To model dynamical systems on networks with higher order (non-pairwise) interactions, we recently introduced a new class of ODEs on hypernetworks. Here we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking  steady state branch “reluctant” if it is tangent to a synchrony space, but does not lie inside it.
We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady state branch.

§ INTRODUCTION

Recent advances in a large variety of research fields have highlighted the importance of non-pairwise interactions for the collective dynamical behavior of complex network systems. These so-called higher order interactions turn out to be crucial in problems from, e.g.,  neuroscience (see <cit.>), social science (see <cit.>), and ecology (see <cit.>). Higher order interaction networks have found their way into various recent mathematical studies as well. We mention in particular the theoretical papers  <cit.>,
which investigate synchronisation in classes of networks with non-pairwise, nonlinear interaction in their equation of motion. We also refer to the excellent surveys <cit.> and references therein, for an in-depth discussion of higher order networks, and numerous examples of higher order network systems  arising in applications.

This paper builds on previous work of the authors   <cit.>, in which we generalised the notion of a coupled cell network,   introduced by Golubitsky, Stewart, Field et al. <cit.>, to the context of higher order networks.
We did this by introducing a class of “hypernetworks” and defining their “admissible” maps and ODEs, thus
formalising the notion of a dynamical system on a higher order interaction network.
We also introduced balanced colorings <cit.> of hypernetworks, and hypergraph fibrations <cit.>, and  used these concepts to classify the robust synchrony patterns, that is, the synchrony spaces that are invariant under every admissible map, to  hypernetwork dynamical systems.

Perhaps the most surprising result in <cit.> is the observation that the robust synchrony spaces of a hypernetwork system are not determined by linear terms in its equations of motion. This distinguishes hypernetworks from classical (dyadic) coupled cell networks, for which it was proved in <cit.> that a synchrony space is  invariant under every admissible map, if and only if it is invariant under every linear admissible map, see also <cit.>.
On the contrary, we prove in <cit.> that a synchrony space of a hypernetwork system is robustly invariant, whenever it is invariant under all  polynomial admissible maps of a specific degree, which depends on the order of the hyperedges in the hypernetwork.
Examples moreover show that our estimate for this polynomial degree is sharp.

As a consequence, a hypernetwork-admissible map of sufficiently low polynomial degree may admit “ghost” synchrony spaces that are not supported by general, e.g., higher degree polynomial admissible maps. These ghost synchrony spaces may have a profound effect on the dynamics of the hypernetwork system. In particular, the final section of <cit.> presents numerical evidence that they can give rise to a remarkable new type of synchrony breaking bifurcation.

We in fact observed this type of bifurcation in a one-parameter family of admissible ODEs for the hypernetwork depicted in <Ref>, meaning that these ODEs are of the form given in <Ref> below.
<Ref> displays two numerically obtained branches of steady states that emerge in  a bifurcation in a particular system of this form. The steady state branches were found by forward integrating the equations of motion – so they are asymptotically stable. We see that $y_0=y_1$ for negative values of the bifurcation parameter $\lambda$, so on the negative branch $y_0$ and $y_1$ are synchronous. On the positive branch, $y_0$ and $y_1$ are nonsynchronous, i.e., for positive values of $\lambda$ it holds that $y_0\neq y_1$. However, the difference $y_0-y_1$ only increases very slowly as a function of $\lambda$. In other words: the branch is tangent to the  synchrony space $\{y_0=y_1\}$.
In <cit.>, we called this phenomenon “reluctant synchrony breaking”. A more detailed numerical analysis, see <Ref>, suggests that $y_1-y_0 \sim \lambda^3$, i.e., that the branch has a third order tangency to the synchrony space. In <Ref>, we will prove that this is indeed the case.

Numerically obtained steady state branches that emerge in a synchrony breaking bifurcation in an admissible ODE for the hypernetwork depicted in <Ref>. For $\lambda<0$ the branch satisfies $y_0=y_1$. We will prove that $y_1-y_0 \sim \lambda^3$ for $\lambda>0$.
Figure taken from <cit.>.

The goal of  this paper is to show that reluctant synchrony breaking is ubiquitous in hypernetworks. The main result that we present is <Ref>, which states that reluctant synchrony breaking occurs generically in one-parameter bifurcations in a large class of hypernetworks. These so-called augmented hypernetworks are constructed by coupling new nodes to an existing network or hypernetwork by means of specific higher order interactions. The hypernetwork depicted in <Ref> is just one example of such an augmented hypernetwork. This means that the anomalous bifurcation that was discovered in <cit.> and described above is not a numerical artefact. Instead, reluctant synchrony breaking is a generic phenomenon in ODEs of the form (<ref>). To illustrate our main result, we present several more examples in this paper. We also argue (see <Ref>) that one may design augmented hypernetworks which admit reluctant synchrony breaking bifurcation branches with an arbitrarily high order of reluctancy, i.e., an arbitrarily high order of tangency to a synchrony space.

Structure of the article. In  <Ref>, we illustrate reluctant synchrony breaking by studying the example presented in <cit.> in more detail. In <Ref>, we recall our general mathematical framework for dynamical systems defined on hypernetworks, and we give the definition of an augmented hypernetwork.
In <Ref>, we prove our main result, <Ref>, which states  that reluctant synchrony breaking occurs generically in augmented hypernetworks, and  which provides a formula for the order of reluctancy of the synchrony breaking steady state branch. We also apply the theorem to the example discussed in this introduction. In <Ref>, the main theorem is illustrated by three more examples.

The hypernetwork that supports the unusual “reluctant” synchrony breaking steady-state branch shown in <Ref>.

§ A FIRST EXAMPLE

We now provide more details on the example that was briefly discussed in the introduction, and that was  introduced and studied numerically in <cit.>.
As mentioned above,  this example concerns  the hypernetwork shown in <Ref>.
In <cit.>, we  introduced the class of admissible ODEs of a hypernetwork (see also <Ref>). The admissible ODEs associated to the hypernetwork in <Ref> are all the ODEs of the form

\begin{equation}\label{vectorfieldex1}
\begin{array}{ll}
\dot x_0 & = G(x_0, \textcolor{uhhdarkgrey}{x_0}, \textcolor{grey}{x_0})\, , \\
\dot x_1 & = G(x_1, \textcolor{uhhdarkgrey}{x_1}, \textcolor{grey}{x_0}) \, , \\
\dot x_2 & = G(x_2, \textcolor{uhhdarkgrey}{x_1}, \textcolor{grey}{x_2})\, , \\
\dot y_0 & = F(\textcolor{uhhsteingrau}{y_0}, \textcolor{mix}{(} \textcolor{red}{x_0}, \textcolor{blue}{x_1} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_1}, \textcolor{blue}{x_2} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_2}, \textcolor{blue}{x_0} \textcolor{mix}{)} )\, ,\\
\dot y_1 & = F(\textcolor{uhhsteingrau}{y_1}, \textcolor{mix}{(} \textcolor{red}{x_0}, \textcolor{blue}{x_2} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_1}, \textcolor{blue}{x_0} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_2}, \textcolor{blue}{x_1} \textcolor{mix}{)} )\, ,\\
\end{array}
\end{equation}

for certain smooth functions $F$ and $G$. We assume for now that the variables $x_i, y_j$ take values in $\R$, so that $F: \R \times \R^2\times \R^2\times \R^2 \to \R$ and $G:\R\times \R\times \R \to \R$.
The brackets in $F$ serve to distinguish the two-dimensional inputs from hyperedges of order two (the purple arrows in the figure.)
The assumption that these hyperedges are   identical, translates into the requirement that $F$ is invariant under all permutations of these  pairs of variables.
That is, we require that for all $Y, X_1, \dots, X_6 \in \R$,

\begin{equation}
\label{exampleFhype}
\begin{split}
& F(Y, (X_0, X_1), (X_2, X_3), (X_4, X_5)) =  \\
&  F(Y, (X_2, X_3), (X_0, X_1), (X_4, X_5)  ) = \\
&  F(Y,(X_0, X_1), (X_4, X_5), (X_2, X_3) )\, .
\end{split}
\end{equation}

To study bifurcations within this class of admissible ODEs, we parameterize the response functions $F$ and $G$ in  (<ref>) by a scalar variable $\lambda$ taking values in some open neighborhood $\Omega \subseteq \R$ of the origin.
This gives the one-parameter family of hypernetwork admissible ODEs

\begin{equation}\label{systembifwithpara}
\begin{array}{ll}
\dot x_0 & = G(x_0, \textcolor{uhhdarkgrey}{x_0}, \textcolor{grey}{x_0}; \lambda)\, , \\
\dot x_1 & = G(x_1, \textcolor{uhhdarkgrey}{x_1}, \textcolor{grey}{x_0}; \lambda) \, , \\
\dot x_2 & = G(x_2, \textcolor{uhhdarkgrey}{x_1}, \textcolor{grey}{x_2}; \lambda)\, , \\
\dot y_0 & = F(y_0, \textcolor{mix}{(} \textcolor{red}{x_0}, \textcolor{blue}{x_1} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_1}, \textcolor{blue}{x_2} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_2}, \textcolor{blue}{x_0} \textcolor{mix}{)}; \lambda)\, ,\\
\dot y_1 & = F(y_1, \textcolor{mix}{(} \textcolor{red}{x_0}, \textcolor{blue}{x_2} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_1}, \textcolor{blue}{x_0} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_2}, \textcolor{blue}{x_1} \textcolor{mix}{)}; \lambda)\, .\\
\end{array}
\end{equation}

To guarantee that the system (<ref>) is admissible for every fixed value of $\lambda \in \Omega$, we  assume $F$ satisfies Equation (<ref>) for any fixed value of $\lambda$.
Let us in addition assume

\begin{equation*}
F(0,(0,0), (0,0), (0,0);0) =  G(0,0,0;0) = 0\, ,
\end{equation*}

so that the system (<ref>) has a steady-state point at the origin for $\lambda = 0$. We may study the persistence of this steady-state by investigating the Jacobian at the origin of the system for $\lambda = 0$.
To this end, we write

\begin{equation}\label{expressionsforFnnfG}
\begin{split}
& F(Y, (X_0, X_1), (X_2, X_3), (X_4, X_5); \lambda )  \\
& = aY + bX_0 + cX_1 + bX_2 + cX_3 + bX_4 + cX_5 + d\lambda \\
& \phantom{=} + \mathcal{O}(\|(Y,X_0, \dots, X_5;\lambda)\|^2) \, ,
\end{split}
\end{equation}

\begin{equation}\label{expressionsforFnnfG2}
G(X_0,X_1,X_2;\lambda) = AX_0+BX_1+CX_2 + \mathcal{O}(|\lambda| + \|(X_0,X_1,X_2)\|^2)\, ,
\end{equation}

with $a, \dots, d, A, B, C \in \R$, to specify the linear terms. The multiple occurrence of the terms $b$ and $c$ in (<ref>) is due to the invariance properties of $F$ in (<ref>). In terms of these coefficients, the Jacobian matrix of the right hand side of <Ref> at $(x, \lambda)=(0,0)$ with respect to the spatial variables $x$ is

$$\left( \begin{array}{ccccc}
A+\textcolor{uhhdarkgrey}{B}  + \textcolor{grey}{C}& 0 & 0 & 0 & 0 \\
\textcolor{grey}{C} & A+\textcolor{uhhdarkgrey}{B} & 0 & 0 & 0 \\
0 & \textcolor{uhhdarkgrey}{B} & A+\textcolor{grey}{C} & 0 & 0 \\
\textcolor{red}{b}+\textcolor{blue}{c} & \textcolor{red}{b}+\textcolor{blue}{c}  & \textcolor{red}{b}+\textcolor{blue}{c} & a & 0 \\
\textcolor{red}{b}+\textcolor{blue}{c} & \textcolor{red}{b}+\textcolor{blue}{c} & \textcolor{red}{b}+\textcolor{blue}{c} & 0  & a
\end{array}
\right)\, .

The eigenvalues of this Jacobian are $A+B+C$, $A+B$ and $A+C$ (all with multiplicity $1$), and $a$ (with geometric multiplicity $2$).
To allow for a steady state bifurcation to occur at $\lambda=0$, we consider the case $A+B = 0$. We moreover assume the generic conditions $a, A+B+C, A+C \not= 0$ to hold.

We claim that as $\lambda$ is varied near $0$, two branches of steady states will generically emerge from the origin. These can be found by first focusing on the subnetwork given by the three nodes of the same type. That is, we first solve

\begin{equation*}%\label{systembifwithparacore}
\begin{split}
&G(x_0, {x_0}, {x_0}; \lambda) = 0 \, ,\\ \nonumber
&G(x_1, {x_1}, {x_0}; \lambda)= 0\, ,\\ \nonumber
&G(x_2, {x_1}, {x_2}; \lambda) = 0\, .
\end{split}
\end{equation*}

A direct calculation shows that, for generic values of the first and second degree Taylor coefficients of $G$, one of the steady state branches is locally given by

\begin{equation}\label{firstbranchs}
x_0(\lambda) = x_1(\lambda) = x_2(\lambda) = x(\lambda) = D_0\lambda + \mathcal{O}(|\lambda|^2) \, ,
\end{equation}

while another branch is given by

\begin{equation}\label{secondbranchs}
\begin{split}
&x_0(\lambda) = D_0\lambda + \mathcal{O}(|\lambda|^2) \, , \ x_1(\lambda) = D_1\lambda + \mathcal{O}(|\lambda|^2) \, , \\
&x_2(\lambda) = D_2\lambda + \mathcal{O}(|\lambda|^2)\, ,
\end{split}
\end{equation}

for certain non-zero and mutually  distinct $D_0, D_1, D_2 \in \R$.
We omit the computation of these branches.
For a  detailed exposition on how to compute steady state bifurcation branches in  so-called feedforward networks (i.e. networks with no loops other than self-loops), we refer to <cit.>.

We now turn to computing the values of $y_0$ and $y_1$ along the bifurcation branches.
We start by looking at the first branch, given by <Ref>. Restricted to this branch, the steady state equation $\dot{y}_0 = 0$ becomes

\begin{equation}\label{y0is0}
F(y_0, (x(\lambda), x(\lambda)), (x(\lambda), x(\lambda)), (x(\lambda), x(\lambda)); \lambda ) = 0\, .
\end{equation}

Combining   (<ref>) and (<ref>) this can be expanded as

\begin{equation*}
ay_0 + (3D_0(b+c) + d)\lambda
+ \mathcal{O}(\|(y_0;\lambda)\|^2) = 0\, ,
\end{equation*}

which by the implicit function theorem has a unique solution given by

\begin{equation*}
y_0(\lambda) = \frac{-3D_0(b+c) - d}{a}\lambda  + \mathcal{O}(|\lambda|^2)\, .
\end{equation*}

Setting $\dot{y}_1 = 0$ gives precisely the same  equation to solve as (<ref>), but with $y_0$ replaced by $y_1$.
Hence, we find $y_0(\lambda) = y_1(\lambda)$ along this first branch, which we will therefore refer to as the synchronous branch of system (<ref>).

We now turn to the second branch of steady states, of which the asymptotics of the $x$-variables is given by <Ref>.
Combining (<ref>) with (<ref>), we find that $\dot{y}_0 = 0$ is equivalent to

\begin{equation*}
ay_0 + ((b+c)(D_0 + D_1 + D_2) + d)\lambda
+ \mathcal{O}(\|(y_0;\lambda)\|^2) = 0\, .
\end{equation*}

It follows again from the implicit function theorem that locally precisely one solution exists, given by

\begin{equation}\label{y0non-synns}
y_0(\lambda) = \frac{-(b+c)(D_0 + D_1 + D_2) - d}{a}\lambda  + \mathcal{O}(|\lambda|^2)\, .
\end{equation}

In exactly the same way, we find that $\dot{y}_1 = 0$ is solved for by

\begin{equation}\label{y1non-synns}
y_1(\lambda) = \frac{-(b+c)(D_0 + D_1 + D_2) - d}{a}\lambda  + \mathcal{O}(|\lambda|^2)\, .
\end{equation}

Note that these expressions for $y_0(\lambda)$ and $y_1(\lambda)$ agree up to first order in $\lambda$.
However, unlike for the synchronous branch, there is no reason to conclude that $y_0(\lambda) = y_1(\lambda)$ along this branch, as the equations for $\dot{y}_0$ and $\dot{y}_1$ in (<ref>) are  different for distinct $x_0, x_1$ and $x_2$:

\begin{align*}
F(y_0, (x_0(\lambda), x_1(\lambda)), (x_1(\lambda), x_2(\lambda)), (x_2(\lambda), x_0(\lambda)); \lambda ) &= 0\, ,\\
F(y_1, (x_0(\lambda), x_2(\lambda)), (x_1(\lambda), x_0(\lambda)), (x_2(\lambda), x_1(\lambda)); \lambda ) &= 0\, .
\end{align*}

We will refer to the branch of steady states given by <Ref> as the reluctant  branch of (<ref>).

<Ref> demonstrates numerically that the reluctant branch is truly non-synchronous. The figure was taken from <cit.>, and it shows a numerically obtained plot of the asymptotically stable bifurcation branches that emerge in a bifurcation in  a particular realisation of system (<ref>), namely for the choices

\begin{align}
& \nonumber
G(X_0,X_1,X_2, \lambda) =  -X_0+X_1-X_2+8\lambda X_0+4X_0^2 \ \text{ and } \\
& \nonumber
F(Y, (X_0, X_1), (X_2, X_3), (X_4, X_5), \lambda ) =  -5Y + 14 \lambda
\\ \nonumber
&-h(10X_0-12X_1)-h(10X_2-12X_3)-h(10X_4-12X_5)   \,
\end{align}

in which

\begin{equation}%\label{num:res-h}
h(x) = \sin(x)+\cos(x)-1  \, .
\end{equation}

<Ref> shows the synchronous branch for $\lambda<0$ and the reluctant branch for $\lambda>0$ –
indeed, $y_0$ and $y_1$ agree for $\lambda<0$,
and quite clearly do not for $\lambda>0$. This becomes even more  visible in <Ref>, which shows $y_0-y_1$ as a function of $\lambda$. The logarithmic plot in <Ref> suggests that $y_0(\lambda) - y_1(\lambda) \sim \lambda^3$.

In <Ref>, we prove that this is truly the case.

For the numerics of <Ref>, we have set

\begin{align}
\label{num:res-g} &G(X_0,X_1,X_2;\lambda) =  -X_0+X_1-X_2+8\lambda X_0+4X_0^2 \, \text{ and }\\
\label{num:res-f} &F(Y, (X_0, X_1), (X_2, X_3), (X_4, X_5); \lambda ) =  -5Y + 14 \lambda\\ \nonumber
\end{align}

\begin{equation}\label{num:res-h}
h(x) = \sin(x)+\cos(x)-1 = \sqrt{2}\sin(x+\frac{\pi}{4}) - 1\, .
\end{equation}

These specifics are chosen to guarantee stable synchronous and reluctant branches for the correct values of $\lambda$, to produce clear pictures and to illustrate that the reluctant behavior $y_0(\lambda) - y_1(\lambda) \sim \lambda^3$ is not merely a consequence of (low-order) polynomial response functions.
The figures are obtained by forward integrating the system for each of $600$ fixed values of $\lambda \in [-0.03, 0.03]$ (from $[0.0005, 0.03]$ for <Ref>), using Euler's method with time steps of $0.1$.
For each value of $\lambda$ integration was performed up to $t=2000$ and starting from the point $(x_0, x_1, x_2, y_0, y_1) = (0.1, -0.2, 0.3, 0.4, 0.5)$ in phase space. Note that only stable branches can be visualised in this way.

The difference between the $y$-nodes along the steady state branches.

A log–log plot of the difference between the $y$-nodes, for $\lambda > 0$.

More details for the steady state branches depicted in <Ref>. The black line segment in the log-log plot has slope $3$ and was added to show that $y_0(\lambda) - y_1(\lambda) \sim \lambda^3$.
Figures taken from <cit.>.

In this article, we rigorously prove the existence of reluctant steady state branches in bifurcations in a large class of hypernetwork systems. Specifically, we will show that  reluctant bifurcation branches appear in generic one-parameter synchrony breaking steady state bifurcations in the admissible ODEs for such hypernetworks.
We also provide a formula for the order in $\lambda$ with which the “reluctant nodes” separate.

Let $\NN$ be a given hypernetwork with $k+1 \geq 3$ nodes $\{v_0 \dots, v_k\}$, all of identical type.
The augmented hypernetwork  $\NN^\diamondsuit$ is obtained from $\NN$ by adding two additional nodes $w_0$ and $w_1$, their self-loops and $(k+1)!$ hyperedges of degree $k$.
The two additional nodes are of the same type, which is different from that of the $v_i$, and the additional hyperedges are likewise of a same, new type.
As in the examples considered in Section <ref>, the new hyperedges are labelled by the elements of the symmetric group on  $k+1$ elements, $S_{k+1}$.
Given $\sigma \in S_{k+1}$, the hyperedge $h_{\sigma}$ satisfies

\begin{equation}
s(h_{\sigma}) = (v_{\sigma(1)}, \dots, v_{\sigma(k)})\, , \qquad
t(h_{\sigma}) = w_{\sgn(\sigma)}\, ,
\end{equation}

where as before, $S_{k+1}$ acts on the ordered set $\{0, \dots, k\}$ and $v_{\sigma(0)}$ is therefore the only node in $\NN$ that is not a source of $h_{\sigma}$.
We will refer to $\NN$ as the core of the augmented hypernetwork $\NN^\diamondsuit$.

Note that $\NNd$ contains $\NN$ as a subnetwork. An example of an augmented network is given by Example <ref>, where $k=2$ and the core is given by the round cells in Figure <ref>.
We also point out that $\NN$ need not be connected, so that the examples considered in Section <ref> are likewise all instances of augmented hypernetworks. In this case the core consists of all disconnected nodes. It follows that much of the notation from Section <ref> is applicable in this section. In particular, given $x = (x_0, \dots, x_k) \in \R^{k+1}$ and $\sigma \in S_{k+1}$, we write  ${\bf x}_{\sigma} = (x_{\sigma(1)}, \dots x_{\sigma(k)}) \in \mathbb{R}^k$.

§ PRELIMINARIES

We now briefly introduce our formalism for hypernetwork dynamical systems.
See <cit.> for a more detailed exposition.
In fact, in the latter paper, we define a  hypernetwork to be a collection ${\rm \bf N} = (V, H, s, t)$ consisting of a finite set of vertices or nodes $V$, a finite set of hyperedges $H$, and source and target maps $s$ and $t$ defined on $H$.
Given an edge $h \in H$, its target $t(h) \in V$ is a single vertex, whereas the source $s(h) = (s_1(h), \dots, s_{k_h}(h)) \in V^{k_h}$ is an ordered ${k_h}$-tuple of vertices.
The number ${k_h}>0$ depends on the hyperedge $h$, and is called its order.
To avoid cluttered notation though, we often suppress the dependence of ${k_h}$ on $h$ when it is clear from context, and simply write $k$.
Note that the $k_h$ vertices in $s(h)$ are not required to be distinct.
The order of a hypernetwork ${\rm \bf N}$ is then defined as the maximum of the orders of its hyperedges, so that the hypernetworks of order $1$ are precisely the classical (dyadic) networks.

In addition to the data that is explicitly given in ${\rm \bf N} = (V, H, s, t)$, we also specify equivalence relations on both the nodes $V$ and the hyperedges $H$. We typically refer to both as the color or type relation.
The reason that these relations are not specified in ${\rm \bf N}$ is because they will apply to all hypernetworks at once.
That is, it will make sense for two nodes in different hypernetworks to have the same color, and likewise for multiple hyperedges across different hypernetworks.
This allows us to define node- and hyperedge-type preserving maps between different hypernetworks, which in turn give rise to semi-conjugacies between the dynamics, see <cit.> for more details.
Intuitively, this color-relation conveys whether two nodes correspond to comparable or incomparable agents in a real-world system modelled by the
hypernetwork, and similarly whether or not two hyperedges specify the same influence.
As is suggested by this interpretation, the vertex- and hyperedge-types have to satisfy certain consistency conditions. These are:

* if two nodes $v_0$ and $v_1$ are of the same type, then there exists a hyperedge-type preserving bijection between the set of hyperedges targeting $v_0$ and those targeting $v_1$;

* two hyperedges $h_0, h_1$ of the same type have the same order $k$, and for each $i \in \{1, \dots, k\}$ the nodes $s_i(h_0)$ and $s_i(h_1)$ are of the same vertex-type.

As mentioned, the conditions above can be seen as  natural for real-world coupled systems.
Their motivation also comes from the fact that they allow us to define dynamical systems that reflect the given hypernetwork structure.

These systems are specified by so-called admissible vector fields. To introduce these, we fix an internal phase space $\R^{n_v}$ for each node $v \in V$, which will be identical for nodes of the same type.
Each node $v$ is given a state variable $x_v \in \R^{n_v}$. The  total phase space of the hypernetwork dynamical system describes the states of all these variables, and  is thus the direct sum $ \bigoplus_{v\in V} \mathbb{R}^{n_v}$.
We also specify, for each hyperedge $h \in H$, the vector of its source variables
\[ {\bf x}_{s(h)} = \left(x_{s_1(h)}, \ldots, x_{s_{k_h}(h)}\right) \, \in \, \bigoplus_{i = 1} ^ {k_h} \mathbb{R}^{n_{s_i(h)}}. \]

Next, we choose for each node $v \in V$ its response function

\begin{align}
F_v \colon   \bigoplus_{h\, :\, t(h)=v} \bigoplus_{i = 1} ^ {k_h} \, \, \mathbb{R}^{n_{s_i(h)}} \to \R^{n_v}\, .
\end{align}

These functions must satisfy certain conditions that reflect our intuitive idea that hyperedges of the same type encode identical influence, as well as the notion that nodes of the same type respond to their input in the same way.
In words, we require that the variables of identical-type hyperedges may be freely interchanged in $F_v$, as well as that $F_v$ and $F_w$ are the same when the nodes $v$ and $w$ are of the same type, after an appropriate identification of their domains.
We may capture both requirements in one succinct condition as follows: given nodes $v$ and $w$ of the same type, for any hyperedge-type preserving bijection $\alpha: t^{-1}(v) \to t^{-1}(w)$ we have

\begin{equation}
\label{eq:bijection_symmetry}
F_{w}\left( \bigoplus_{t(h_2)=w} {\bf x}_{s(h_2)} \right) = F_{v}\left(\bigoplus_{t(h_1)=v} {\bf x}_{s(\alpha(h_1))} \right) \, ,
\end{equation}

for all $x = \bigoplus_{v \in V} x_v \in \bigoplus_{v\in V} \mathbb{R}^{n_v}$.
Recall that at least one such $\alpha$ exists when $v$ and $w$ are of the same type.
Finally, we define the hypernetwork admissible vector field
$$f^{\bf N}: \bigoplus_{v\in V} \mathbb{R}^{n_v} \to  \bigoplus_{v\in V} \mathbb{R}^{n_v}\, ,$$

on the total phase space, to be given component-wise by

$$f^{\bf N}_v(x) = F_v\left(  \bigoplus_{h\, :\, t(h)=v} {\bf x}_{s(h)}  \right)\,  $$

for all $v \in V$ and $x \in \bigoplus_{v\in V} \mathbb{R}^{n_v}$.

Of particular interest in this paper are so-called augmented hypernetworks, also introduced in <cit.>.
Their definition involves the symmetric group on $k+1$ elements, denoted by $S_{k+1}$, which acts on the ordered set $\{0, \dots, k\}$ by permutations.
We  denote by $S_{k+1}^0$ and $S_{k+1}^1$ the subsets of even and odd permutations, respectively, and denote by $\sgn(\sigma) \in \{0,1\}$ the sign of a permutation $\sigma \in S_{k+1}^{\sgn(\sigma)}$.

[Definition 5.1 in <cit.>]
Let ${\rm \bf N}$ be a hypernetwork with $k+1 \geq 3$ nodes $v_0, \dots, v_k$ of the same type. We define the augmented hypernetwork with core ${\rm \bf N}$, denoted by $\NN^\diamondsuit$, as the hypernetwork obtained by adding two additional nodes $w_0, w_1$, one self-loop for each $w_i$ and $(k+1)!$ new hyperedges to  ${\rm \bf N}$.
The new nodes are of the same type, which differs from that of the $v_i$.
Likewise, we construct a new hyperedge-type which we assign to the $(k+1)!$ additional hyperedges.
Necessarily the self-loops on the new nodes are of a same, new type too.
The $(k+1)!$ new hyperedges are indexed by the symmetric group $S_{k+1}$, so that we may denote them by $h_{\sigma}$ for $\sigma \in S_{k+1}$.
We define their sources and targets by

\begin{align}
t(h_{\sigma}) &= w_{\sgn(\sigma)} \text{ and } \\ \nonumber
s(h_{\sigma}) &= (v_{\sigma(1)}, \dots, v_{\sigma(k)}) \, ,
\end{align}

where $S_{k+1}$ acts on the ordered set $\{0, \dots, k\}$.
Note that $v_{\sigma(0)} \in\{v_0, \dots, v_k\}$ is therefore the only $v$-node not in the source of $h_{\sigma}$, and that these hyperedges all have order $k$.

The hypernetwork discussed in <Ref> and depicted in <Ref>, is an example of an augmented hypernetwork. Here the core consists of the $k+1 = 3$ circular nodes in the center and the arrows between them.  This core in fact forms a classic (dyadic) network.
The two “added” nodes are depicted as the square ones.
<Ref> gives the form of a general admissible vector field for this augmented hypernetwork.
Recall that the response function $F$ in <Ref> is invariant under permutations of the three pairs of inputs, which reflects that the six   “added” hyperedges are all of the same type.

By assumption, all nodes in the  core $\NN$ of an augmented hypernetwork are of the same type, so that $\NNd$ has precisely two node-types.
This means that two response functions are required to describe an admissible vector field $f^{\NNd}$  for $\NN^\diamondsuit$.
We will usually denote these by $F$ and $G$, where $G$ is used for the nodes in $\NN$ and $F$ for the two additional nodes.
Likewise, we see that the total phase space is determined by two vector spaces: one for the internal dynamics of the $v$-nodes, $\R^{n_v}$, and one for that of the $w$-nodes, $\R^{n_w}$.
We will later set both equal to $\R$.
Note that $F$ takes one argument from $\R^{n_w}$, corresponding to the self-loop, and $(k+1)!/2$ entries from $\bigoplus^k \R^{n_v}$ for the remaining hyperedges.
As these latter hyperedges are indexed by (half of) the symmetric group, we may see the response function as

$$  F \colon\R^{n_w} \oplus \hspace{-3pt}\bigoplus_{\sigma \in S^0_{k+1}}  \hspace{-5pt} \bigg(\bigoplus^k \R^{n_v}\bigg) \, \to\, \R^{n_w}\, ,$$

with the property that the $(k+1)!/2$ entries with values in $\bigoplus^k \R^{n_v}$ may be freely interchanged.
Note that we simply index these entries by $S^0_{k+1}$ to emphasize that they correspond to the hyperedges that are indexed by (part of) the symmetric group.
We could have also used $S^1_{k+1}$ and, because we may freely interchange these entries, we do not have to give an explicit identification between $S^0_{k+1}$ and $S^1_{k+1}$.
In particular, in any augmented hypernetwork the dynamics of the $w$-nodes may simply be written as

\begin{equation}%\label{eq:factorisation}
\dot{y}_{0} = F\left({y}_{0}, \bigoplus_{\sigma \in S^0_{k+1}} {\bf x}_{\sigma} \right) \text{ and }\, \dot{y}_{1} = F\left({y}_{1}, \bigoplus_{\sigma \in S^1_{k+1}} {\bf x}_{\sigma} \right)\, ,
\end{equation}

where ${\bf x}_{\sigma} = (x_{\sigma(1)}, \dots, x_{\sigma(k)})$, and where we write $y_i$ for the state of node $w_i$ and $x_j$ for the state of node $v_j$.

We may generalise the definition of an augmented hypernetwork by connecting the auxiliary nodes $w_0$ and $w_1$ only to a subset $\mathcal{S}$ of nodes of the same type of the core, consisting of $k+1$ nodes of the same type. In this case, the nodes in the core that are not elements of $\mathcal{S}$ may in fact be of different type.
We can then add $(k+1)!$ hyperedges of order $k$, precisely as in <Ref>, but now with all source nodes in $\mathcal{S}$.
It will be clear that our results also hold for such hypernetworks, but to avoid a cluttered exposition we will mostly work with <Ref>.
See also <Ref>.

§ RELUCTANT SYNCHRONY-BREAKING

We now show that the reluctant synchrony breaking observed in <Ref> is not a peculiarity of systems of the form (<ref>), but can occur in any augmented hypernetwork.
In fact, we shall give natural conditions on the core $\NN$ that guarantee that reluctant synchrony  breaking occurs generically in the augmented hypernetwork $\NN^{\diamondsuit}$. Moreover, in <Ref> below we  give a precise expression for the degree (in the bifurcation parameter) at which the reluctant synchrony breaking occurs.
We start by introducing some useful notation and conventions.

As it is sometimes convenient to make explicit the dependence of an admissible vector field on its response functions, we will often write $f^{\NNd}_{(F,G)}$ and $f^{\NN}_{(G)}$ for the admissible vector fields of $\NNd$ and $\NN$, respectively.
Furthermore, because in this section we are mainly interested in bifurcations, we will often use $f^{\NN}$ (and  $f^{\NNd}$, $f^{\NNd}_{(F,G)}$ etc.) to denote parameter families of admissible vector fields.
This means $f^{\NN}$ is an admissible vector field for any fixed value of the bifurcation parameter, as in <Ref>.

Throughout this section we will investigate asymptotics and power series in $\lambda$ for bifurcation branches. Some of these might involve fractional powers of $\lambda$, meaning that such branches are only defined for positive or negative values of $\lambda$. To keep this section as readable as possible, we assume from here on out that all branches are defined for positive values of $\lambda$, so that we may always write $\lambda^p$ for any  power $p \geq 0$.
The corresponding results for negative values of $\lambda$ follow easily by redefining $\lambda$ as $-\lambda$.

Let $\NN$ be a hypernetwork with $n$ nodes and denote by $f^{\NN} \colon \R^n \times \Omega \rightarrow \R^n$ a one-parameter family of admissible vector fields for $\NN$, where each node has a one-dimensional internal phase space.
In this paper, a locally defined branch of steady states $x(\lambda) = (x_1(\lambda), \dots, x_n(\lambda))$ for $f^{\NN}$ is called fully synchrony-breaking if for all $i,j \in \{1, \dots, n\}$ with $i>j$ there exist numbers $p_{i,j}>0$ and $D_{i,j}\not=0$ such that

\begin{equation}\label{differ4nces}
x_i(\lambda) - x_j(\lambda) = D_{i,j}\lambda^{p_{i,j}} + \text{ ``higher order terms'' in } \lambda\, .
\end{equation}

Given a fully synchrony-breaking steady state branch, we define its order of asynchrony as the number

\begin{equation}
\bar{p} = \sum_{\substack{i,j = 1 \\ i>j}}^n p_{i,j}\, .
\end{equation}

In what follows, we will need to make some assumptions about the asymptotics and regularity of the branches.
To avoid a detailed exposition, we instead use a formal ansatz.
From here on out, we will always assume a fully synchrony-breaking branch $x(\lambda) = (x_1(\lambda), \dots, x_n(\lambda))$ to come with a finite set of numbers $\Upsilon \subseteq (0, \bar{p}\,]$ such that we may write

\begin{equation}\label{eq:formalpowerseries}
x_i(\lambda) = \sum_{p \in \Upsilon} A_{i,p} \lambda^{p} + \mathcal{O}(|\lambda|^{\bar{p}+\epsilon})\,
\end{equation}

for some $\epsilon > 0$, and with $A_{i,p} \in \R$.
Note that, in contrast to the $D_{i,j}$ in Equation (<ref>), these $A_{i,p}$ may very well vanish.
As we may write

\begin{align}\label{eq:formalpowerseriesdifff}
x_i(\lambda)-x_j(\lambda) &= \sum_{p \in \Upsilon} (A_{i,p}-A_{j,p}) \lambda^{p} + \mathcal{O}(|\lambda|^{\bar{p}+\epsilon}) \\ \nonumber
&= D_{i,j}\lambda^{p_{i,j}} + \text{ ``higher order terms''} \, ,
\end{align}

and because $p_{i,j} \leq \bar{p}$, we see that necessarily $p_{i,j} \in \Upsilon$ for all $i>j$.

In the theorem below, we assume all admissible vector fields correspond to one-dimensional internal dynamics for each node.

Let $\NNd$ be an augmented hypernetwork with core $\NN$, the latter consisting of $k+1 \ge 3$ nodes, and let $f^{\NN}_{(G)}: \R^{k+1} \times \Omega \rightarrow \R^{k+1}$ be a one-parameter family of admissible vector fields for $\NN$, corresponding to some response function $G$.
Assume $f^{\NN}_{(G)}$  admits a fully synchrony-breaking branch of steady states $x(\lambda) = (x_0(\lambda),   \dots, x_{k}(\lambda))$ with order of asynchrony $\bar{p}$.

Then for a generic
$\lambda$-dependent response function $F$, the system $f^{\NNd}_{(F,G)}: \R^{k+3} \times \Omega \rightarrow \R^{k+3}$ admits a steady state branch

\[ z(\lambda) = (x_0(\lambda), \dots, x_k(\lambda), y_0(\lambda), y_1(\lambda))\]

for which

\begin{equation*}
y_0(\lambda) - y_1(\lambda) = E \lambda^{\bar{p}} + \mathcal{O}({|\lambda|^{\bar{p}+\epsilon}})\,  ,
\end{equation*}

for some non-zero $E\in \R$.

Moreover, $z(\lambda)$ is a branch of steady states for $f^{\NNd}_{(-F,G)}$ as well, and if $x(\lambda)$ is stable for $f^{\NN}_{(G)}$, then $z(\lambda)$ is stable for either $f^{\NNd}_{(F,G)}$ or $f^{\NNd}_{(-F,G)}$.

Before proving the theorem, we first apply it to our running example.

In <Ref>, we investigated a bifurcation scenario with a fully synchrony-breaking branch in the core, given by
$x_i(\lambda) = D_i\lambda + \mathcal{O}(|\lambda|^2)$ for $i \in \{0,1,2\}$ and with mutually distinct $D_i$.
It follows that
$x_i(\lambda)- x_j(\lambda) = (D_i-D_j)\lambda + \mathcal{O}(|\lambda|^2)$ for all $i,j$. As $D_i-D_j\ne0$ for $i>j$, we obtain
$p_{3,1} = p_{3,2} = p_{2,1} = 1$ and hence

\begin{equation*}
\bar{p} = 1+1+1 = 3\, .
\end{equation*}

<Ref> therefore predicts a bifurcation branch

$$z(\lambda) = (x_0(\lambda), x_1(\lambda), x_2(\lambda), y_0(\lambda), y_1(\lambda) )$$

in the augmented system $f^{\NNd}_{F,G}$, for generic choice of $F$ and with any $G$ supporting the aforementioned fully synchrony-breaking branch in the core, which satisfies

\begin{equation*}
y_0(\lambda) - y_1(\lambda) \sim \lambda^3\, .
\end{equation*}

This is indeed what we found in our numerical investigation, see <Ref>.

The proof of <Ref> requires some machinery from <cit.>. There we introduced the polynomials $P_{(k)}$, given by

\begin{equation}\label{eq:powersumsym1}
\begin{split}
&P_{(k)} \colon \bigoplus_{\sigma \in S^0_{k+1}} \mathbb{R}^k \to  \mathbb{R} \\
&P_{(k)}\left(\bigoplus_{\sigma \in S^0_{k+1}}  {\bf X}_{\sigma}\right) = \sum_{\sigma \in S^0_{k+1}}X_{\sigma,1}^{1}X_{\sigma,2}^{2}\dotsm X_{\sigma,k}^{k}\, ,
\end{split}
\end{equation}

for $k \in \N$, and where ${\bf{X}}_{\sigma} = (X_{\sigma,1}, \dots, X_{\sigma,k}) \in \R^k$ for $\sigma \in S^0_{k+1}$.
We also state the following result, a proof of which can be found in <cit.>.

\begin{equation*}
Q: \bigoplus_{\sigma \in S_{k+1}^0} \R^k \rightarrow \R
\end{equation*}

be a polynomial function that is invariant under all permutations of its $\#S_{k+1}^0$ entries from $\R^k$.
Then there exists a polynomial $S \colon \R^{k+1} \rightarrow \R$ such that

\begin{equation}\label{eq:factorisation}
Q\left(\bigoplus_{\sigma \in S^0_{k+1}} {\bf x}_{\sigma} \right) - Q\left(\bigoplus_{\sigma \in S^1_{k+1}} {\bf x}_{\sigma} \right) = S(x) \prod_{\substack{i,j=0 \\ i > j }}^k (x_i - x_j)
\end{equation}

for all $x = (x_0, \dots, x_k) \in \R^{k+1}$, where ${\bf x}_{\sigma} = (x_{\sigma(1)}, \dots, x_{\sigma(k)})$ for all $\sigma \in

It can readily be seen that for any polynomial $Q$ satisfying the conditions of <Ref>, (<ref>) is actually equivalent to the fact that

\begin{equation}\label{eq:similiartwothesame}
Q\left(\bigoplus_{\sigma \in S^0_{k+1}} {\bf x}_{\sigma} \right) = Q\left(\bigoplus_{\sigma \in S^1_{k+1}} {\bf x}_{\sigma} \right)
\end{equation}

whenever $(x_0, \dots, x_k) \in \R^{k+1}$ satisfies $x_i = x_j$ for some distinct $i,j \in \{0,,\dots, k\}$.
This observation still holds when $Q$ is not polynomial, see Lemma 5.5 of <cit.>. The latter fact actually  underlies the proof of <Ref> that is given in <cit.>.

The polynomials $P_{(k)}$ defined in (<ref>) satisfy

\begin{equation}\label{eq:factorisation_spec}
P_{(k)}\left(\bigoplus_{\sigma \in S^0_{k+1}} {\bf x}_{\sigma} \right) - P_{(k)}\left(\bigoplus_{\sigma \in S^1_{k+1}} {\bf x}_{\sigma} \right) = \prod_{\substack{i,j=0 \\ i > j }}^k (x_i - x_j)
\end{equation}

for all $x = (x_0, \dots, x_k) \in \R^{k+1}$.

By <Ref> we have

\begin{equation*}%\label{eq:factorisationcopy}
P_{(k)}\left(\bigoplus_{\sigma \in S^0_{k+1}} {\bf x}_{\sigma} \right) - P_{(k)}\left(\bigoplus_{\sigma \in S^1_{k+1}} {\bf x}_{\sigma} \right) = S(x)\prod_{\substack{i,j=0 \\ i > j }}^k (x_i - x_j)
\end{equation*}

for some polynomial $S$.
It remains to show that $S = 1$.
To this end, note that both the left and right hand side of <Ref> has total degree $1+\dots+k = \frac{k(k+1)}{2}$.
This means $S$ is a constant polynomial.
As both sides of <Ref> contain a term $1\cdot x_1x_2^2\dots x_k^k$, we see that $S = 1$ and the result follows.

Before we move on to the proof of <Ref>, we first have a closer look at the set of powers $\Upsilon$. Recall that we may write

\begin{equation}\label{eq:formalpowerseries2}
x_i(\lambda) = \sum_{p \in \Upsilon} A_{i,p} \lambda^{p} + \mathcal{O}(|\lambda|^{\bar{p}+\epsilon})\,
\end{equation}

for all the components $x_i(\lambda)$ of a fully synchrony-breaking branch.
By adding zero-coefficients $A_{i,p}$ to Expression (<ref>) and by decreasing $\epsilon$ if needed, we may assume that for all $p,q \in \Upsilon$, we have

\begin{equation}\label{ex:suminvofupsilon}
\begin{array}{rclclcl}
p+q &\in& \Upsilon &\text{ if }& \, p+q &\leq& \bar{p}\,; \\
p+q &\geq& \bar{p}+\epsilon  &\text{ if }& \, p+q &>& \bar{p}\,,
\end{array}
\end{equation}

and also that

\begin{equation*}
\begin{array}{rclclcl}
1 &\in& \Upsilon &\text{ if }&\, 1 &\leq& \bar{p}\,; \\
1 &\geq& \bar{p}+\epsilon  &\text{ if }&\, 1 &>& \bar{p}\,.
\end{array}
\end{equation*}

More precisely, we can add to $\Upsilon$ all non-zero sums $s= c+\sum_{p \in \Upsilon}c_p p $ with non-negative integer coefficients $c,c_p$, such that $s \leq \bar{p}$.
Note that this adds a finite number of elements to $\Upsilon$, as necessarily $c_p \leq \ceil*{\frac{\bar{p}}{p}}$ and  $c \leq \ceil*{\bar{p}}$. It then follows from <Ref> that $\bar{p} \in \Upsilon$, as we have $p_{i,j} \in \Upsilon$ for all $i>j$.
This allows us to iteratively investigate coefficients corresponding to (possibly non-integer) powers of $\lambda$ in the branches, as well as in polynomial expressions involving the components of these branches.
For instance, if $x_1(\lambda)$ and $x_2(\lambda)$ are given by <Ref>, then we may likewise write

\begin{align*}
x_1(\lambda)x_2(\lambda) &= \sum_{p \in \Upsilon} A'_{p} \lambda^{p} + \mathcal{O}(|\lambda|^{\bar{p}+\epsilon}) \text{ and } \\
\lambda x_1(\lambda) &= \sum_{p \in \Upsilon} A''_{p} \lambda^{p} + \mathcal{O}(|\lambda|^{\bar{p}+\epsilon})\,
\end{align*}

for some $A'_{p}, A''_{p} \in \R$.
Finally, whenever

\begin{equation*}%\label{eq:formalpowerseries2w}
w(\lambda) = \sum_{p \in \Upsilon} A_{p} \lambda^{p} + \mathcal{O}(|\lambda|^{\bar{p}+\epsilon})\,
\end{equation*}

for some locally defined map $w\colon \R_{\geq 0} \mapsto \R$ and with $A_p \in \R$, then for $q \in \Upsilon$ we may write

\begin{equation*}
[w(\lambda)]_{\leq q} := \sum_{\substack{p \in \Upsilon \\ p \leq q }} A_{p} \lambda^{p} \, \text{ and } \,
[w(\lambda)]_{< q} := \sum_{\substack{p \in \Upsilon \\ p < q }} A_{p}\lambda^{p}
\end{equation*}

for the truncated power series.

By assumption, $x(\lambda) = (x_0(\lambda), \dots,  x_k(\lambda))$ locally solves
$(f^{\NNd}_{(F,G)}(x,y;\lambda))_{v} = (f^{\NN}_{(G)}(x;\lambda))_v = 0$ for all nodes $v$ in the core
$\NN$ and all $y = (y_0, y_1) \in \R^2$.
To solve for the $y$-components, let $K\in \N$ be such that

\begin{equation}
(K+1)\min(p \mid p \in \Upsilon) > \bar{p}\, .
\end{equation}

We expand a general response function $F$ as

\begin{equation}\label{eq:iterativesolve0}
F(Y,{\bf{X}};\lambda) = aY + \sum_{\ell,m=0}^K Q_{\ell,m}\left({\bf{X}}\right)Y^\ell\lambda^m + \mathcal{O}( \| \left({\bf{X}},Y;\lambda\right)\|^{K+1}) \,
\end{equation}

for $Y \in \R$, $\lambda \in \R_{\geq 0}$, and where

\begin{equation*}
{\bf{X}} := \bigoplus_{\sigma \in S^0_{k+1}} {\bf{X}}_{\sigma}
\end{equation*}

with ${\bf{X}}_{\sigma} \in \R^k$.
Here each $Q_{\ell,m}$ is a polynomial of degree at most $K$ that is invariant under all permutations of the vectors ${\bf{X}}_{\sigma}$, which follows from the fact that $F$ is invariant under permutations of these vectors.
Our  assumption (which is necessary for a bifurcation) that $F(0,0;0) = 0$ implies that $Q_{0,0}(0) = 0$.
Moreover, by setting the number $a \in \R$ equal to the derivative of $F$ at $(0,0;0)$ in the $Y$-direction, we may assume that  $Q_{1,0}(0) = 0$.

For $s \in \{0,1\}$, the equation $\dot{y}_s = 0$ gives

\begin{equation}\label{eq:iterativesolve}
ay_s + \sum_{\ell,m=0}^K Q_{\ell,m}\left(\bigoplus_{\sigma \in S^s_{k+1}} {\bf x}_{\sigma}(\lambda)\right)y_s^\ell\lambda^m + \mathcal{O}(\|(x(\lambda),y_s;\lambda)\|^{K+1}) = 0 \, ,
\end{equation}

where ${\bf x}_{\sigma}(\la) = (x_{\sigma(1)}(\la), \dots, x_{\sigma(k)}(\la))$ for all $\sigma \in
If we assume $a\not= 0$ then by the implicit function theorem, this equation locally has a unique solution $y_s(\lambda)$, which can be written as

\begin{equation*}%\label{eq:formalpowerseries2y}
y_s(\lambda) = \sum_{p \in \Upsilon} B_{s,p} \lambda^{p} + \mathcal{O}(|\lambda|^{\bar{p}+\epsilon})\, .
\end{equation*}

The coefficients $B_{s,p} \in \R$ can iteratively be solved for from the equation

\begin{equation}\label{eq:iterativesolve2}
ay_s(\lambda) + \sum_{\ell,m=0}^K Q_{\ell,m}\left(\bigoplus_{\sigma \in S^s_{k+1}} {\bf x}_{\sigma}(\lambda)\right)y_s^\ell(\lambda)\lambda^m + \mathcal{O}(|\lambda|^{\bar{p}+\epsilon}) = 0 \, ,
\end{equation}

which is (<ref>) applied to the solution branch $(x_0(\la),\dotsc,x_k(\la),y_0(\la),y_1(\la))$.
We now want to show that

\begin{equation}\label{eq:toshowdiffpowersyy}
y_0(\lambda) - y_1(\lambda) = \mathcal{O}(|\lambda|^{\bar{p}})
\end{equation}

for these unique solutions. We will do so by proving for all $q \in \Upsilon$ with $q < \bar{p}$ that

\begin{equation}\label{eq:toshowimplic}
[y_0]_{<q} = [y_1]_{<q} \implies [y_0]_{\leq q} = [y_1]_{\leq q}\, .
\end{equation}

Note that for $q = \min(p \mid p \in \Upsilon)$ we have $[y_0]_{<q} = [y_1]_{<q} = 0$.
Hence, iterated use of Implication (<ref>) indeed proves <Ref>.
To show that the statement in (<ref>) holds, we subtract <Ref> for $s=1$ from the one for $s=0$, which gives us

\begin{align}%\label{eq:iterativesolve3}
a(y_0(\lambda)-y_1(\lambda)) &+ \sum_{\ell,m=0}^K \sum_{s = 0}^1 (-1)^s Q_{\ell,m}\left(\bigoplus_{\sigma \in S^s_{k+1}} {\bf x}_{\sigma}(\lambda)\right)y_s^\ell(\lambda)\lambda^m \\ \nonumber
&+ \mathcal{O}(|\lambda|^{\bar{p}+\epsilon}) = 0 \, .
\end{align}

Given $q \in \Upsilon$ satisfying $q < \bar{p}$, let $q^{+}$ denote the smallest element in $\Upsilon$ such that $q^{+} > q$, i.e. $q^{+}$ is the “next power” to consider.
Note that $q < \bar{p}$ means $q^{+} \leq \bar{p}$ exists. It follows that

\begin{equation}\label{eq:iterativesolve4}
\begin{split}
0 &=   a([y_0(\lambda)]_{\leq q}-[y_1(\lambda)]_{\leq q}) \\  &\phantom{=}+ \sum_{\ell,m=0}^K \sum_{s = 0}^1 (-1)^s Q_{\ell,m}\left(\bigoplus_{\sigma \in S^s_{k+1}} {\bf x}_{\sigma}(\lambda)\right)[y_s(\lambda)]_{<q}^\ell\lambda^m + \mathcal{O}(|\lambda|^{q^+})  \, .
\end{split}
\end{equation}

Here we have used that

\begin{align}
&Q_{\ell,m}\left(\bigoplus_{\sigma \in S^s_{k+1}} {\bf x}_{\sigma}(\lambda)\right)[y_s(\lambda)]_{\leq q}^\ell\lambda^m \\ \nonumber
= \, &Q_{\ell,m}\left(\bigoplus_{\sigma \in S^s_{k+1}} {\bf x}_{\sigma}(\lambda)\right)[y_s(\lambda)]_{< q}^\ell\lambda^m + \mathcal{O}(|\lambda|^{q^+}) \, ,
\end{align}

which is clear whenever $\ell = 0$, $\ell > 1$ or $m >0$. For $(\ell, m) = (1,0)$ it holds because $Q_{1,0}(0) = 0$, so that $Q_{1,0}$ has no constant term and hence

$$Q_{1,0}\left(\bigoplus_{\sigma \in S^s_{k+1}}{\bf x}_{\sigma}(\lambda)\right)$$

is divisible by $\lambda^r$ for $r = \min(p \mid p \in \Upsilon)$.
We now assume $[y_0(\lambda)]_{< q} = [y_1(\lambda)]_{< q}$, so that $[y_s(\lambda)]_{< q} = [y_0(\lambda)]_{< q}$ for both choices of $s$. Using <Ref>, <Ref> becomes

\begin{equation}\label{eq:iterativesolve5}
\begin{split}
0 &=   a([y_0(\lambda)]_{\leq q}-[y_1(\lambda)]_{\leq q}) \\  &+ \sum_{\ell,m=0}^K \sum_{s = 0}^1 (-1)^s Q_{\ell,m}\left(\bigoplus_{\sigma \in S^s_{k+1}} {\bf x}_{\sigma}(\lambda)\right)[y_s(\lambda)]_{<q}^\ell\lambda^m + \mathcal{O}(|\lambda|^{q^+}) \\
&=   a([y_0(\lambda)]_{\leq q}-[y_1(\lambda)]_{\leq q}) \\  &+ \sum_{\ell,m=0}^K [y_0(\lambda)]_{<q}^\ell\lambda^m\sum_{s = 0}^1 (-1)^s Q_{\ell,m}\left(\bigoplus_{\sigma \in S^s_{k+1}} {\bf x}_{\sigma}(\lambda)\right) + \mathcal{O}(|\lambda|^{q^+}) \\
&=   a([y_0(\lambda)]_{\leq q}-[y_1(\lambda)]_{\leq q}) \\  &+ \sum_{\ell,m=0}^K [y_0(\lambda)]_{<q}^\ell\lambda^m
S_{\ell, m}(x(\lambda)) \prod_{\substack{i,j=0 \\ i > j }}^k (x_i(\lambda) - x_j(\lambda)) + \mathcal{O}(|\lambda|^{q^+})\, ,
\end{split}
\end{equation}

for some polynomials $S_{\ell, m}$. As it is clear that

\begin{equation*}
\prod_{\substack{i,j=0 \\ i > j }}^k (x_i(\lambda) - x_j(\lambda)) = \mathcal{O}(|\lambda|^{\bar{p}})\, ,
\end{equation*}

<Ref> simplifies to

\begin{equation*}%\label{eq:iterativesolve6}
a([y_0(\lambda)]_{\leq q}-[y_1(\lambda)]_{\leq q}) = \mathcal{O}(|\lambda|^{q^+})\, .
\end{equation*}

Using again the assumption that $a\not= 0$, this indeed gives $[y_0(\lambda)]_{\leq q}=[y_1(\lambda)]_{\leq q}$.

By induction, (<ref>) holds true as outlined above.
It follows that we may write

\begin{equation*}
y_0(\lambda) - y_1(\lambda) = E \lambda^{\bar{p}} + \mathcal{O}({|\lambda|^{\bar{p}+\epsilon}})\,  .
\end{equation*}

for some $E= B_{0, \bar{p}} - B_{1,\bar{p}} \in \R$.
We next want to show that $E\not= 0$ generically.
To this end, recall that $B_{s,\bar{p}}$ can be solved for from <Ref>.
In fact, for fixed values of $a\not= 0$ and the power series coefficients of each $x_i(\lambda)$, we may express  $B_{s,\bar{p}}$ as a polynomial in the coefficients of the various $Q_{\ell,m}$.
Therefore, we may likewise express $E= B_{0, \bar{p}} - B_{1,\bar{p}}$ as such a polynomial.
Now, any polynomial on a finite dimensional vector space is either identically zero, or vanishes only on the complement of an open dense set.
Therefore, the proof is complete if we can give at least one response function $F$ for which $E \not= 0$.
To this end, consider

\begin{equation}%\label{eq:iterativesolve01}
F(Y,{\bf{X}};\lambda) = aY  + P_{(k)}({\bf{X}})\, .
\end{equation}

Using <Ref> we get

\begin{equation}%\label{eq:iterativesolve01}
\begin{split}
0 &=  F\left(y_0(\la), \bigoplus_{\sigma \in S^0_{k+1}} {\bf x}_{\sigma}(\lambda), \lambda\right) - F\left(y_1(\la), \bigoplus_{\sigma \in S^1_{k+1}} {\bf x}_{\sigma}(\lambda), \lambda\right) \\
&= a(y_0(\la) - y_1(\la)) +  P_{(k)}\left(\bigoplus_{\sigma \in S^0_{k+1}} {\bf x}_{\sigma}(\lambda) \right) - P_{(k)}\left(\bigoplus_{\sigma \in S^1_{k+1}} {\bf x}_{\sigma}(\lambda) \right) \\
&= a(y_0(\la) - y_1(\la)) + \prod_{\substack{i,j=0 \\ i > j }}^k (x_i(\lambda) - x_j(\lambda)) \\
&= a(y_0(\la) - y_1(\la)) + \bigg(\prod_{\substack{i,j=0 \\ i > j }}^k D_{i,j}\bigg)\lambda^{\bar{p}} + \mathcal{O}(|\lambda|^{\bar{p} + \epsilon})\, .
\end{split}
\end{equation}

Hence, for this particular choice of response function we obtain

\begin{equation*}
E = -\frac{1}{a}\prod_{\substack{i,j=0 \\ i > j }}^k D_{i,j} \not= 0\, ,
\end{equation*}

which shows that $E$ is indeed generically non-vanishing.

Finally, as $(f^{\NNd}_{(-F,G)})_{w} = -(f^{\NNd}_{(F,G)})_{w}$ for the two nodes $w$ outside the core, we see that $z(\lambda)$ is a branch of steady-states for $f^{\NNd}_{(F,G)}$, if and only if it is for $f^{\NNd}_{(-F,G)}$.
<Ref> shows that the branch

\begin{equation}
z(\lambda) = (x_0(\lambda), \dots,  x_k(\lambda),  y_0(\lambda), y_1(\lambda))
\end{equation}

is stable if $x(\lambda)$ is for $f^{\NN}_{(G)}$, and if in addition $a<0$. This last condition holds for either $f^{\NNd}_{(F,G)}$ or $f^{\NNd}_{(-F,G)}$, which completes the proof.

The condition that $x(\lambda)$ is fully synchrony-breaking is really essential in <Ref>.
More precisely, it follows from <Ref> that if $x_i(\lambda) = x_j(\lambda)$ for some distinct $i,j$, then the equations $\dot{y}_0 = 0$ and $\dot{y}_1 = 0$ give identical solutions $y_0(\lambda)$ and $y_1(\lambda)$.

Recall from <Ref> that we may generalize the definition of an augmented hypernetwork to allow only hyperedges between the two additional nodes and a subset $\mathcal{S}$ of nodes of the same type of the core.
It is clear that the results of <Ref> still hold for this construction.
More precisely, we then need a steady-state bifurcation branch in the core that has different components for the nodes in $\mathcal{S}$.
We may then define the order of asynchrony as in <Ref>, but comparing only states in $\mathcal{S}$.
As in <Ref> we will then generically have a steady-state bifurcation in a corresponding admissible vector field for the augmented hypernetwork, with the difference between the $w$-nodes growing in $\lambda$ raised to the power of the order of asynchrony.
Stability of this branch for the augmented hypernetwork can be guaranteed if the relevant branch for the core is stable.

§ MORE EXAMPLES

In this section, we present
three more examples that illustrate  <Ref>.

Consider the augmented hypernetwork shown in <Ref>. Its admissible ODEs are given by

\begin{equation}\label{systembifwithpara2c}
\begin{array}{ll}
\dot x_0 & = G(x_0, \textcolor{grey}{x_1}; \lambda)\, , \\
\dot x_1 & = G(x_1, \textcolor{grey}{x_0}; \lambda)\, , \\
\dot x_2 & = G(x_2, \textcolor{grey}{x_2}; \lambda)\, , \\
\dot y_0 & = F(y_0, \textcolor{mix}{(} \textcolor{red}{x_0}, \textcolor{blue}{x_1} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_1}, \textcolor{blue}{x_2} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_2}, \textcolor{blue}{x_0} \textcolor{mix}{)}; \lambda)\, ,\\
\dot y_1 & = F(y_1, \textcolor{mix}{(} \textcolor{red}{x_0}, \textcolor{blue}{x_2} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_1}, \textcolor{blue}{x_0} \textcolor{mix}{)}, \textcolor{mix}{(} \textcolor{red}{x_2}, \textcolor{blue}{x_1} \textcolor{mix}{)}; \lambda)\, ,\\
\end{array}
\end{equation}

where $F$ has the usual symmetry properties and where we assume each node to have a one-dimensional phase space.
The grey box in <Ref> denotes the core of this hypernetwork, which is a disconnected, classical first-order network, and whose dynamics corresponds to that of the $x$-variables in (<ref>).
Generically, a one-parameter bifurcation in an admissible system for the core is either given by the product of two saddle-nodes or by a pitchfork bifurcation.
Only the latter of these involves a fully synchrony-breaking branch, and so we focus on that case, which we realise by choosing

\begin{equation}\label{eq:Gin2ndex}
G(X_0,X_1;\lambda) =  -X_0-X_1+\lambda X_0-X_0^3 \, .
\end{equation}

It follows that for $\lambda<0$, the only (stable) steady state branch is given by $x_0(\lambda) = x_1(\lambda) = x_2(\lambda) = 0$. For $\lambda>0$ we find (apart from some unstable branches) two stable, fully synchrony-breaking branches, given by

\begin{equation*}
x_0(\lambda) = - x_1(\lambda) = \pm \lambda^{1/2}, \, x_2(\lambda) = 0\, .
\end{equation*}

For each of these two latter branches we have

\begin{align*}
x_2(\lambda) - x_0(\lambda) &=  \mp \lambda^{1/2}, \, \,
x_2(\lambda) - x_1(\lambda) =  \pm \lambda^{1/2}, \\
x_1(\lambda) - x_0(\lambda) &=  \mp 2\lambda^{1/2}\, ,
\end{align*}

from which we see that

\begin{equation*}
\bar{p} = \frac{1}{2}+\frac{1}{2}+\frac{1}{2} = \frac{3}{2}\, .
\end{equation*}

Hence, for the choice of response function $G$ given by (<ref>), <Ref> predicts the full system (<ref>) to undergo a bifurcation where $y_0(\lambda) - y_1(\lambda) \sim \lambda^{3/2}$ for $\lambda > 0$, for generic choice of $F$.

A numerical investigation corroborates this result, see <Ref>. These figures are obtained using Euler's method for the system (<ref>) with $G$ given by <Ref> and where we use

\begin{align}\label{num:res-F}
&F(Y, (X_0, X_1), (X_2, X_3), (X_4, X_5); {\lambda} ) \\ \nonumber =  &-5Y + 14 \lambda
\end{align}

in which

\begin{equation}\label{num:res-h}
h(x) = \sin(x)+\cos(x)-1 = \sqrt{2}\sin(x+\frac{\pi}{4}) - 1\, .
\end{equation}

This function $F$ is chosen to facilitate sufficient non-linearity for the result of <Ref> to hold, while also satisfying the required symmetry condition.
We forward integrated the system (<ref>) for each of $600$ equidistributed values of $\lambda \in [-0.03, 0.03]$.
For each fixed value of $\lambda$, integration was performed up to $t=5000$ with time steps of $0.1$, and starting from the point $(x_0, x_1, x_2, y_0, y_1) = (0.1, -0.2, 0.3, 0.4, 0.5)$.
For the log-log plot of <Ref>, we instead chose $600$ values of $\lambda \in [0.0005, 0.03]$, such that the values of $\ln(\lambda)$ are equidistributed.

An augmented hypernetwork with a disconnected core, shown within the grey box. We have left out self-loops corresponding to self-influence of each node.

The stable branches of a synchrony-breaking bifurcation.

The difference between the $y$-nodes along the stable branches.

A log–log plot of the difference between the $y$-nodes.

Numerically obtained bifurcation diagram for a system of the form (<ref>), corresponding to the augmented hypernetwork shown in <Ref>. The black line segment in the log-log plot has fixed slope $\frac{3}{2}$ for comparison, indicating that $y_0(\lambda) - y_1(\lambda) \sim \lambda^{\frac{3}{2}}$.

We next turn to the augmented hypernetwork depicted in <Ref>. The core of this hypernetwork, shown in the grey box, is an example of a classical (dyadic) network that itself shows reluctant synchrony-breaking. More precisely, admissible systems for this core are of the form

\begin{equation}\label{system5fund}
\begin{array}{ll}
\dot x_0 & = G(x_0, \textcolor{uhhdarkgrey}{x_1}, \textcolor{grey}{x_0}; \lambda)\, , \\
\dot x_1 & = G(x_1, \textcolor{uhhdarkgrey}{x_2}, \textcolor{grey}{x_0}; \lambda) \, , \\
\dot x_2 & = G(x_2, \textcolor{uhhdarkgrey}{x_2}, \textcolor{grey}{x_0}; \lambda)\, . \\
\end{array}
\end{equation}

These ODEs are special instances of those of the more general form

\begin{equation}\label{system5fund2}
\begin{array}{ll}
\dot x_0 & = H(x_0, \textcolor{uhhdarkgrey}{x_1}, \textcolor{grey}{x_0}, \textcolor{weirdgrey}{x_1}, \textcolor{notevengrey}{x_2}; \lambda)\, ,  \\
\dot x_1 & = H(x_1, \textcolor{uhhdarkgrey}{x_2}, \textcolor{grey}{x_0}, \textcolor{weirdgrey}{x_1}, \textcolor{notevengrey}{x_2}; \lambda) \, , \\
\dot x_2 & = H(x_2, \textcolor{uhhdarkgrey}{x_2}, \textcolor{grey}{x_0}, \textcolor{weirdgrey}{x_1}, \textcolor{notevengrey}{x_2}; \lambda)\, , \\
\end{array}
\end{equation}

obtained by setting $H(x,y,z,u,v; \lambda) = G(x,y,z; \lambda)$.
Alternatively, one may think of (<ref>) as denoting all admissible systems for a network obtained from the core in <Ref> by adding six additional arrows.
The first three of these are from node 1 to all nodes in the core (including an additional self-loop for node 1), and are all of a single, new type.
The last three are from node 2 to all nodes in the core, likewise all of one new type.
The reason for adding these new arrow-types is that we can rigorously compute generic steady state bifurcations in systems of the form (<ref>), using center manifold reduction. See <cit.> for a detailed exposition of the techniques used.

For the sake of this example, it is enough to know that in  Subsection 7.2 of <cit.> it is shown that the system (<ref>) generically undergoes a steady-state bifurcation involving a synchrony-breaking branch

\begin{equation}\label{branches5cnetwork}
\begin{split}
x(\lambda) &= (x_0(\lambda), x_1(\lambda), x_2(\lambda)) \\
&= (D_0\lambda + \mathcal{O}(|\lambda|^2), \, D_1\lambda + \mathcal{O}(|\lambda|^2), \, D_1\lambda + \mathcal{O}(|\lambda|^2))\, ,
\end{split}
\end{equation}

\begin{equation}\label{branches5cnetwork-2}
x_2(\lambda) - x_1(\lambda) = D_{2,1}\lambda^2 + \mathcal{O}(|\lambda|^3))\,
\end{equation}

and with $D_0, D_1, D_0-D_1, D_{2,1} \not= 0$.
As this branch diverges from the synchrony space $\{x_1 = x_2\}$ at only quadratic leading order, we may again speak of reluctant synchrony breaking.
As opposed to the reluctant synchrony breaking we have considered in augmented hypernetworks though, the space $\{x_1 = x_2\}$ is actually robust for systems of the form (<ref>), and so for the special cases (<ref>) as well.
The synchrony-breaking branch (<ref>) can furthermore take over stability from a fully synchronous one as $\lambda$ increases through zero, see Table 2.1 in <cit.>.
We therefore predict such a bifurcation to occur in the special system (<ref>) as well.

An augmented hypernetwork with a core that itself shows reluctant synchrony breaking. As before, we have left out self-loops corresponding to self-influence of each node.

<Ref> reveals that this is indeed the case.
<Ref> shows the components of the stable branches for the augmented hypernetwork of <Ref>.
However, as the core is a subnetwork of its augmented hypernetwork, we see that the $x$-variables depend only on each other and so depict a bifurcation in the three-node system (<ref>) as well.
<Ref> shows a reluctant separation happening between the nodes $v_1$ and $v_2$, corresponding to the variables $x_1$ and $x_2$, with <Ref> indicating this occurs as $\sim \lambda^2$.

It follows from (<ref>) and (<ref>) that we have

\begin{equation*}
x_2(\lambda)-x_0(\lambda) \sim \la, \, x_1(\lambda)-x_0(\lambda) \sim \la \text{ and } x_2(\lambda)-x_1(\lambda) \sim \la^2\,
\end{equation*}

so that

\begin{equation*}
\bar{p} = 1 + 1 + 2 = 4\, .
\end{equation*}

By <Ref> this implies that the augmented hypernetwork system generically exhibits the highly reluctant synchrony-breaking asymptotics

\begin{equation*}
y_0(\lambda) - y_1(\lambda) \sim \lambda^4\, .
\end{equation*}

The numerics in <Ref> corroborates this surprising asymptotics – see in particular the branches corresponding to the $y$-variables in <Ref>.

The details of the numerics are the same as for the previous example, except that time ran until $t=5000$ for <Ref>, and until $t=15000$ for <Ref>. The response function for the $x$-variables was chosen to be

\begin{equation*}%\label{eq:Gin3ndexd}
G(X_0,X_1,X_2;\lambda) =  -0.55X_1+0.25X_2+1.5\lambda X_0-0.1X_0^2 \, ,
\end{equation*}

with $F$ given by <Ref>.

The stable branches of a synchrony-breaking bifurcation.

The difference between $x_2$ and $x_1$ along the stable branches.

A log–log plot of the difference between between $x_2$ and $x_1$.

The difference between the  $y$-nodes along the stable branches.

A log–log plot of the difference between the $y$-nodes.

Numerically obtained bifurcation diagram for the augmented hypernetwork system shown in <Ref>, whose core is of the form (<ref>). The black line segment in (c) has fixed slope $2$ for comparison, indicating that $x_2(\lambda) - x_1(\lambda) \sim \lambda^{2}$. The black line segment in (e) has fixed slope $4$, which indicates that $y_0(\lambda) - y_1(\lambda) \sim \lambda^{4}$.

An augmented hypernetwork with a core consisting of a classical feed-forward network with four nodes, shown within the grey box. We have left out self-loops corresponding to self-influence of each node. In contrast to the previous examples, for this hypernetwork we have $k+1=4$, so that there are $(k+1)!=24$ hyperedges of order $k=3$.

Finally, we consider the augmented hypernetwork of <Ref>, which has as its core a classical feed-forward network with four nodes.
More precisely, the nodes in the core evolve according to the ODEs

\begin{equation}\label{system4feedfor}
\begin{array}{ll}
\dot x_0 & = G(x_0, \textcolor{uhhdarkgrey}{x_1}, \textcolor{grey}{x_2}; \lambda)\, , \\
\dot x_1 & = G(x_1, \textcolor{uhhdarkgrey}{x_2}, \textcolor{grey}{x_3}; \lambda) \, , \\
\dot x_2 & = G(x_2, \textcolor{uhhdarkgrey}{x_3}, \textcolor{grey}{x_3}; \lambda)\, , \\
\dot x_3 & = G(x_3, \textcolor{uhhdarkgrey}{x_3}, \textcolor{grey}{x_3}; \lambda)\, . \\
\end{array}
\end{equation}

It is known that this system generically supports steady-state bifurcations in which stability passes from a fully synchronous branch to one in which

\begin{align*}
x_0(\la) \sim \la^{1/4}\, , \,\,
x_1(\la) \sim \la^{1/2} \text{ and } \,
x_2(\la), x_3(\la), x_2(\la)- x_3(\la) \sim \la\, ,
\end{align*}

see <cit.> and <cit.>.  This unusually fast rate of synchrony breaking is also referred to as amplification.
It follows that

\begin{equation*}
\bar{p} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{2} + \frac{1}{2} + 1 = \frac{11}{4}\, ,
\end{equation*}

so that <Ref> predicts a reluctant steady state branch with

\begin{equation*}
y_0(\lambda) - y_1(\lambda) \sim \lambda^{\frac{11}{4}}\, .
\end{equation*}

This unusual growth rate is verified numerically in <Ref>, which  was obtained by numerically integrating the augmented system for

\begin{equation*}
G(X_0, X_1, X_2, \la) =  10X_1 -20X_2 + 15\la X_0 -100X_0^2\,
\end{equation*}

\begin{align*}
F\left(Y, \bigoplus_{\sigma \in S^0_{4}}  {\bf X}_{\sigma};\la\right) &= -0.01\sum_{\sigma \in S^0_{4}}
h(120 X_{\sigma,1}+40X_{\sigma, 2}-100X_{\sigma,3})
-5Y-\la \, ,
\end{align*}

where ${\bf{X}}_{\sigma} = (X_{\sigma,1}, \dots, X_{\sigma,4}) \in \R^4$ and with $h$ given by  <Ref>.

<Ref> shows the components of the stable branches, whereas <Ref> is a log-log plot of the difference of the $y$-components, for positive values of $\la$. In this latter picture, the dashed black line segment has fixed slope $11/4$, indicating that indeed $y_0(\lambda) - y_1(\lambda) \sim\lambda^{\frac{11}{4}}$.
For comparison we also plotted the dotted black line segment, which has fixed slope $10/4$, and which does not fit as well for low values of $\lambda$. Both figures were made by forward integrating the vector field for the augmented system from the point
$$(x_0, \dots, x_3, y_0, y_1) = (-0.001, -0.002, -0.003, -0.004, 0.001, 0.002)$$
in phase space, for various values of $\lambda$ and with time steps of $0.1$.
For <Ref> this was done up to $t=2000$ and with $600$ equidistant values of $\la \in [-0.03, 0.03]$.
For <Ref> this was up to $t=20000$ and for $100$ equidistant values of $\ln(\la) \in [\ln(0.00003), \ln(0.03)]$.

The stable branches of a synchrony-breaking bifurcation.

A log–log plot of the difference between the $y$-nodes.

Numerically obtained bifurcation diagram for an augmented hypernetwork with core given by a four-node feed-forward network, see <Ref>. The dashed black line segment in the log-log plot has fixed slope $\frac{11}{4}$, whereas the dotted segment has fixed slope $\frac{10}{4}$.

<Ref> shows that even classical (dyadic) networks may generically support  reluctant synchrony breaking bifurcations.
However, we are not aware of any method to design networks that, for instance, break synchrony up to some prescribed degree in $\lambda$.
For hypernetworks, the augmented hypernetwork construction makes this design problem  more tractable. In fact, we show below that one may construct hypernetworks which support generic reluctant synchrony breaking to arbitrarily high order.
We  foresee possible applications of this observation for understanding the phenomenon of homeostasis <cit.>.

To illustrate how one can create hypernetworks with an arbitarily high order of reluctant synchrony breaking, we return to the five-node augmented hypernetwork of <Ref>.
In <Ref>, we observed that the three-node core of this hypernetwork supports a steady state branch in which $x_i(\la) = D_i\la + \mathcal{O}(|\lambda|^2)$ for some mutually distinct $D_i \in \R$.
Using the same notation as in  <Ref>, we   expand the response function $F$ for the $y$-nodes as

\begin{equation}\label{expressionsforFnnfG--2}
\begin{split}
& F(Y, (X_0, X_1), (X_2, X_3), (X_4, X_5); \lambda )  \\
& = aY + bX_0 + cX_1 + bX_2 + cX_3 + bX_4 + cX_5 + d\lambda \\
& \phantom{=} + \mathcal{O}(\|(Y,X_0, \dots, X_5;\lambda)\|^2) \, ,
\end{split}
\end{equation}

and we recall that we found a reluctant steady state branch in the augmented hypernetwork with asymptotics

\begin{equation}\label{y0non-synns--2}
y_0(\lambda) = \frac{-(b+c)(D_0 + D_1 + D_2) - d}{a}\lambda  + \mathcal{O}(|\lambda|^2)\,
\end{equation}

\begin{equation}\label{y1non-synns--2}
y_1(\lambda) = \frac{-(b+c)(D_0 + D_1 + D_2) - d}{a}\lambda  + \mathcal{O}(|\lambda|^2)\, .
\end{equation}

To this augmented hypernetwork we can now add another node of the same type as the $y$-nodes, with corresponding variable $y_2$.
We couple it to the nodes in the core in such a way that

\begin{equation}\label{y2eeq--2}
\dot{y_2} = F(y_2, (x_0, x_1), (x_1, x_0), (x_0, x_0), \la)\, .
\end{equation}

The aforementioned branch of steady states is then supported by this larger hypernetwork as well, where in addition

\begin{equation}\label{y2non-synns--2}
y_2(\lambda) = \frac{-(b+c)(D_0 + D_1 + D_0) - d}{a}\lambda  + \mathcal{O}(|\lambda|^2)\, ,
\end{equation}

as can be seen using <Ref>.
To summarise, we now have a branch where the three $y$-nodes satisfy

\begin{align}
y_0(\la) &= E_0\la + \mathcal{O}(|\la|^2) \, , \, \, y_1(\la) = E_0\la + \mathcal{O}(|\la|^2) \, ,\\ \nonumber
y_2(\la) &= E_2\la + \mathcal{O}(|\la|^2) \text{ and } \, y_0(\la) - y_1(\la) \sim \la^p
\end{align}

for some $p>1$ (in this particular case $p=3$).
Moreover, from Equations (<ref>), (<ref>) and (<ref>) we see that $E_0 \not= E_2$, as $D_0 \not= D_2$ by assumption.

We may now use the three $y$-nodes as the core for another augmented hypernetwork, say by adding two $z$-nodes of a new type (c.f. <Ref>).
We also add a third $z$-node, precisely as we did with the third $y$-node.
That is, we set

\begin{equation}\label{z2eeq--2}
\begin{split}
\dot{z_0} &= \tilde{F}(z_0, (y_0, y_1), (y_1, y_2), (y_2, y_0), \la)\, ,\\
\dot{z_1} &= \tilde{F}(z_1, (y_0, y_2), (y_1, y_0), (y_2, y_1), \la)\, ,\\
\dot{z_2} &= \tilde{F}(z_2, (y_0, y_1), (y_1, y_0), (y_0, y_0), \la)\, ,
\end{split}
\end{equation}

where $\tilde{F}$ is a response function for the $z$-nodes. Just as before, we will then find

\begin{align}
z_0(\la) &= E'_0\la + \mathcal{O}(|\la|^2) \, , \, \, \, z_1(\la) = E'_0\la + \mathcal{O}(|\la|^2) \, ,\\ \nonumber
z_2(\la) &= E'_2\la + \mathcal{O}(|\la|^2) \text{ and } z_0(\la) - z_1(\la) \sim \la^{p+2}
\end{align}

for some generically nonzero $E'_0, E'_2 \in \R$. The new power ${p+2}$ follows from <Ref>, as

\begin{align}
&y_2(\la) - y_0(\la) \sim \la\, , \, \, \, y_2(\la) - y_1(\la) \sim \la \, ,\\ \nonumber
&y_0(\la) - y_1(\la) \sim \la^p\, ,
\end{align}

which holds because $E_0 \not= E_2$. We may also argue that $E'_0 \not= E'_2$ in precisely the same way that we argued that $E_0 \not= E_2$.

This shows that by iteratively growing the augmented hypernetwork, we may increase the order (in $\lambda$) of reluctancy of the reluctant steady state branch. In other words, we may design hypernetworks with an arbitrarily high order of reluctant synchrony breaking. Concretely, our example shows that we can arrange for $p = 3, 5, 7, \dotsc$.
It is also clear from this construction that the resulting reluctant branch may be assumed stable.

§ ACKNOWLEDGEMENTS

We thank Martin Golubitsky and Ian Stewart for enlightening discussions. S.v.d.G. was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)––453112019. E.N. was partially supported by the Serrapilheira Institute (Grant No. Serra-1709-16124). B.R. gratefully acknowledges the hospitality and financial support of the Sydney Mathematical Research Institute.

§ DATA AVAILABILITY

The code used to generate the numerical results can be found in <cit.>.

§ PROOF OF CONJECTURE <REF> FOR $K=2,\DOTSC,6$

In Section <ref> we have shown that, in a hypernetwork with hyperedges of degree $k$, if we restrict ourselves to polynomial admissible maps of degree less than $k(k+1)/2$, there can be more invariant synchrony spaces than for general admissible maps. We suspect the following to hold:

[Conjecture 1]
Given a hypernetwork with hyperedges of degree $k$ or less. If a poly-diagonal space is invariant for all polynomial admissible vector-fields of degree $k(k+1)/2$ and less, then it corresponds to a balanced partition and is hence an invariant synchrony space for all admissible maps.

If this conjecture is true, then we can extend Theorem <ref> with the following bullet point iv), which would then be equivalent to its current bullet points ${\it i)}$, ${\it ii)}$ and ${\it iii)}$.

iv) ${\rm Syn}_P$ is  invariant under any polynomial N-admissible map of degree at most $k(k+1)/2$, where $k$ is the order of the hypernetwork.

Note that the implication ${\it iii)} \implies {\it iv)}$ is obvious, so the difficulty lies in proving that ${\it iv)} \implies {\it iii)}$.

Recall that in the proof that iii) implies i) in  <Ref> we chose the polynomial admissible map (<ref>), consisting of monomial terms
(<ref>),  of degree $2^k-1$. This choice allowed us to count hyperedges with a prescribed pattern of source colours.
Our strategy for proving Conjecture <ref> would be as follows. Let $k \geq 2$ be an integer and let $\{1, 2, \ldots, C\}$ be a finite set of colours. We denote by $s: \{1, \ldots, k\}\to \{1, \ldots, C\}$ a “colouring”.

Next, we denote by $\mathbb{R}[X_1, \dots, X_k]$ the ring of polynomials over $\mathbb{R}$ in $k$ variables. Its elements have the interpretation of hyperedge response functions. Similarly, we denote  by $\mathbb{R}[Y_1, \ldots, Y_C]$ the ring of polynomials over $\mathbb{R}$ in $C$ variables.  The colouring $s$ now induces a ring homomorphism $\psi_s$ from $\mathbb{R}[X_1, \dots, X_k]$ to $\mathbb{R}[Y_1, \ldots, Y_c]$, by replacing $X_i$ by $Y_{s(i)}$ for all $i \in \{1, \dots, k\}$.
The interpretation is that we  evaluate the polynomial on the polydiagonal subspace in which $X_i=X_j$ if $s(i)=s(j)$.
In other words, $\psi_s$ is defined (on monomials) by
$$\psi_s(X_1^{w_1} \cdots X_k^{w_k}) := Y_1^{W_1}\cdots Y_C^{W_C} $$

where $W_c = W_c(w) = \sum_{i: s(i)=c} w_i$.
In the proof of Theorem <ref> we used the fact that for the monomial

$$p = p(X) = X_1 X_2^2 X_3^4 \cdots X_k^{2^{k-1}}$$

the map $w \mapsto W(w)$ is injective — because two sums of powers of $2$ can only be equal if they consist of the same terms. This is true independent of the colouring $s$. However, it would have sufficed to choose a different monomial for each colouring $s$. Similarly, to prove <Ref> it   suffices to prove the following:

For any  colouring $s$, there exists a monomial $p_s \in \break \mathbb{R}[X_1, \dots, X_k]$ of degree $k(k+1)/2$ or less such that $\psi_s(p_s) = \psi_t(p_s)$   implies $s = t$.

Conjecture 2 implies Conjecture <ref> as the monomial $p_s$ allows us to count how many hyperedges have an input-sequence of nodes that form the sequence $s$ when the corresponding poly-diagonal subspace is considered. We next prove Conjecture 2 for $k \in \{2, \dots, 6\}$.

§.§ $k=2$

For $k=2$, the only sequences to consider are, in essence, $aa, ab, ba$ and $bb$. Consider $p_{aa} = p_{ab} =  p_{ba} =  p_{bb} = p = X_1X_2^2$. Setting  $s = aa, ab, ba$ and $bb$ gives $\psi_s(p) = a^3, ab^2, a^2b$ and $b^3$, respectively. Note that $p$ is a polynomial of degree $2(2+1)/2 = 3$. Because $\psi_s(p)$ is different for each $s$, this choice of monomial proves Conjecture 2 for $k=2$.

In general, we will not be able to find a single monomial for every sequence simultaneously as we did for $k=2$. At this point we note that there is an action of $S_k$ on both the set of symbol-sequences of length $k$ and the set of monomials in $\mathbb{R}[X_1, \dots, X_k]$. These respect each other in the sense that $\psi_{\sigma\cdot s}(p) = \psi_{s} (\sigma\cdot p)$ for all $s \in C^k$, $\sigma \in S_k$ and $p \in \mathbb{R}[X_1, \dots, X_k]$. If $p_s$ is a monomial satisfying $\psi_s(p_s) = \psi_t(p_s) \implies s = t$, then $\psi_{\sigma \cdot s}(\sigma^{-1} \cdot p_s) = \psi_t(\sigma^{-1} \cdot p_s)$ gives $\psi_{ s}(p_s) = \psi_{\sigma^{-1} \cdot t}(p_s)$. Hence, we find $s = \sigma^{-1} \cdot t$ and so $t = \sigma \cdot s$, showing that a monomial as in Conjecture 2 exists for $\sigma \cdot s$ as well (namely $\sigma^{-1} \cdot p_s$). For this reason we only have to consider symbol-sequences up to a reordering. As a renaming of the symbols makes no essential difference as well, we conclude that we only have to check a sequence for each partition of $k$.

§.§ $k=3$

For $k = 3$, if we have $s = aaa$ we let $p = X_1X_2X_3$. Clearly $\psi_t(p) = a^3$ if and only if $s = t$. In fact, any monomial $p = X_1^{n_1}X^{n_2}_2X^{n_3}_3$ with $n_1, n_2, n_3 > 0$ and $n_1 + n_2 + n_3 \leq 3(3+1)/2 = 6$ would have worked.

For $s = aab$, we take $p = X_1X_2X_3^3$. It follows that $\psi_s(p) = a^2b^3$. If we have $\psi_t(p) = a^2b^3$ then necessarily $t(3) = b$, after which it readily follows that $t(1) = t(2) = a$. Hence we have $s=t$. These kinds of combinatorial puzzles (dubbed `playing hang-man') will be used a lot.

For $s = abc$ we take $p = X_1X_2^2X_3^3$, so that $\psi_s(p) = ab^2c^3$. In case $\psi_t(p) = ab^2c^3$ we necessarily have $t(1) = a$. After that we see that $t(2) = b$ and finally $t(3) = c$. Hence, we find $t = s$. This completes the case for $k = 3$.

Before we move on, we collect some general cases.

[Lemma 1]  For $s = aa\dots a$ we may choose $p_s = X_1X_2 \dots X_k$.

[Proof] We find $\psi_s(p_s) = a^k$ and clearly $\psi_t(p_s) \not= a^k$ if $t \not= s$. Note that $p_s$ has degree $k < k(k+1)/2$. As in the special case $k=3$ (and $k=2$), we see that many other choices of $p_s$ are possible.

[Lemma 2]  For $s = aa\dots abb\dots b$, where we have $l$ times $a$ and $k-l$ times $b$, we may choose $p_s = X_1X_2 \dots X_l X_{l+1}^{l+1} \dots X_{k}^{l+1}$.

[Proof] We have $\psi_s(p_s) = a^lb^{(k-l)\cdot(l+1)}$. The only way to get $\psi_t(p_s) = a^lb^{(k-l)\cdot(l+1)}$ is when we build $t$ by starting with $l$ copies of $a$ and then filling up the remaining slots with $b$s. Note that the degree of $p_s$ is indeed low enough.

[Lemma 3]  Given $l \leq k$, consider $s = a_1a_2\dots a_{l-1}a_la_l \dots a_l$ for some distinct $a_1$ through $a_l$. For any such sequence we may choose $p_s = X_1X_2^2 \dots X_k^k$.

[Proof] We get $\psi_s(p_s) = a_1^1a_2^2 \dots a_{l-1}^{l-1}a_l^{l+(l+1) + \dots + k}$. Whenever $\psi_s(p_s)= \psi_t(p_s)$ we see that necessarily $t(1) = a_1$ and $t(i) \not= a_1$ for all $i \not= 1$. After that, we similarly conclude $t(i) = a_2$ if and only if $i = 2$, then $t(i) = a_3$ if and only if $i = 3$ and so forth, up to $t(i) = a_{l-1}$ if and only if $i = l-1$. After this we see that $t(l) = t(l+1) = \dots = t(k) = a_l$, so that indeed $t = s$.

§.§ $k=4$

$s = aaaa$ follows from Lemma 1.

$s=aaab, aabb$ and $abbb$ follow from Lemma 2.

$s=abcc$ and $abcd$ follow from Lemma 3.

§.§ $k=5$

$s = aaaaa$ follows from Lemma 1.

$s=aaaab, aaabb, aabbb$ and $abbbb$ follow from Lemma 2.

$s=abccc$ follows from Lemma 3.

For $s=abbcc$, choose $p_s = X_1X_2^2X_3^2X_4^3X_5^3$. We get $\psi_s(p_s) = ab^4c^6$. If $\psi_s(p_s) = \psi_t(p_s)$ then we first conclude that $t(1)  = a$. It then follows that $t(2) = t(3) = b$ and $t(4) = t(5) = c$, so that $t = s$.

$s=abcdd$ and $abcde$ follow from Lemma 3.

§.§ $k=6$

$s = aaaaaa$ follows from Lemma 1.

$s=aaaaab, aaaabb, aaabbb, aabbbb$ and $abbbbb$ follow from Lemma 2.

$s=abcccc$ follows from Lemma 3.

For $s=aabbcc$, choose $p = X_1X_2X_3^3X_4^3X_5^4X_6^4$. We get $\psi_s(p) = a^2b^6c^{8}$. If $\psi_s(p) = \psi_t(p)$ then we first conclude that $t(1) = t(2) = a$. It then follows that $t(3) = t(4) = b$ and $t(5) = t(6) = c$, so that $t = s$.

For $s=abbccc$, choose $p = X_1X_2^2X_3^2X_4^3X_5^3X_6^3$. We get $\psi_s(p) = ab^4c^{9}$. If $\psi_s(p) = \psi_t(p)$ then we first conclude that $t(1) = a$. It then follows that $t(2) = t(3)  = b$ and $t(4) = t(5) = t(6) = c$, so that $t = s$.

$s=abcddd$ follows from Lemma 3.

For $s=abccdd$, choose $p = X_1X_2^2X_3^3X_4^3X_5^4X_6^4$. We get $\psi_s(p) = ab^2c^{6}d^8$, and one verifies that  $\psi_t(p) = ab^2c^{6}d^8$ gives $t = abccdd$.

$s=abcdee$ and $abcdef$ follow from Lemma 3.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Society for Neuroscience

abstractThe ability of cortical neurons to perform temporally accurate computations has been shown to be important for encoding of information in the cortex; however, cortical neurons are expected to be imprecise temporal encoders because of the stochastic nature of synaptic transmission and ion channel gating, dendritic filtering, and background synaptic noise. Here we show for the first time that fast local spikes in basal dendrites can serve to improve the temporal precision of neuronal output. Integration of coactivated, spatially distributed synaptic inputs produces temporally imprecise output action potentials within a time window of several milliseconds. In contrast, integration of closely spaced basal inputs initiates local dendritic spikes that amplify and sharpen the summed somatic potential. In turn, these fast basal spikes allow precise timing of output action potentials with submillisecond temporal jitter over a wide range of activation intensities and background synaptic noise. Our findings indicate that fast spikes initiated in individual basal dendrites can serve as precise “timers” of output action potentials in various network activity states and thus may contribute to temporal coding in the cortex.
journaltitleJournal of Neuroscience
titleSubmillisecond Precision of the Input-Output Transformation Function Mediated by Fast Sodium Dendritic Spikes in Basal Dendrites of CA1 Pyramidal Neurons

Springer International Publishing

booktitleHigher-Order Systems
seriesUnderstanding Complex Systems
titleConsensus Dynamics and Opinion Formation on Hypergraphs

abstractA central assumption in most ecological models is that the interactions in a community operate only between pairs of species. However, two species may interactively affect the growth of a focal species. Although interactions among three or more species, called higher-order interactions, have the potential to modify our theoretical understanding of coexistence, ecologists lack clear expectations for how these interactions shape community structure. Here we analytically predict and numerically confirm how the variability and strength of higher-order interactions affect species coexistence. We found that as higher-order interaction strengths became more variable across species, fewer species could coexist, echoing the behavior of pairwise models. If interspecific higher-order interactions became too harmful relative to self-regulation, coexistence in diverse communities was destabilized, but coexistence was also lost when these interactions were too weak and mutualistic higher-order effects became prevalent. This behavior depended on the functional form of the interactions as the destabilizing effects of the mutualistic higher-order interactions were ameliorated when their strength saturated with species' densities. Last, we showed that more species-rich communities structured by higher-order interactions lose species more readily than their species-poor counterparts, generalizing classic results for community stability. Our work provides needed theoretical expectations for how higher-order interactions impact species coexistence in diverse communities.
journaltitleProceedings of the National Academy of Sciences of the United States of America
titleCoexistence in diverse communities with higher-order interactions
ibbs, Levin et al 2022 - Coexistence in diverse communities.pdf:application/pdf

abstractIn the study of dynamical systems on networks or graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that, instead of microscopic details of the individual nodes or vertices, rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here, we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitrary higher-order interactions between nodes. As, for instance, in the theory of coupled map lattices, we study Laplace-type interaction structures in detail. Since the spectral theory of Laplacians on hypergraphs is richer than on graphs, we see the possibility of different dynamical phenomena. More generally, our arguments provide a blueprint for how to generalize dynamical structures and results from graphs to hypergraphs.
journaltitlePhysical Review E
titleCoupled dynamics on hypergraphs: Master stability of steady states and synchronization

abstractWe consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogs of structures seen in related network models.
titleConsensus on simplicial complexes: Results on stability and synchronization

abstractA common approach for analyzing hypergraphs is to consider the projected adjacency or Laplacian matrices for each order of interactions (e.g., dyadic, triadic, etc.). However, this method can lose information about the hypergraph structure and is not universally applicable for studying dynamical processes on hypergraphs, which we demonstrate through the framework of cluster synchronization. Specifically, we show that the projected network does not always correspond to a unique hypergraph structure. This means the projection does not always properly predict the true dynamics unfolding on the hypergraph. Additionally, we show that the symmetry group consisting of permutations that preserve the hypergraph structure can be distinct from the symmetry group of its projected matrix. Thus, considering the full hypergraph is required for analyzing the most general types of dynamics on hypergraphs. We show that a formulation based on node clusters and the corresponding edge clusters induced by the node partitioning, enables the analysis of admissible patterns of cluster synchronization and their effective dynamics. Additionally, we show that the coupling matrix projections corresponding to each edge cluster synchronization pattern, and not just to each order of interactions, are necessary for understanding the structure of the Jacobian matrix and performing the linear stability calculations efficiently.
titleAnalyzing states beyond full synchronization on hypergraphs requires methods beyond projected networks
ouza 2021 - Analyzing states beyond full synchronization:Attachments/Salova, D'Souza 2021 - Analyzing states beyond full synchronization.pdf:application/pdf

abstractWe develop the formalism to analyze cluster synchronization for dynamical elements coupled via hypergraphs beyond pairwise interactions. We introduce the notion of edge partitions and show how node and edge partitions allow us to identify admissible cluster synchronization states and simplify their linear stability calculations. The analysis in terms of node and edge partitions provides a principled way to track dynamics on hypergraphs, and the projected Laplacian matrices based on each edge cluster are essential to block diagonalizing the Jacobian in order to reduce the dimensionality of the stability analysis. Our work enables detailed analysis of patterns of synchronization beyond full synchronization and beyond dyadic interactions.
titleCluster synchronization on hypergraphs
ouza 2021 - Cluster synchronization on hypergraphs:Attachments/Salova, D'Souza 2021 - Cluster synchronization on hypergraphs.pdf:application/pdf

abstractWe consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.
journaltitleChaos (Woodbury, N.Y.)
titleDynamical systems defined on simplicial complexes: Symmetries, conjugacies, and invariant subspaces
ijholt, DeVille 2022 - Dynamical systems defined on simplicial (2).pdf:application/pdf

titleNetwork Dynamics with Higher-Order Interactions: Coupled Cell Hypernetworks for Identical Cells and Synchrony
family=von der Gracht,

journaltitleSIAM Journal on Applied Mathematics
titleHypernetworks: Cluster Synchronization Is a Higher-Order Effect

journaltitlePhysics Reports
titleNetworks beyond pairwise interactions: Structure and dynamics

Springer International Publishing

booktitleEmerging Frontiers in Nonlinear Science
seriesNonlinear Systems and Complexity
titleNonlinearity + Networks: A 2020 Vision

journaltitleSIAM Review
titleThe Why, How, and When of Representations for Complex Systems

journaltitleSIAM Review
titleWhat Are Higher-Order Networks?
ick, Gross et al 2023 - What Are Higher-Order Networks.pdf:application/pdf
family=Del Genio,

journaltitlePhysics Reports
titleThe structure and dynamics of networks with higher order interactions
occaletti, Lellis et al 2023 - The structure and dynamics.pdf:application/pdf

journaltitleJournal of Nonlinear Science
titleSome Curious Phenomena in Coupled Cell Networks

abstractA formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the `input sets'. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of `robust' patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood.
journaltitleBull. Amer. Math. Soc.
titleNonlinear dynamics of networks: The groupoid formalism

journaltitleDynamical Systems
titleCombinatorial dynamics

journaltitleDiscrete Mathematics
titleFibrations of graphs

journaltitleJournal of the European Mathematical Society
titleModular dynamical systems on networks

journaltitleSIAM J. Appl. Dyn. Syst.
titlePatterns of Synchrony in Coupled Cell Networks with Multiple Arrows

journaltitleJournal of Nonlinear Science
titleThe Lattice of Synchrony Subspaces of a Coupled Cell Network: Characterization and Computation Algorithm
family=von der Gracht,

titleAmplified steady state bifurcations in feedforward networks
on der Gracht, Nijholt et al. 2022 - Amplified steady state bifurcations.pdf:application/pdf

journaltitleSIAM Review
titleCenter manifolds of coupled cell networks

journaltitleSIAM Journal on Applied Dynamical Systems
titleAmplified Hopf bifurcations in feed-forward networks

abstractHomeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter [Formula: see text] varies over some interval. We discuss two main aspects of homeostasis, both related to the effect of coordinate changes on the input-output map. The first is a reformulation of homeostasis in the context of singularity theory, achieved by replacing 'approximately constant over an interval' by 'zero derivative of the output with respect to the input at a point'. Unfolding theory then classifies all small perturbations of the input-output function. In particular, the 'chair' singularity, which is especially important in applications, is discussed in detail. Its normal form and universal unfolding [Formula: see text] is derived and the region of approximate homeostasis is deduced. The results are motivated by data on thermoregulation in two species of opossum and the spiny rat. We give a formula for finding chair points in mathematical models by implicit differentiation and apply it to a model of lateral inhibition. The second asks when homeostasis is invariant under appropriate coordinate changes. This is false in general, but for network dynamics there is a natural class of coordinate changes: those that preserve the network structure. We characterize those nodes of a given network for which homeostasis is invariant under such changes. This characterization is determined combinatorially by the network topology.
journaltitleJournal of Mathematical Biology
titleHomeostasis, singularities, and networks

journaltitleJournal of mathematical biology
titleThe structure of infinitesimal homeostasis in input–output networks
family=von der Gracht,

§ AN UPPER BOUND

We now show that if a synchrony space is invariant under all polynomials admissible maps of degree $2^k-1$, then the corresponding partition is balanced. In fact, it suffices to use admissible maps defined by

$$f_v(x) = f^W_v(x) := \sum_{t(h)=v} x_{s_1(h)}^{W_1} \cdots x_{s_k(h)}^{W_k} $$

for the weights

$$W=(0, 1, 2, 4, 8, \ldots, 2^{k-1})\, .$$

These maps clearly have polynomial degree $2^k-1$.
Robustness implies that $f_v(x) = f_w(x)$ for $v, w$ in the same element of the partition, i.e.,

$$  \sum_{t(h)=v} X_{1}^{n_1(h)} \cdots X_C^{n_C(h)}  =  \sum_{t(h)=w} X_{1}^{n_1(h)} \cdots X_C^{n_C(h)}  \, ,$$

$$n_c(h) = \sum_{i \, : \, s_i(h) \in P_c} W_i = \sum_{i=1}^k \delta_{\{s_i(h) \in P_c\}}\cdot W_i\, .$$

The equality of the polynomials implies that there is a bijection $\alpha: t^{-1}(v)\to t^{-1}(w)$ between targeting hyperedges so that

$$n_c(h)= n_c(\alpha(h)) \mbox{for each}\ h\in t^{-1}(v), $$

that is, for each $h$ we have

\begin{align}\label{equality1}
\sum_{i \, : \, s_i(h) \in P_c} W_i  =  \sum_{i \, : \, s_i(\alpha(h)) \in P_c} W_i  \, .
\end{align}

Now we are interested in the question whether this implies that for each $c$ we have

\begin{align}\label{equality2}
\{i \, : \, s_i(h) \in P_c\} = \{i \, : \, s_i(\alpha(h)) \in P_c\}\ \mbox{for every}\ c\, .
\end{align}

If so, then the partition is balanced and the proof is done.

We say that $W$ is an input-distinguishing weight vector if the map
$$w :\{1, \ldots, C\}^k \to \{0, 1, 2, \ldots\}^C $$

defined by $w(P)_c =   \sum_{i\in P_c} W_i$ is injective. Here, a partition of $\{1, \ldots, k\}$ into $C$ subsets is encoded by a map from $\{1, \ldots, k\}$ to $\{1, \ldots, C\}$. The map $w$ assigns to each partition into $C$ subsets (of hyperedge inputs) the collection of weights for each subset.

The vector $W = (1, 2, \ldots, 2^{k-1})$ is input-distinguishing.

Obvious: $w(P)$ consists of  binary expansions of the $w(S)$ with $S\in P$. This binary expansion is unique.

The vector $W = (1,2)$ is input-distinguishing.

We have $k=2$ and we shall assume that $C=2$ (the only interesting case). The only partition to consider is then $\{1\} \cup \{2\}$.
We have $w_c = (m_1+2m_2, n_1 + 2n_2)$ where $n_1, n_2 \in \{0,1\}$ are the number of times the color $c$ is assigned to input 1 respectively input 2.

The vector $W = (1, 2, 3)$ is $C$-distinguishing for each $C$.

For $C=1$ we have $w(P) = \sum W_i$.
Suppose that $a + 2b + 3c = w$ for $a,b,c\in \{0,1\}$. If $w=0$ then $a=b=c=0$. If $w=1$ then $a=1, b=c=0$. If $w=2$ then $a=c=0$ and $b=2$. The only tricky case arises when $w=3$. Then there are solutions $a=b=1, c=0$ and $a=b=0, c=1$.

§ A COMBINATORIAL PROBLEM

Here we describe the combinatorial problem of finding the bijection between hyperedges that must be solved to prove that a polynomially invariant synchrony space corresponds to a balanced partition.

Say that $v_1, v_2$ are two cells both targeted by $N$ hyperedges of the one particular type (we ignore the hyperedges of another type that also target them). Robustness of a synchrony space given by $x_{v} = X_c$ (where the $c$ label the colours of the partition) under polynomial admissible maps implies that for both cells the

$$\sum_{(c_1, \ldots, c_k)} \sharp (c_1, \ldots, c_k)\, X_{c_1}^{W_1}\cdots X_{c_k}^{W_k}$$

are equal. Here $(c_1, \ldots, c_k)$ denotes the possible“colourings” of a hyperedge, i.e., the vector of colours of inputs of a hyperedge. All we want to know is that every such colouring appears equally often among the hyperedges targeting $v_1$ and those targeting $v_2$. We want to know how we can infer the frequency of these colourings from this information.

For example, if all colours are different, i.e., $c_i \neq c_j$ for all $i\neq j$, then we can take $W_1=1, W_2=2, \ldots, W_k=k$ and we obtain

$$\sum_{(c_1, \ldots, c_k)} \sharp (c_1, \ldots, c_k)\, X_{c_1}^{1}\cdots X_{c_k}^{k}$$

From this we can infer the frequency of these colour-input-types: they are equal for $v_1$ and $v_2$, and given by the coefficient of the corresponding monomial.

Let $k \in \mathbb{N}$ and let $C$ be a finite set. An ordered tuple ${\bf c} = (c_1, \ldots, c_k) \in C^k$ is called an hyperedge input pattern. A weight vector $W=(W_1, \ldots, W_k)\in \mathbb{N}_0^k$ is called ${\bf c}$-distinguishing if from the polynomial

$$\sum_{h} X_{c(s_1(h))}^{W_1}\cdots X_{c(s_k(h))}^{W_k}$$

we can determine the frequency of the hyperedge input pattern.

We just saw that the weight $(1,2, \ldots, k)$ is ${\bf c}$-distinguishing  if ${\bf c}$ is a list of all different colours.

§ THE FOLLOWING ARE JUST NOTES

It follows to a certain extent the known proof for networks (=hypernetworks of order 1). For two vertices $v_1, v_2\in V$ let us write $v_1 \sim v_2$ if they have the same colour. Similarly, for $h_1, h_2  \in H$ we write $h_1\sim h_2$ if they have the same colour.

Now let $v_1, w_1 \in V$ and $h_1 \in H$ satisfy $t(h_1) = v_1$ and $s_i(h_1) = w_1$. We shall assume that $h_1$ is a hyperedge of indegree $k$, which is less than or equal to the order of the hypernetwork. Note that every every vertex $v \sim v_1$ is targeted by exactly the same number of hyperedges $h\sim h_1$. We denote this number by

$$ n := \sharp \{h\in H \ |\ h \sim h_1 \ \mbox{and} \ t(h)=v \ \mbox{for any} \ v \sim v_1\} \, .$$

For every such hyperedge we will count how many of their sources vertices $v$ are synchronous

We now choose a nonzero $X \in \mathbb{R}^{n_{w_1}}$
and a linear map $A: \mathbb{R}^{n_{w_1}}  \to \mathbb{R}$ that satisfies
$A(X) =  1$.
We use use this linear map to define, for any $v\in V$, and for every $1\leq m \leq k$ the homogeneous degree-$m$ polynomial map $f: \bigoplus_{v\in V} \mathbb{R}^{n_v} \to \bigoplus_{v\in V} \mathbb{R}^{n_v}$ by

$$f_v(x) = F_v\left( \bigoplus_{t(h)=v} x_{s(h)} \right) := \sum_{\scriptsize{\begin{array}{c} t(h)=v \\ h \sim h_1 \end{array}}} \left( \sum_{s_j(h)\sim  w_1} A(x_{s_j(h)}) \right)^m\, Y\, .$$

Here, $Y \in \mathbb{R}^{n_{v_1}}$ is an arbitrary nonzero vector, and $\sim$ denotes being of the same colour  for both vertices and hyperedges.  By construction, this $f$ is an admissible vector field. In fact, we see that $f_v=0$ if $v$ and $v_1$ have a different colour. Also, when $v$ and $v_1$ have the same colour, then the sums defining $f_v(x)$ and $f_{v_1}(x)$ have equally many terms, because there they have equally many hyperedges of each colour targeting them. Also the sums inside the $m$-th power all have equally many terms because every hyperedge that appears has exactly the same number of inputs of each colour.

We will now evaluate the $f_v$ at a special point $x^*$ of ${\rm Syn}_P$ that satisfies

x^*_{w}= \left\{ \begin{array}{ll} X  & \mbox{if}\ w \ \mbox{in the same element of}\ P \ \mbox{as}\ w_1,\\
0 & \mbox{otherwise.}
\end{array}\right.

$$f_v(x^*) =  \left( \sum_{h\sim H, t(h)=v } \left( \sharp  \{ i \, |\, s_i(h)  \ \mbox{in the same element of}\ P\ \mbox{as}\ w_1 \} \right)^m \right) Y $$

So essentially $f_v(x^*)$ counts the sum over the hyperedges targeting $v$ of the $m$-th power of the number of sources of the hyperedge that are  synchronous to $w_1$ in ${\rm Syn}_P$.
Our assumption that ${\rm Syn}_P$ is invariant under polynomials of degree at most the order of the hypernetwork, implies that $f_{v_1}(x) = f_{v_2}(x)$ for any $x\in {\rm Syn}_P$.  So these sums of $m$-th powers are equal for all vertices.

Let $0 \leq a_1, \ldots, a_n, b_1, \ldots, b_n \leq k$ be integers satisfying

$$a_1^m+ \ldots + a_n^m = b_1^m+ \ldots + b_n^m\ \mbox{for all} \ 1 \leq m \leq k\, .$$

Then after relabelling we have $a_1=b_1, \ldots a_n=b_n$.

Let us define

$$A_l = \sharp\{i \ | \ a_i=l\}\ \mbox{and}\ B_l = \sharp\{i \ | \ b_i=l\}\ \mbox{for}\ l=0, \ldots, k\, .$$

It  suffices to prove that $A_0=B_0, A_1=B_1, \ldots, A_k=B_k$.
Note that our assumptions imply that

\begin{align}\nonumber
\begin{array}{lcr}
0 A_0 + 1 A_1 + 2A_2 + \ldots + k A_k & =  & 0 B_0 + 1 B_1 + 2B_2 + \ldots + k B_k \\
0^2 A_0 + 1^2 A_1 + 2^2A_2 + \ldots + k^2 A_k & = & 0^2 B_0 + 1^2 B_1 + 2^2 B_2 + \ldots + k^2 B_k \\
\vdots & \vdots & \vdots \\
0^k A_0 + 1^k A_1 + 2^kA_2 + \ldots + k^k A_k & = & 0^k B_0 + 1^k B_1 + 2^k B_2 + \ldots + k^k B_k
\end{array}
\end{align}

The square matrix

$$\left( \begin{array}{cccc}
1 & 2 & \hdots & k \\
1^2 & 2^2 & \hdots & k^2 \\
\vdots & & \ddots & \vdots \\
1^k & 2^k & \hdots  & k^k
\end{array}\right)

is a so-called Vandermonde matrix, and is therefore invertible. Its determinant is a a so-called Vandermonde determinant, which in this case equals
$$1\cdot 2 \cdot \ldots \cdot k \cdot (k-1)\cdot (k-2)^2 \cdot \ldots \cdot 1^{k-1} \neq 0\, .$$

This implies that $A_1=B_1$, $A_2=B_2, \ldots, A_k=B_k$. Because $A_0+A_1+\ldots + A_n=B_0+B_1+\ldots+B_n=n$ it follows that also  $A_0=B_0$. This proves the proposition.

§ COMPOSITION

To see if our definition covers what we want, we may ask what is the hyperedge network corresponding to $f\circ g$ if $f$ and $g$ have hypernetwork structures ${\rm \bf N}_1$ and ${\rm \bf N}_2$ respectively? Do we get  hyperedges inside hyperedges and can we properly describe them with this formalism? It is clear how to make composite hyperedges: if $v_2 = s_i(v_1)$ of a hyperedge $h_1$ in ${\rm \bf N}_1$ and $v_3=s_j(v_2)$ of a hyperedge $h_2$ in ${\rm \bf N}_2$ then $v_3=s_{(i,j)}(v_1)$ for some new hyperedge $h_3$ is the composed hypernetwork in ${\rm \bf N}_1 \circ {\rm \bf N}_2$.  Here we replaced each source $v_2$ of  $h_1$ targeting $v_1$ by the entire collection of hyperedges that target $v_2$. This yields a new hyperedge, with new internal symmetries that form a wreath product.

§.§ First definition

A hypernetwork  is a tuple ${\rm \bf N} = (V, H, s, t)$ where $V$ is a finite set of vertices and $H$ are a finite set of hyperedges. The map $t: H \to V$ assigns to each hyperedge a unique target vertex. The map $s: H\to \cup_{k=1}^{\infty} V^k$ assigns to each hyperedge a finite list of sources. If $s(h)\in V^k$ then we call $k$ the inputdegree of the hyperedge.

Note that a hyperedge of inputdegree $1$ is a “normal” edge with only one source. Hyperedges can be used to encode nonlinear interaction:

We say that a map or vector field $f: \bigoplus_{v\in V} \mathbb{R}^{n_v} \to  \bigoplus_{v\in V} \mathbb{R}^{n_v}$ is ${\rm \bf N}$-admissible if the inequality

$$\frac{\partial^k f_{t}}{\partial x_{s_1} \cdots \partial x_{s_k} } \not \equiv 0 $$

implies that there is an $h\in H$ with with inputdegree $k$, $t(h)=t$  and  $s_1(h)=s_1, \ldots, s_k(h)=s_k$.

This definition is problematic because it implies that if there is a normal edge from $v_1$ to $v_2$ then there should also be a hyperedge from $(v_1, v_1)$ to $v_2$ etc. Otherwise, the dynamics of $v_2$ can only depend linearly on the state of $v_1$. Artificially including such hyperedges is not desirable.

Nevertheless, a hypernetwork gives rise to a class of admissible maps / vector fields with nonlinearities that respect the hypernetwork structure. We also have the following alternative definition that is probably more general, less intuitive and more useful in practice.

We say that a map or vector field $f: \bigoplus_{v\in V} \mathbb{R}^{n_v} \to  \bigoplus_{v\in V} \mathbb{R}^{n_v}$ is ${\rm \bf N}$-admissible if it is of the form

$$f_v(x) = F_v\left( \bigoplus_{t(h)=v} x_{s(h)} \right) $$

for some functions $F_v$ depending on possibly many variables repeated. Here, $x_{s(h)} = (x_{s_1(h)}, \ldots, x_{s_k(h)})$ with $k$ the inputdegree of $h$.

Our definition implies that not all hypernetworks give rise to a nonempty class of admissible maps and vector fields. In particular, the definition implies that the hypergraph is input-symmetric: if we permute the sources of a hyperedge  $h\in H$ then we obtain another hyperedge in $H$:

For $h\in H$ with inputdegree $k$ and $\sigma: \{1,\ldots,k\}\to \{1, \ldots, k\}$ a permutation, define the new hyperedge ${\sigma}_*h$ as the hyperedge with the same inputdegree, the same target, and

$$s_j(\sigma_*h) := \sigma(s_j(h))\, .$$

The formula $(\tau \sigma)_*h=\tau_*\sigma_*h$ explains the notation with the lower $*$. With the property that $h\in H \implies \sigma_*h\in H$ showing up, it would perhaps have been more sensible to define the source of a hyperedge as a multiset (=unordered list possibly with repetition) of vertices rather than a list of vertices.

We also necessarily have the property that, if $h$ is a hyperedge, then also all its faces are hyperedges:

If $h \in H$ and $s(h) = (s_1(h), \ldots, s_k(h))$. For $1\leq i\leq k$, the $i$-th face of $h$ is the hyperedge $h^{(i)}$ with $t(h^{(i)})=t(h)$ and

$$s_j(h^{(i)}) = \left\{ \begin{array}{ll}
s_j(h) & \mbox{when}\ 1\leq  j < i \, , \\
s_{j+1}(h) & \mbox{when}\ i \leq j \leq k-1\, .
\end{array} \right.

Indeed, if a higher order partial derivative is nonzero, then so are the lower order partial derivatives occurring in it.

Thus, the natural hypernetworks to consider so that Definition <ref> makes sense, have the following two properties:

i) When $h\in H$ then $h^{(i)}\in H$ for all $1\leq  i \leq k$ where $k$ is the inputdegree of $h$.

ii) When $h\in H$ then ${\sigma}_*h\in H$ where $\sigma$ represents any permutation of the sources of $h$.

Interestingly, the first of these two property is reminiscent of a simplicial complex (in that the faces of a simplex are also simplices).

§.§ Second definition

In this section we try to give a different definition of hypernetworks and admissible maps that allows to talk about balanced colourings etc.

A coloured hypernetwork is a hypernetwork ${\rm \bf N}$ in which all vertices and hyperedges are assigned a colour (chosen from some finite set of colours), such that

1. if two hyperedges $h_1, h_2\in H$ have the same colour, then their targets $t(h_1)$ and $t(h_2)$ have the same colour, the inputdegrees of $h_1$ and $h_2$ are equal, and the sources $s_j(h_1)$ and $s_j(a)$ have the same colour for each $j$.

2. if two vertices $v_1, v_2\in V$ have the same colour, then there is at least one colour-preserving bijection $\beta :t^{-1}(v_1)\to t^{-1}(v_2)$ between the hyperedges that target $v_1$ and $v_2$.

3. For any colour preserving bijection $\beta: t^{-1}(v_1)\to t^{-1}(v_2)$  between the hyperedges that target $v_1$ and $v_2$,
we have
$$\beta(h)^{(i)} = \beta(h^{(i)})\, .$$

So $\beta$ respects the faces of each hyperedge.

In ${\rm \bf N}$ is a coloured network, then we say that a map or vector field $f: \bigoplus_{v\in V} \mathbb{R}^{n_v} \to  \bigoplus_{v\in V} \mathbb{R}^{n_v}$ is ${\rm \bf N}$-admissible if it is admissible as in the previous definition, and
moreover, if $v_1$ and $v_2$ have the same colour, and $\beta:t^{-1}(v_1)\to t^{-1}(v_2)$ is one of the colour-preserving bijections between their targeting hyperedges, we have the formula

$$F_{v_1}\left(\bigoplus_{t(h)=v_1} x_{s(\beta(h))} \right) = F_{v_2}\left( \bigoplus_{t(h)=v_2} x_{s(h)} \right) $$

§ EXAMPLE

$$\dot x_1 = b x_2^2 + b x_3 x_4$$

that we would like to consider a hyperedge network with two hyperedges: $h_1$ from $(2, 2)$ to $1$ and $h_2$ from $(3, 4)$ to $1$. Thus we would like to write the RHS as

$$F_1(x_2, x_2, x_3, x_4)$$

where the interchangeability of the hyperedges $h_1$ and $h_2$ corresponds to the interchangeability of the first two variables with the second two variables. But also the first two can be interchanged, and the second two. Any linear system of this form is of the type
$$F_1(x_2, x_2, x_3, x_4) = a( 2 x_2 +  x_3 +  x_4)$$

which is in fact a linear fully  symmetric function  of $(x_2, x_2, x_3, x_4)$. Thus it naturally has a larger groupoid.

So what we see here is that there is a group ($S_4$) of colour preserving bijections of the faces of the hyperedges that is broken into a wreath product $\mathbb{Z}_2 \wr \mathbb{Z}_2$ by adding the hyperedges.

So instead we could write

$$F_1 = F_1(x_2, x_2, x_3, x_4, x_2, x_2, x_3, x_4) = a(2x_2 + x_3 + x_4) + b(x_2^2 + x_3 x_4) $$

where first four dependencies are through edges and the second four through hyperedges. The symmetries are

$$F_1(\overline{x_2, x_2, x_3, x_4}, \overline{ \overline{x_2, x_2}, \overline{x_3, x_4}} )$$