adjacency matrix.

(1) In figure 13, the 4-cell network has three distinct eigenvalues, namely ±1 and 0. Because the valency of this network is 1, it is just possible to construct lifts that are not λ-preserving if and only if λ=0. The 5-cell network in the same figure is an example of a lift that is not 0-preserving.

(2) In figure 14, the 7-cell network on the left is a quotient of the 10-cell network on the right. It is clear that the lift is not λ-preserving, for every eigenvalue λ of the strongly connected component distinct from the source of the condensation of the quotient. Note that −1 is also an eigenvalue of the source of the condenstion.

(3) In figure 15, the 7-cell network on the left is a quotient of the 11-cell network on the right. It is clear that the lift is not λ-preserving, for every eigenvalue λ of the source of the condensation of the quotient. Note, as before, that −1 is an eigenvalue of both strongly connected components of the quotient. In figure 16, the 9-cell network is another example of a lift of the 7-cell network that is not −1-preserving.

(4) In figure 17, the 4-cell network is a quotient of the other two networks. The 9-cell lift on the right is not λ-preserving, for every eigenvalue λ of the 4-cell network.

## 5. Conclusion

This work is motivated by a paper of Aguiar *et al.* [6] about the lifting of codimension-1 synchrony-breaking (steady-state or Hopf) bifurcations occurring in coupled cell systems. To be precise, in [6], a one-parameter steady-state or Hopf bifurcation problem occurring in a coupled cell system consistent with the structure of a given regular network is considered; also, considering a lift of this network, it asks how does the bifurcation lift to the overall space? The study suggests some other important questions, such as: is it always possible to find, for a given regular network, lifts that just exhibit the bifurcating branches determined by the quotient? Is it possible to identify network features associated with the existence of lifts that may exhibit additional bifurcating branches? What is the role of the network architecture in these constructions?

Dias & Moreira [5] studied this problem for non-uniform quotient networks, motivated by a problem raised by Stewart & Golubitsky [9]. In this paper, we study this problem when the quotient networks are uniform.

We have shown that rings and their lifts play a special role in the study of this problem. We have also proved that, given a valency-*v* regular uniform network and an eigenvalue λ of the corresponding adjacency matrix,

(1) there are always non-trivial λ-preserving lifts and

(2) there are lifts that are not λ-preserving if and only if λ=0 or

*v*>1.

As a consequence, assuming that a codimension-1 synchrony-breaking bifurcation occurs for a coupled cell system associated with a given regular uniform network, we showed that generically there are non-trivial lifts that just exhibit the bifurcating branches determined by the quotient network. We also identified all generic situations where lifts exist that may exhibit bifurcating branches not predicted by the quotient.

## Funding statement

Research supported by the FCT (Fundação para a Ciência e a Tecnologia) grant no. SFRH/BPD/64844/2009. Research partially funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese government through the FCT under projects PEst-C/MAT/UI0144/2013 and PTDC/MAT/100055/2008.

## Acknowledgements

The author thanks Ana Paula Dias for all helpful comments and valuable suggestions.

- Received March 24, 2014.
- Accepted June 3, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.