# Heteroclinic Dynamics of Localized Frequency Synchrony: Heteroclinic Cycles for Small Populations

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## Abstract

Many real-world systems can be modeled as networks of interacting oscillatory units. Collective dynamics that are of functional relevance for the oscillator network, such as switching between metastable states, arise through the interplay of network structure and interaction. Here, we give results for small networks on the existence of heteroclinic cycles between dynamically invariant sets on which the oscillators show localized frequency synchrony. Trajectories near these heteroclinic cycles will exhibit sequential switching of localized frequency synchrony: a population oscillators in the network will oscillate faster (or slower) than others and which population has this property sequentially changes over time. Since we give explicit conditions on the system parameters for such dynamics to arise, our results give insights into how network structure and interactions (which include higher-order interactions between oscillators) facilitate heteroclinic switching between localized frequency synchrony.

## Keywords

Oscillator networks Phase oscillators Higher-order interactions Weak chimera Symmetry Heteroclinic cycle## Mathematics Subject Classification

34C15 34C28 34C37 34D06 37C29 37C80## 1 Introduction

Networks of interacting oscillatory units can give rise to dynamics where the system appears to be in one metastable state before “switching” to another in a rapid transition. Such dynamics are in particular believed to be of functional relevance for neuronal networks where one observes sequential switching between patterns involving localized activity or synchrony (Ashwin and Timme 2005; Britz et al. 2010; Tognoli and Scott Kelso 2014). One approach is to capture these dynamics on a macroscopic scale: One assigns each pattern an activity variable whose dynamics are then described by kinetic equations (Rabinovich et al. 2006). The resulting equations are of generalized Lotka–Volterra type which support stable heteroclinic cycles, that is, cycles of hyperbolic equilibria which are connected by heteroclinic trajectories. The dynamics near such heteroclinic cycles now resemble sequential switching dynamics of activity patterns. Indeed, heteroclinic cycles and networks have been long studied in their own right; see Weinberger and Ashwin (2018) for a recent review.

However, such a qualitative approach fails to capture the dynamics on the level of single, nonlinearly interacting oscillators. In particular, it does not necessarily illuminate what ingredients of network topology and the interactions between oscillators (Stankovski et al. 2017) facilitate switching dynamics. If one assumes weak coupling, phase reduction provides a powerful tool to describe the dynamics of an oscillator network; in this reduction, each oscillator is represented by a single phase variable on the torus \({\mathbf {T}}:= {\mathbb {R}}/2\pi {\mathbb {Z}}\) and the dynamics of the phases are described by a phase oscillator network. Simple networks of globally and identically coupled phase identical oscillators support heteroclinic cycles and networks (Ashwin et al. 2007; Bick et al. 2016). The equilibria involved in these cycles are phase-locking patterns with oscillators in different clusters which have a constant phase difference. The symmetry properties of these networks, however, imply that all oscillators rotate with the same speed (frequency) on average—the network is globally frequency synchronized. In a neural network, this corresponds to all neurons firing at the same average rate while the exact timing of firing changes.

By contrast, even networks of identical phase oscillators that are organized into different populations can give rise to dynamics where frequency synchrony is local to a population rather than global across the whole network. In other words, the interactions in a network of identical oscillators cause some units to evolve faster (or slower) than others. Dynamically invariant sets with this property relate to “chimeras” (Panaggio and Abrams 2015; Schöll 2016; Omel’chenko 2018) which have—as patterns with localized frequency synchrony—been hypothesized to play a functional role in the context of neuroscience (Shanahan 2010; Tognoli and Scott Kelso 2014; Bick and Martens 2015). From a mathematical point of view, the notion of a weak chimera (Ashwin and Burylko 2015; Bick and Ashwin 2016; Bick 2017) formalizes the definition of a dynamically invariant set with localized frequency synchrony for finite networks of identical phase oscillators.

Here we prove the existence of robust heteroclinic cycles between invariant sets with localized frequency synchrony in small phase oscillator networks with higher-order interactions. In contrast to attracting sets with localized frequency synchrony, the dynamics here induce sequential switching dynamics: Which population of oscillators oscillates at a faster (or slower) rate will change over time. These results are of interest from several distinct perspectives. First, they illuminate how the interplay of network structure and functional interactions between units gives rise to heteroclinic dynamics in phase oscillator networks: We explicitly relate the network coupling parameters to the existence of heteroclinic cycles. Second, the results highlight how higher-order network interactions shape the (global) network dynamics; apart from higher harmonics in the phase interaction function, the higher-order interactions also include nonadditive interactions between oscillator phases which arise naturally in phase reductions of generically coupled identical oscillators (Ashwin and Rodrigues 2016) or other resonant interactions (Komarov and Pikovsky 2013). Here, the interplay of higher-order interactions and nontrivial network topology induces dynamics beyond (full) synchrony. Third, our results provide new examples of heteroclinic cycles in network dynamical systems relevant for applications. We highlight how these examples are distinct from situations previously considered in the literature.

This work is organized as follows. In this paper, we build on results in a recent brief communication (Bick 2018) to prove the existence of robust heteroclinic cycles between localized frequency synchrony; in a companion paper (Bick and Lohse 2019) we give a detailed discussion of the stability of such heteroclinic cycles (which may be embedded into larger heteroclinic structures). The remainder of this paper is organized as follows. In Sect. 2 we review some preliminaries on heteroclinic cycles and phase oscillator networks. In Sect. 3 we show existence of a heteroclinic cycle between localized patterns of frequency synchrony in networks consisting of three populations of two oscillators. In Sect. 4 we consider networks which consist of three populations of three oscillators and show the existence of a heteroclinic cycle of localized frequency synchrony; here, there are continua of saddle connections in two-dimensional invariant subspaces. Finally, in Sect. 5, we give some numerical evidence that these phenomena persist in networks with more generic interactions before giving some concluding remarks.

## 2 Preliminaries

### 2.1 Heteroclinic Cycles

Let \({\mathcal {M}}\) be a smooth *d*-dimensional manifold and let *X* be a smooth vector field on \({\mathcal {M}}\). For a hyperbolic equilibrium \(\xi \in {\mathcal {M}}\) let \(W^ s (\xi )\) and \(W^ u (\xi )\) denote its stable and unstable manifold, respectively.

### Definition 2.1

*heteroclinic cycle*\({\mathbf {\mathsf {C}}}\) consists of a finite number of hyperbolic equilibria \(\xi _q\in {\mathcal {M}}\), \(q=1,\cdots ,Q\), together with heteroclinic trajectories

*Q*.

For simplicity, we write \({\mathbf {\mathsf {C}}}=(\xi _1, \cdots , \xi _Q)\). If \({\mathcal {M}}\) is a quotient of a higher-dimensional manifold and \({\mathbf {\mathsf {C}}}\) is a heteroclinic cycle in \({\mathcal {M}}\), we also call the lift of \({\mathbf {\mathsf {C}}}\) a heteroclinic cycle.

While heteroclinic cycles are in general a nongeneric phenomenon, they can be robust in dynamical systems with symmetry. Let \(\Gamma \) be a finite group which acts on \({\mathcal {M}}\). For a subgroup \(H\subset \Gamma \) define the set \({{\,\mathrm{Fix}\,}}(H) = \left\{ \, x\in {\mathcal {M}}\,\left| \;\gamma x=x\ \forall \gamma \in H\right. \right\} \) of points fixed under *H*; any \({{\,\mathrm{Fix}\,}}(H)\) is invariant under the flow generated by *X*. For \(x\in {\mathcal {M}}\) let \(\Gamma x=\left\{ \, \gamma x\,\left| \;\gamma \in \Gamma \right. \right\} \) denote its group orbit and \(\Sigma (x) = \left\{ \, \gamma \in \Gamma \,\left| \;\gamma x = x\right. \right\} \) its *isotropy subgroup*. Now assume that the smooth vector field *X* is \(\Gamma \)-equivariant vector field on \({\mathcal {M}}\), that is, the action of the group commutes with *X*.

Now let \({\mathbf {\mathsf {C}}}=(\xi _1, \cdots , \xi _Q)\) be a heteroclinic cycle with the following properties. For an isotropy subgroup \(\Sigma _q\subset \Gamma \) write \(P_q={{\,\mathrm{Fix}\,}}(\Sigma _q)\). Now suppose that there exist \(\Sigma _q\) (and thus \(P_q\)) such that \(\xi _q, \xi _{q+1}\in P_q\), \(\xi _{q+1}\) is a sink in \(P_q\), and \([\xi _q\rightarrow \xi _{q+1}]\subset P_q\). Then \({\mathbf {\mathsf {C}}}\) is *robust* with respect to \(\Gamma \)-equivariant perturbations of *X*, that is, \(\Gamma \)-equivariant vector fields close to *X* will have a heteroclinic cycle close to \({\mathbf {\mathsf {C}}}\); see Krupa (1997) for details.

#### 2.1.1 Dissipative Heteroclinic Cycles

Trajectories close to a heteroclinic cycle can show switching dynamics: Qualitatively speaking, the trajectory will spend time close to one saddle \(\xi _q\) before a rapid transition to \(\xi _{q+1}\). This is in particular the case when the heteroclinic cycle is attracting (in some sense); see for example Weinberger and Ashwin (2018) for a more elaborate discussion.

*X*at \(\xi \) ordered such that

*saddle value*(or saddle index) \(\nu (\xi )=-\frac{{{\,\mathrm{Re}\,}}\lambda ^{(l)}}{{{\,\mathrm{Re}\,}}\lambda ^{(d)}}\) compares the rates of minimal attraction and maximal expansion close to \(\xi \); cf. Shilnikov et al. (1998) and Afraimovich et al. (2016). In particular, we say that \(\xi \) is

*dissipative*if \(\nu (\xi )>1\). For a heteroclinic cycle \({\mathbf {\mathsf {C}}}=(\xi _1, \cdots , \xi _Q)\), write \(\nu _q\, {:=}\,\nu (\xi _q)\).

### Definition 2.2

A heteroclinic cycle \({\mathbf {\mathsf {C}}}\) is *dissipative* if \(\nu ({\mathbf {\mathsf {C}}})\, {:=}\, \prod _q\nu _q>1.\)

Intuitively speaking, a heteroclinic cycle is dissipative if there is more contraction of phase space than expansion close to the saddle points. Obviously, a heteroclinic cycle is dissipative if all its equilibria are dissipative. Subject to suitable additional assumptions, we may expect a dissipative heteroclinic cycle to be asymptotically stable:

### Proposition 2.3

Here we restrict ourselves to show the existence of dissipative heteroclinic cycles; we address the problem of stability explicitly in the companion paper (Bick and Lohse 2019).

#### 2.1.2 Cyclic Heteroclinic Chains

Definition 2.1 of a heteroclinic cycle makes no assumptions on the number of heteroclinic trajectories between equilibria. Indeed, if there are unstable manifolds of dimension larger than one, there may be continua of heteroclinic trajectories. In that case, the question about stability is more challenging as discussed by Ashwin and Chossat (1998), in particular because the condition in Remark 2.3 does not allow any set of points (however small) on the unstable manifold of one saddle to lie outside of the stable manifold of the next saddle.

*cyclic*if each vertex has unique edges entering and leaving it.

### Definition 2.4

*associated heteroclinic chain*

*cyclic heteroclinic chain*.

In contrast to Definition 2.1, the heteroclinic chain associated to a heteroclinic cycle now contain all heteroclinic trajectories which connect individual equilibria. Note that heteroclinic chains do not need to be closed in \({\mathcal {M}}\): Some part of \(W^ u (\xi _q)\) for some *q* may lie outside of the heteroclinic chain.

### 2.2 Phase Oscillator Networks with Nonpairwise Interactions

*k*in population \(\sigma \). Hence, the state of the oscillator network is determined by \(\theta =(\theta _{1}, \cdots , \theta _{M})\in {\mathbf {T}}^{NM}\) where \(\theta _\sigma = (\theta _{\sigma , 1}, \cdots , \theta _{\sigma , N})\in {\mathbf {T}}^N\) is the state of population \(\sigma \). Let \(g_2, g_4: {\mathbf {T}}\rightarrow {\mathbb {R}}\) be smooth \(2\pi \)-periodic functions and

^{1}. For these network dynamics, the phase interactions within populations are determined by the coupling (or phase interaction) function \(g_2\) evaluated at phase differences of oscillator pairs. By contrast, the interactions between populations, given by (1), are mediated by the nonpairwise interaction function \(g_4\) evaluated at linear combination of four of the oscillators’ phases. The parameter \(K^{-}>0\) determines the coupling strength to the previous population, whereas \(K^{+}>0\) determines the coupling strength to the previous population. Here we assume \(K\,{:=}\, K^{-} = K^{+}>0\) for simplicity. For \(g_4 = \cos \), the equations (2) approximate the dynamics of a phase oscillator networks with mean-field mediated bifurcation parameters up to rescaling of time as outlined in Bick (2018).

#### 2.2.1 Symmetries and Invariant Sets

*phase synchrony*and

*splay phase*configuration—typically we call any element of the group orbit \({\mathbf {S}}_N\mathrm {D}\) a splay phase. For a network of interacting populations, we use the shorthand notation

The network interactions in (2), which include nonpairwise coupling, induce symmetries. More precisely, the equations (2) are \(({\mathbf {S}}_N\times {\mathbf {T}})^M\rtimes {\mathbb {Z}}_M\)-equivariant. Each copy of \({\mathbf {T}}\) acts by shifting all oscillator phases of one population by a common constant while \({\mathbf {S}}_N\) permutes its oscillators. The action of \({\mathbb {Z}}_M\) permutes the populations cyclically. These actions do not necessarily commute.

To reduce the phase-shift symmetry \({\mathbf {T}}^{M}\), we rewrite (2) in terms of phase differences \(\psi _{\sigma ,k}\, {:=}\, \theta _{\sigma , k+1} - \theta _{\sigma , 1}\), \(k=1, \cdots , N-1\). Hence, with \(\psi _\sigma \in {\mathbf {T}}^{N-1}\) we also write for example \(\psi _1\mathrm {S}\cdots \mathrm {S}\) (or simply \(\psi \mathrm {S}\cdots \mathrm {S}\) if the index is obvious) to indicate that all but the first population is phase synchronized.

The symmetries yield invariant subspaces on \({\mathbf {T}}^{MN}\) for the dynamics given by (2). In particular, the \({\mathbf {S}}_N\) permutational symmetries within each population imply that the sets (5) are invariant (Ashwin and Swift 1992). Moreover, any set of the form \(\theta _1\cdots \theta _M\) with \(\theta _\sigma \in \left\{ \mathrm {S},\mathrm {D}\right\} \) is an equilibrium relative to the continuous \({\mathbf {T}}^{M}\) symmetry, that is, the corresponding \(\psi _1\cdots \psi _M\) is an equilibrium in the reduced dynamics.

#### 2.2.2 Frequencies and Localized Frequency Synchrony

*instantaneous angular frequency*of oscillator \((\sigma , k)\), define the

*asymptotic average angular frequency*of oscillator \((\sigma , k)\) by

### Definition 2.5

*localized frequency synchrony*if for any \(\theta ^0\in A\) we have \(\Omega _{\sigma ,k} = \Omega _{\sigma }\) and there exist indices \(\sigma \ne \tau \) such that

### Remark 2.6

Note that a chain-recurrent set *A* with localized frequency synchrony is a *weak chimera* as defined by Ashwin and Burylko (2015).

### Lemma 2.7

(Theorem 1 in Ashwin and Burylko 2015) The system symmetries imply \(\Omega _{\sigma ,k} = \Omega _{\sigma ,j}\).

## 3 Heteroclinic Cycles for Two Oscillators per Population

The phase space of (9) is organized by invariant subspaces as sketched in Fig. 1. For completeness, we characterize \({\mathrm {S}\mathrm {S}\mathrm {S}}\) and \({\mathrm {D}\mathrm {D}\mathrm {D}}\) before we focus on sets with localized frequency synchrony.

### 3.1 Saddle Invariant Sets with Localized Frequency Synchrony

In the following we estimate the asymptotic average frequencies of \({\mathrm {D}\mathrm {S}\mathrm {S}}\), \({\mathrm {D}\mathrm {D}\mathrm {S}}\), \({\mathrm {S}\mathrm {D}\mathrm {S}}\), \({\mathrm {S}\mathrm {D}\mathrm {D}}\), \({\mathrm {S}\mathrm {S}\mathrm {D}}\), \({\mathrm {D}\mathrm {S}\mathrm {D}}\), and \({\mathrm {D}\mathrm {S}\mathrm {S}}\). Note that it suffices to consider \({\mathrm {D}\mathrm {S}\mathrm {S}}\), \({\mathrm {D}\mathrm {D}\mathrm {S}}\) since the latter four are their images under the \({\mathbb {Z}}_M\) action which permutes populations.

### Lemma 3.1

### Proof

By Lemma 2.7 we have \(\Omega _{\sigma } = \Omega _{\sigma ,k}\) for all \(k=1, \cdots ,N\), that is, all oscillators within a single population have the same asymptotic average angular frequency.

If the populations are uncoupled, \(K=0\), we have \(\Omega _{1}(\theta ^0) = \omega +g_2(0)\) for \(\theta ^0\in \mathrm {S}\psi _2\psi _3\) and \(\Omega _{2}(\theta ^0) = \omega +g_2(\pi )\) for \(\theta ^0\in \psi _1\mathrm {D}\psi _3\). This implies that for \(K\ge 0\) and coupling (15) we have \(\left| g_2(0)-g_2(\pi )\right| -2K = \left| 2\sin (\alpha _2)\right| -2K \le \left| \Omega _{1}(\theta ^0)-\Omega _{2}(\theta ^0)\right| \) for \(\theta ^0\in \mathrm {S}\mathrm {D}\psi _3\). Consequently, \({\mathrm {D}\mathrm {S}\mathrm {S}}\), \({\mathrm {D}\mathrm {D}\mathrm {S}}\) and their symmetric counterparts have localized frequency synchrony on \({\mathbf {T}}^{MN}\) if (C\(\Omega \)N2) is satisfied. \(\square \)

Note that this is clearly only a sufficient condition; it suffices for our purpose but can be improved by evaluating the asymptotic average angular frequencies explicitly.

### 3.2 Heteroclinic Cycles

In the previous section, we evaluated the local properties of the dynamically invariant sets \({\mathrm {D}\mathrm {S}\mathrm {S}}\) and \({\mathrm {D}\mathrm {D}\mathrm {S}}\). Heteroclinic cycles require conditions on the local stability as well as the existence of global saddle connections.

### Lemma 3.2

### Proof

### Remark 3.3

Note that here the local conditions (C\(\lambda \)N2’) suffice to guarantee the existence of global saddle connections. This indicates that more than two harmonics are needed for generic bifurcation behavior; cf. Corollary 1 in Ashwin et al. (2016).

By replacing (C\(\lambda \)N2’) with a stricter set of conditions, we immediately obtain the following statement.

### Lemma 3.4

This leads to the main result of this section.

### Theorem 3.5

### Proof

Note that if (C\(\lambda \)N2’)—or (C\(\lambda \)N2)—holds, the heteroclinic trajectories in Lemma 3.2 are source-sink connections in an invariant subspace forced by symmetry. Hence, to prove the assertion it suffices to show that there are indeed parameter values such that (C\(\Omega \)N2), (C\(\lambda \)N2), and (C\(\nu \)N2) are satisfied simultaneously.

Again, conditions (C\(\Omega \)N2), (C\(\lambda \)N2), and (C\(\nu \)N2) for the existence of a dissipative robust heteroclinic cycle are sufficient. Figure 2 illustrates the region in parameter space that is given by these conditions.

### Remark 3.6

For \({\mathbf {\mathsf {C}}}_2\) set \(\xi _1 = {\mathrm {D}\mathrm {S}\mathrm {S}}, \xi _2={\mathrm {D}\mathrm {D}\mathrm {S}}, \cdots , \xi _6={\mathrm {D}\mathrm {S}\mathrm {D}}\). We have \(W^ u (\xi _q)\smallsetminus \left\{ \xi _q\right\} \subset W^ s (\xi _{q+1})\) and thus the heteroclinic chain associated with \({\mathbf {\mathsf {C}}}_2\) is closed.

## 4 Heteroclinic Cycles for Three Oscillators per Population

In this section, we show that there are dissipative robust heteroclinic cycles for networks of \(M=3\) populations of \(N=3\) oscillators with coupling (24). Indeed, we proceed as before and derive conditions for which there are heteroclinic source-sink connections on invariant subspaces forced by symmetry. The resulting heteroclinic network will be robust. Thus, for the remainder of the section we set \((\alpha _2,\alpha _4)=(\frac{\pi }{2},\pi )\) rather writing down the conditions in full generality.

A stability analysis of the phase configurations \({\mathrm {S}\mathrm {S}\mathrm {S}}\) and \({\mathrm {D}\mathrm {D}\mathrm {D}}\) can be done as in the previous section. Moreover, we restrict ourselves to \({\mathrm {D}\mathrm {S}\mathrm {S}}\) and \({\mathrm {D}\mathrm {D}\mathrm {S}}\) because of symmetry.

### 4.1 Local Dynamics

We first establish conditions for \({\mathrm {D}\mathrm {S}\mathrm {S}}\) and \({\mathrm {D}\mathrm {D}\mathrm {S}}\) to be invariant sets with localized frequency synchrony which are suitable saddles in the reduced system.

### Lemma 4.1

### Proof

The proof is essentially the same as for Lemma 3.1. Frequency synchrony with populations is given by Lemma 2.7.

For \(K=0\) we have \(\Omega _{1}(\theta ^0) = \omega +2g_2(0)=\omega +2\) for \(\theta ^0\in {\mathrm {S}\psi _2\psi _3}\) and \(\Omega _{2}(\theta ^0) = \omega +g_2(2\pi /3)+g_2(4\pi /3)=\omega -1\) for \(\theta ^0\in {\psi _1\mathrm {D}\psi _3}\). This implies that for \(K\ge 0\) and coupling (24) we have \(3-\frac{4}{3}K\le \left| \Omega _{1}(\theta ^0)-\Omega _{2}(\theta ^0)\right| \) Thus \({\mathrm {D}\mathrm {S}\mathrm {S}}\), \({\mathrm {D}\mathrm {D}\mathrm {S}}\) have localized frequency synchrony on \({\mathbf {T}}^{MN}\) if (C\(\Omega \)N3) is satisfied. \(\square \)

### Lemma 4.2

### Proof

Note that expansion at \({\mathrm {D}\mathrm {S}\mathrm {S}}\) is determined by the double real eigenvalue \(\lambda _2^{\mathrm {D}\mathrm {S}\mathrm {S}}\) forced by symmetry. The eigenvalues of the linearization (25) and (26) also give insight into the local bifurcations as parameters are varied. For example, \({\mathrm {D}\mathrm {D}\mathrm {S}}\) undergoes a Hopf bifurcation as the parameter *r* goes through zero.

### 4.2 Global Dynamics

In the previous section we established the existence of suitable saddle invariant sets. To obtain a heteroclinic cycle, these invariant sets have to be joined by a heteroclinic trajectory.

### Lemma 4.3

### Proof

*V*are depicted in Fig. 3a.) That is, we have

We will first consider the dynamics on \({\mathrm {D}\psi \mathrm {S}}\) and show that there is a source-sink heteroclinic trajectory \([{\mathrm {D}\mathrm {S}\mathrm {S}}\rightarrow {\mathrm {D}\mathrm {D}\mathrm {S}}]\). By assumption (C\(\lambda \)N3), we have that \({\mathrm {D}\mathrm {S}\mathrm {S}}\) is a source in \({\mathrm {D}\psi \mathrm {S}}\) with \(W^ u ({\mathrm {D}\mathrm {S}\mathrm {S}})\subset {\mathrm {D}\psi \mathrm {S}}\). In the following, we derive conditions on \(K\) and *r* such that *V* is a (Lyapunov-like) potential function on \({\mathcal {C}}\) which guarantee that *V* is strictly increasing along trajectories in \({\mathcal {C}}\). Thus, any trajectory in \({\mathcal {C}}\) converges to \(\mathrm {D}\in {\mathcal {C}}\) which yields a heteroclinic trajectory \([{\mathrm {D}\mathrm {S}\mathrm {S}}\rightarrow {\mathrm {D}\mathrm {D}\mathrm {S}}]\).

*V*increases along trajectories if \(\dot{V}>0\) on \({\mathcal {C}}\smallsetminus \mathrm {D}\). Rearranging terms, \(\dot{V}>0\) is equivalent to

*Q*has a unique maximum on \({\mathcal {C}}\) at \(\psi =\mathrm {D}\) with \(Q(\mathrm {D}) = 1\). Thus, \(\dot{V} > 0\), and trajectories in \({\mathcal {C}}\) approach \(\mathrm {D}\) asymptotically.

*R*is removable since \(\langle {{\,\mathrm{grad}\,}}V, X_r\rangle \) also vanishes on \({\mathcal {C}}\smallsetminus \mathrm {D}\) at the same order. Hence, \(R(\psi )\) is a bounded function on \({\overline{{\mathcal {C}}}}\) and evaluating minima and maxima on \({\overline{{\mathcal {C}}}}\) yields \(\left\| R\right\| _{\mathcal {C}}<15\); cf. Fig. 3b. This implies that \(\dot{V} > 0\) on \({\mathcal {C}}\smallsetminus \mathrm {D}\) if

### Lemma 4.4

### Proof

### Theorem 4.5

The network of \(M=3\) populations of \(N=3\) phase oscillators with dynamics (2) and coupling functions (24) has a robust dissipative heteroclinic cycle Open image in new window between dynamically invariant sets with localized frequency synchrony.

### 4.3 Nonclosed Heteroclinic Chains

For networks of \(N=3\) oscillators per population, there are continua of heteroclinic connections between equilibria. In particular, the heteroclinic chain associated with \({\mathbf {\mathsf {C}}}_3\) contains all heteroclinic trajectories \([{\mathrm {D}\mathrm {S}\mathrm {S}}\rightarrow {\mathrm {D}\mathrm {D}\mathrm {S}}]\), \([{\mathrm {D}\mathrm {D}\mathrm {S}}\rightarrow {\mathrm {S}\mathrm {D}\mathrm {S}}]\) of trivial isotropy. While the resulting associated heteroclinic chain \({\mathfrak {H}}({\mathbf {\mathsf {C}}}_3)\) is cyclic, it is not closed. In particular, the condition of Remark 2.3 is not satisfied.

To formalize this observation, we adapt some terminology that was recently introduced in Ashwin et al. (2018).

### Definition 4.6

- (i)
\(\xi _q\) is

*complete in*\({\mathfrak {H}}({\mathbf {\mathsf {C}}})\) if \(W^ u (\xi _q)\subset {\mathfrak {H}}({\mathbf {\mathsf {C}}})\), - (ii)
\(\xi _q\) is

*almost complete in*\({\mathfrak {H}}({\mathbf {\mathsf {C}}})\) if \(W^ u (\xi _q)\smallsetminus {\mathfrak {H}}({\mathbf {\mathsf {C}}})\) is of measure zero (with respect to the Lebesgue measure for any volume form on \(W^ u (\xi _q)\)), - (iii)
\(\xi _q\) is

*equable in*\({\mathfrak {H}}({\mathbf {\mathsf {C}}})\) if there is a \(d = d(q)\in {\mathbb {N}}\) such that for all*p*with \(C_{qp}\ne \emptyset \) the set \(C_{qp}\) is a manifold with \(\dim (C_{qp})=d\).

Note that completeness relates to *clean* heteroclinic networks defined in Field (2017). With these notions we obtain the following statement.

### Theorem 4.7

For parameters satisfying (35) and *r* sufficiently small, the heteroclinic chain Open image in new window is cyclic, equable, and almost complete but not complete. The closure of Open image in new window is complete, but neither cyclic nor equable.

### Proof

Note that the heteroclinic chain \({\mathfrak {H}}({\mathbf {\mathsf {C}}}_3)\) associated with \({\mathbf {\mathsf {C}}}_3\) is cyclic—because \(W^ u ({\mathrm {D}\mathrm {S}\mathrm {S}})\subset {\mathrm {D}\psi \mathrm {S}}\) and \(W^ u ({\mathrm {D}\mathrm {D}\mathrm {S}})\subset {\psi \mathrm {D}\mathrm {S}}\)—and thus equable.

*r*sufficiently small by assumption, there are exactly three equilibria \(\xi ^{\mathrm {D}\psi \mathrm {S}}\) on \(\partial {\mathcal {C}}\) (which lie in the same group orbit) with \(\chi \approx 2\arctan (3K)\). These are of saddle type, attracting within \(\partial {\mathcal {C}}\) and transversely repelling (within \({\mathrm {D}\psi \mathrm {S}}\)); see Fig. 3c. Therefore,

## 5 Dynamics of Networks with Noise and Broken Symmetry

With further forced symmetry breaking, \(\delta _\mathrm {sym}, \delta _\mathrm {asym}>0\), the phase oscillator network (37) exhibits irregular switching of localized frequency synchrony even in the absence of noise. These potentially chaotic dynamics arise close to the heteroclinic networks \({\mathbf {\mathsf {C}}}_N\), \(N=2,3\), as shown in Fig. 7.

## 6 Discussion and Conclusions

Phase oscillator networks with higher-order interactions can give rise to heteroclinic cycles between frequency synchrony; in numerical simulations these lead to sequential acceleration and deceleration of oscillator populations. Indeed, because of dissipativity, we expect that the attractor of the deterministic system is a subset of the closure of the associated heteroclinic chain. For networks of \(N=2\) oscillators in each population we calculate the stability of the heteroclinic cycles and their bifurcations explicitly in the companion paper (Bick and Lohse 2019). For \(N=3\) the unstable manifold of each saddle is (at least) two-dimensional and the assumptions to apply existing stability results (Krupa and Melbourne 1995; Ashwin and Chossat 1998) are not satisfied. We will address this question in future research.

Rather than assuming weak coupling between populations, the results presented here rely on the symmetries induced by the nonpairwise higher-order network interaction terms. Our numerical simulations for nearby vector fields where these symmetries were broken indicated the persistence of some residual heteroclinic structure. In this context, it would be desirable to extend the methods of forced symmetry breaking (Sandstede and Scheel 1995; Chossat and Field 1995; Guyard and Lauterbach 1999) to understand the bifurcation behavior for nearby network vector fields with generic interactions.

Numerical simulations indicate that switching dynamics between localized frequency synchrony also arises in networks with \(M=3\) populations of \(N>3\) phase oscillators (Bick 2018). Indeed, the methods used here are likely applicable to such networks as well: Without higher harmonics, \(r=0\), the oscillators are sinusoidally coupled and the phase space \({\mathbf {T}}^{MN}\) is foliated by low-dimensional manifolds (Pikovsky and Rosenblum 2011; Chen et al. 2017) on which we expect to have a similar potential functions as in the proof of Lemma 4.3. While we cannot expect hyperbolicity in this limit due to the degeneracy in the system, suitable network interaction terms with higher harmonics, combined with an approach similar to Vlasov et al. (2016), could give rigorous results to show the existence of heteroclinic networks.

In summary, heteroclinic switching dynamics between localized frequency synchrony may arise in networks of identical phase oscillators with higher-order interactions. Here we gave rigorous results for small oscillator networks, but we anticipate similar approaches to be viable for larger networks. While the heteroclinic switching observed here are distinct from those discussed in Komarov and Pikovsky (2011), where large networks (\(N\ge 1000\)) of nonidentical oscillators are considered, it would be interesting to relate the two.

## Footnotes

- 1.
Without loss of generality, we may set \(\omega \) to any value by going into a suitable co-rotating frame.

## Notes

### Acknowledgements

The author is indebted to P Ashwin, JSW Lamb, A Lohse, and M Rabinovich for helpful conversations.

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