# Continua and persistence of periodic orbits in ensembles of oscillators

R Ronge1, M A Zaks1, T Pereira2,3 1 Institut für Physik, Humboldt-Universität
zu Berlin, 12489 Berlin, Germany  2 Institute of Mathematical and Computer
Sciences, University of São Paulo, Brazil  3 Department of Mathematics,
Imperial College London, London SW7 2AZ, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

Certain systems of coupled identical oscillators like the Kuramoto-Sakaguchi
or the active rotator model possess the remarkable property of being Watanabe-
Strogatz integrable. We prove that such systems, which couple via a global
order parameter, feature a normally attracting invariant manifold that is
foliated by periodic orbits. This allows us to study the asymptotic dynamics
of general ensembles of identical oscillators by applying averaging theory.
For the active rotator model, perturbations result in only finitely many
persisting orbits, one of them giving rise to splay state dynamics. This sheds
some light on the persistence and typical behavior of splay states previously
observed.

††: Nonlinearity

Keywords: Watanabe-Strogatz integrability, phase oscillators, active rotator,
normal hyperbolicity

##  1 Introduction

Studies of synchronization phenomena in ensembles of self-sustained
oscillators or excitable elements have found numerous applications, from
physics [1, 2] over engineering [3, 4] and neurobiology [5, 6] to social
science [7, 8]. For systems of _identical_ elements, an important class of
phase models is of the form

$\displaystyle\dot{\phi}_{j}=f(Z)\rme^{\rmi\phi_{j}}+g(Z)+\bar{f}(Z)\rme^{-\rmi\phi_{j}}$
(1)

where $\phi_{j}\in\mathbb{S}^{1}:=\mathbb{R}/2\pi\mathbb{Z}$ with
$j=1,\dots,N$ and

$\displaystyle Z=\frac{1}{N}\sum_{j=1}^{N}\rme^{\rmi\phi_{j}}$

is the Kuramoto order parameter [9]. Hence every element $\phi_{j}$ couples to
the rest of the ensemble via $Z$. The Kuramoto-Sakaguchi model for identical
phase oscillators [10] as well as the active rotator model by Shinomoto and
Kuramoto [11] belong to this class.

Systems of the form (1) have found particular interest since the seminal work
[12] by Watanabe and Strogatz, showing that these systems possess $N-3$
conserved quantities due to a Möbius group symmetry [13]. Thanks to this
symmetry, the dynamics of (1) is determined by just three coupled differential
equations in the Möbius group parameters for any $N\geq 3$, a phenomenon known
as Watanabe-Strogatz (WS) integrability.

WS-integrability has lead to the understanding of multiple phenomena in
ensembles of identically driven oscillators such as Chimera states [14],
classification of attractors [15], and asymptotic stability of periodic two-
cluster states [16, 17], to name a few. WS-integrable systems can also feature
splay states, which are of special interest because of their specific spatio-
temporal symmetry [18, 19] and are potential candidates for attractors for
systems of type (1).

Equation (1) serves as a phase-reduced model for ensembles of general
oscillatory or excitable elements to first-order Fourier mode [20, 21]. Its
Möbius group symmetry and subsequent partial integrability is the result of
neglecting higher modes, which would otherwise break this symmetry. It remains
an open problem as to how the WS-framework can be applied to general settings.

In this work, we focus on models of the type

$\eqalign{\dot{\phi}_{j}=f(Z)\rme^{\rmi\phi_{j}}+g(Z)+\bar{f}(Z)\rme^{-\rmi\phi_{j}}+\epsilon
h(\phi_{j})\\\ h(\phi_{j})=\sum_{n=2}^{\infty}a_{n}\sin n\phi_{j}+b_{n}\cos
n\phi_{j}}$ (2)

where the symmetry of (1) is broken by introducing higher harmonics in the
_on-site_ dynamics. Note that the Fourier expansion of $h$ starts with the
second modes since any zeroth or first mode perturbations can be absorbed by
$f$ and $g$.

Our motivation comes from the classic model of coupled identical active
rotators

$\dot{\phi}_{j}=\omega-\sin\phi_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\sin(\phi_{k}-\phi_{j})$
(3)

with $\left|\omega\right|<1$ by Shinomoto and Kuramoto [11] where the common
fields in (1) are

$\displaystyle g(Z)=\omega$ $\displaystyle
f(Z)=-\frac{\rmi}{2}\left(1+\kappa\bar{Z}\right).$

In particular, this model is closely related to systems of coupled theta-
neurons [22, 23]. Here, when we speak of an _active rotator_ , we mean an
element that is _excitable_ [24] and not already an oscillator by itself.
Hence the sole interest in the case $\left|\omega\right|<1$. It was argued in
[25] that for $\kappa<-\sqrt{1-\omega^{2}}$, the system (3) gives rise to a
continuum of periodic orbits, i. e., when the coupling becomes sufficiently
_repulsive_. Families of periodic orbits are in fact a common occurrence in
WS-theory as already noted in [12].

The classic active rotator model is a first order phase reduced description
for general systems of coupled _class I excitable elements_ , characterized by
being close to a saddle-node bifurcation on an invariant circle [24]. The
question arises what happens to the continuous family of periodic orbits from
[25] if we break the symmetry of (3) by, e. g., including higher-order Fourier
modes in its on-site dynamics. We therefore investigate how the degenerate
dynamics of (3) change if we consider what we call _generalized_ active
rotators [17, 26] instead, for which the equations of motion read

$\eqalign{\dot{\phi}_{j}=\omega-\sin\phi_{j}+\epsilon
h(\phi_{j})+\frac{\kappa}{N}\sum_{k=1}^{N}\sin(\phi_{k}-\phi_{j})\cr
h(\phi_{j})=\sum_{n=2}^{\infty}a_{n}\sin n\phi_{j}+b_{n}\cos n\phi_{j}}$ (4)

and which are of type (2). Below, we outline our main results.

##  2 Informal statements of main results

We first investigate the geometry of the continuum in section 4 which results
in theorem 4.12 and proposition 4.13. Below, we give an informal statement for
the special case of the active rotator model (3), discussed in section 6.

######  Theorem (Informal statement for active rotators).

Consider a system of classic active rotators (3) with $\left|\omega\right|<1$.
For $\kappa<\kappa_{0}=-\sqrt{1-\omega^{2}}$ and sufficiently large $N$, the
system possesses a continuous family of periodic orbits where the union over
these orbits forms a _normally attracting invariant manifold_ (NAIM). One of
the orbits of this continuum features splay state dynamics.

The orbits of the continuum can be parameterized by $N-3$ functionally
independent cross-ratios $\boldsymbol{\lambda}$ which are preserved under the
flow of (1), see [13]. The splay state corresponds thereby to special value
$\boldsymbol{\lambda}=\boldsymbol{\lambda^{*}}$, defined below. On introducing
symmetry-breaking perturbations in the on-site dynamics in (2), the periodic
orbits of the family generally vanish while the NAIM persists. Averaging the
resulting perturbation term in the now time-dependent cross-ratios over an
orbit of the unperturbed system results in an averaged system for which
hyperbolic fixed points correspond to hyperbolic periodic orbits of the
perturbed system. This leads to our second main result theorem 5.1 for which
we again give an informal version below.

######  Theorem.

The averaged dynamics of the cross-ratios possess a fixed point at
$\boldsymbol{\lambda^{*}}$, corresponding to the splay state in the
unperturbed system.

Whether $\boldsymbol{\lambda^{*}}$ is hyperbolic depends on the specific setup
of the model (2). However, we conjecture that splay state orbits are
generically hyperbolic.

In section 6, we apply our results to the active rotator models where we first
determine the critical coupling strength $\kappa_{0}$ of (3) below which the
NAIM-forming continuum arises and apply the averaging principle to the
dynamics on the NAIM and conduct numerical experiments where we find that
$\boldsymbol{\lambda^{*}}$ is hyperbolic for the averaged dynamics so that the
splay state is robust in (4).

##### Strategy:

We outline the main steps to prove our main theorems 4.12 and 5.1.:

Step 1: Since we are dealing with identically driven units, we can assume the
elements to be in (strict) cyclic order which is preserved by the flow of (1).
In lemma 4.8, we show that there exists a diffeomorphism from the space
$\mathbb{T}^{N}_{\mathrm{ordered}}$ of phases
$\boldsymbol{\phi}\in\mathbb{T}^{N}$ in strict cyclic order to the space of
Möbius group parameters and the $N-3$ conserved cross-ratios of (1) which
allows to write down the dynamics of $\boldsymbol{\phi}(t)$ in terms of these
new coordinates.

Step 2: We show that there exists an open set of cross-ratios, for which the
equations of motion for the system (1) in WS-variables can be approximated in
$C^{1}$-norm by truncating higher-order terms in $N$. We then proceed in
theorem 4.12 by showing that there exists a family of periodic orbits for this
truncated system, whose union possesses the desired NAIM structure by virtue
of the persistence theorems for hyperbolic orbits [29].

Step 3: The dynamics on the NAIM describe the asymptotic dynamics of the
perturbed system (2). For any smooth $h$, applying the method of averaging to
the slowly-varying cross-ratio components of the vector field on the NAIM, we
define an averaged perturbation function $\boldsymbol{F}_{h}$ which governs
the averaged dynamics in $\boldsymbol{\lambda}$. Hyperbolic fixed points of
this system determine persistent periodic orbits for the perturbed system (2).

Step 4: Using cyclic permutation properties of the cross-ratios, we show in
theorem 5.1 for $N=4$ that $\boldsymbol{F}_{h}$ vanishes for smooth choices of
$h$ for the cross-ratios $\boldsymbol{\lambda^{*}}$ that correspond to the
splay state. Afterwards, we show that this statement holds true for general
$N\geq 4$ by summing over the group action of cyclic permutations.

##  3 Preliminaries

###  3.1 Normally attracting invariant manifolds

Our definition of a normally attracting invariant manifold (NAIM) is according
to [30].

######  Definition 3.1 (Normally attracting invariant manifold).

Let $r\geq 1$ and $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ with
$\boldsymbol{x}\in\mathbb{R}^{n}$ and $\boldsymbol{f}\in C^{r}$ be a dynamical
system with flow $\Phi:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$. A
given $C^{r}$-submanifold $\mathcal{M}\subset\mathbb{R}^{n}$ is then called an
attracting invariant manifold of this system if it fulfills the following
three criteria:

  1. 1.

$\mathcal{M}$ is invariant under the flow, i. e.,
$\Phi^{t}(\mathcal{M})=\mathcal{M}$ $\forall t\in\mathbb{R}$.

  2. 2.

There exists a continuous splitting

$T_{\mathcal{M}}\mathbb{R}^{n}=T\mathcal{M}\oplus\mathcal{N}$

of the tangent bundle $T\mathbb{R}^{n}$, restricted to $\mathcal{M}$, into the
tangent bundle $T\mathcal{M}$ and a normal bundle $\mathcal{N}$ with
continuous projections $\pi_{\mathcal{M}}$ and $\pi_{\mathcal{N}}$. This
splitting is invariant under the linearized flow
$\mathrm{D}\Phi^{t}=\mathrm{D}\Phi^{t}_{\mathcal{M}}\oplus\mathrm{D}\Phi^{t}_{\mathcal{N}}$.

  3. 3.

There exist real numbers $a$ and $b$ with $a<r\,b\leq 0$, and $C>0$ such that
the following exponential growth conditions hold on
$T_{\mathcal{M}}\mathbb{R}^{n}$:

$\displaystyle\forall t\leq 0,(\boldsymbol{x},\boldsymbol{\nu})\in
T\mathcal{M}:$
$\displaystyle\left|\left|\mathrm{D}\Phi^{t}_{\mathcal{M}}(\boldsymbol{x})\cdot\boldsymbol{\nu}\right|\right|$
$\displaystyle\leq C\,\rme^{bt}\left|\left|\boldsymbol{\nu}\right|\right|$
$\displaystyle\forall t\geq
0,(\boldsymbol{x},\boldsymbol{\nu})\in\mathcal{N}:$
$\displaystyle\left|\left|\mathrm{D}\Phi^{t}_{\mathcal{N}}(\boldsymbol{x})\cdot\boldsymbol{\nu}\right|\right|$
$\displaystyle\leq C\,\rme^{at}\left|\left|\boldsymbol{\nu}\right|\right|.$

A key feature of NAIMs is that they persist under small perturbations of
$\boldsymbol{f}$, as is stated in the following theorem, cf. Theorem. 4.1
[31].

######  Theorem 3.2 (Persistence of NAIMs).

Let $\mathcal{M}\subset\mathbb{R}^{n}$ be a compact attracting invariant
manifold of the system $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$.
Then, there exists an $\epsilon>0$ such that for any vector field
$\boldsymbol{\tilde{f}}$ with
$\left|\left|\boldsymbol{\tilde{f}}-\boldsymbol{f}\right|\right|_{C^{1}}\leq\epsilon$,
there exists a unique invariant $C^{r}$-manifold $\tilde{\mathcal{M}}$ for
$\boldsymbol{\tilde{f}}$ that is diffeomorphic to $\mathcal{M}$, normally
attracting, and
$\mathcal{O}\left(\left|\left|\boldsymbol{\tilde{f}}-\boldsymbol{f}\right|\right|_{C^{1}}\right)$-close
to $\mathcal{M}$.

We close this section with the following remarks.

######  Remarks.

  1. 1.

If not stated otherwise, we employ the Euclidean norm
$\left|\left|\cdot\right|\right|$ and its induced norm.

  2. 2.

The persistence theorem 3.2 for NAIMs also holds for manifolds with boundary
as long as they are _overflowing invariant_. In our case, the NAIMs have a
boundary which is generally not overflowing invariant. This problem can be
overcome by modifying $\boldsymbol{f}$ in a neighborhood of the boundary
accordingly.

  3. 3.

We use Landau’s big-$\mathcal{O}$ notation for order functions, i. e.,
$f(\epsilon)=\mathcal{O}(g(\epsilon))$ if there exist constants $K\geq 0$ and
$\epsilon_{0}\geq 0$ such that $0\leq\left|f(\epsilon)\right|\leq
K\left|g(\epsilon)\right|$ for all $0\leq\epsilon\leq\epsilon_{0}$.

###  3.2 The averaging method

To investigate the asymptotic dynamics of (2), we employ averaging theory.
This approach relies on Theorem 7.9 in [32] on averaging over periodic orbits,
stated below.

######  Theorem 3.3.

Consider the system

$\eqalign{\dot{\boldsymbol{x}}=\epsilon\;\\!\boldsymbol{F}(\boldsymbol{x},\psi)+\epsilon^{2}\boldsymbol{F}_{2}(\boldsymbol{x},\psi,\epsilon)\cr\dot{\psi}=\Omega(\boldsymbol{x})+\epsilon\,G(\boldsymbol{x},\psi,\epsilon)}$
(5)

where $\boldsymbol{x}\in\mathbb{R}^{n}$ and $\psi\in\mathbb{S}^{1}$ and assume
that the function $\Omega(\boldsymbol{x})$ is bounded away from zero, i. e.,
$\Omega(\boldsymbol{x})>c>0$ for some number $c$ and all
$\boldsymbol{x}\in\mathbb{R}^{n}$. If the averaged system

$\dot{\boldsymbol{y}}=\epsilon\;\\!\boldsymbol{\hat{F}}(\boldsymbol{y})$

with

$\boldsymbol{\hat{F}}(\boldsymbol{y}):=\frac{1}{2\pi}\int_{0}^{2\pi}\boldsymbol{F}(\boldsymbol{y},\psi)\,\mathrm{d}\psi$

possesses a hyperbolic fixed point $\boldsymbol{y}_{0}\in\mathbb{R}^{n}$ and
$\epsilon>0$ is sufficiently small, (5) possesses a periodic solution
$t\mapsto\left(\boldsymbol{x}(t),\psi(t)\right)$ of the same stability type as
$\boldsymbol{y}_{0}$.

###  3.3 Watanabe-Strogatz integrability

Watanabe and Strogatz showed that a class of systems of $N$ identically driven
phase variables possess $N-3$ conserved quantities [12]. Later on, this
finding was explained in a broader geometrical context in terms of Möbius
transformations [13]. More specific, WS-integrable systems can be brought to
the form

$\dot{\phi}_{j}=f\rme^{\rmi\phi_{j}}+g+\bar{f}\rme^{-\rmi\phi_{j}}$ (6)

with smooth functions $f:\mathbb{T}^{N}\to\mathbb{C}$ and
$g:\mathbb{T}^{N}\to\mathbb{R}$ where $\mathbb{T}^{N}:=(\mathbb{S}^{1})^{N}$
denotes the $N$-dimensional torus and a bar denotes the complex conjugate. For
short, we write
$\boldsymbol{\phi}:=(\phi_{1},\dots,\phi_{N})\in\mathbb{T}^{N}$.

The dynamics of the $\phi_{j}(t)$ in (6) can be written as

$\displaystyle\rme^{\rmi\phi_{j}(t)}=G_{\alpha(t),\psi(t)}\left(\rme^{\rmi\theta_{j}}\right),\quad
j=1,\dots,N$ (7)

where

$G_{\alpha,\psi}\left(z\right):=\frac{\alpha+\rme^{\rmi\psi}z}{1+\bar{\alpha}\rme^{\rmi\psi}z}$

for $z\in\mathbb{C}$ with parameters $\psi\in\mathbb{S}^{1}$ and
$\alpha\in\mathbb{D}:=\left\\{z\in\mathbb{C}\;;\,\left|z\right|<1\right\\}$.
$G_{\alpha,\psi}$ is an element of the Möbius group

$\mathcal{G}:=\left\\{G_{\alpha,\psi}:\mathbb{D}\to\mathbb{D}\;;\,(\alpha,\psi)\in\mathbb{D}\times\mathbb{S}^{1}\right\\}.$

where $\mathbb{D}:=\left\\{z\in\mathbb{C}\;;\,\left|z\right|<1\right\\}$
denotes the complex unit disk. Generally, the Möbius group is the group of
biholomorphic automorphisms of the augmented complex plane
$\hat{\mathbb{C}}:=\mathbb{C}\cup\\{\infty\\}$ [28]. In the context of WS-
theory, one considers the subgroup $\mathcal{G}$ of biholomorphic
automorphisms of $\mathbb{D}$. By analytic continuation, these transformations
map the boundary $\partial\mathbb{D}$ bijectively onto itself. In accordance
to the literature on WS-theory we call $\mathcal{G}$ _the_ Möbius group.

Equation (7) means that $\mathcal{G}$ acts diagonally on $\mathbb{T}^{N}$.
Setting in accordance to the literature for any
$\boldsymbol{\theta}=(\theta_{1},\dots,\theta_{N})\in\mathbb{T}^{N}$

$\rme^{\rmi\boldsymbol{\theta}}:=\left(\rme^{\rmi\theta_{1}},\dots,\rme^{\rmi\theta_{N}}\right)$

we write in a slight abuse of notation

$G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\theta}}\right):=\left(G_{\alpha,\psi}\left(\rme^{\rmi\theta_{1}}\right),\dots,G_{\alpha,\psi}\left(\rme^{\rmi\theta_{N}}\right)\right).$
(8)

The time evolution of the group parameters $\alpha$ and $\psi$ is then given
by

$\eqalign{\dot{\alpha}=\rmi\left(f\alpha^{2}+g\alpha+\bar{f}\right)\\\
\dot{\psi}=f\alpha+g+\bar{f}\bar{\alpha}}$ (9)

where $f$ and $g$ now depend implicitly on $\alpha$, $\psi$, and
$\boldsymbol{\theta}$ by virtue of (7).

##  4 NAIMs for WS-integrable systems

The aim of this section is to establish a way in which the dynamics of a
systems of identical phase variables $\boldsymbol{\phi}$ can be written in
terms of the Möbius group parameters and the cross-ratios. The main result of
this section is thus lemma 4.8 on the existence of a diffeomorphism between
the set of all $\boldsymbol{\phi}\in\mathbb{T}^{N}$ in cyclic order and the
space $\mathbb{D}\times\mathbb{S}^{1}\times V$ where $V$ is the
$(N-3)$-dimensional space of conserved quantities.

###  4.1 Watanabe-Strogatz variables

Since all units are identically driven, the order of phases on
$\mathbb{S}^{1}$ is preserved under the flow of (1). We restrict our attention
without loss of generality to the case where the $\phi_{j}$ are in (strict)
cyclic order. This gives rise to the following definition.

######  Definition 4.1.

For fixed $N$, the space
$\mathbb{T}^{N}_{\mathrm{ordered}}\subset\mathbb{T}^{N}$ is defined as

$\mathbb{T}^{N}_{\mathrm{ordered}}:=\left\\{\boldsymbol{\theta}\in\mathbb{T}^{N}\;;\,\theta_{1}<\dots<\theta_{N}<\theta_{1}+2\pi\right\\}.$

The (diagonal) group action (8) of the Möbius group $\mathcal{G}$ induces the
following equivalence relation on $\mathbb{T}^{N}_{\mathrm{ordered}}$.

######  Definition 4.2.

Any two points
$\boldsymbol{\vartheta},\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$
are equivalent if and only if there exists a $G_{\alpha,\psi}\in\mathcal{G}$
such that
$\rme^{\rmi\boldsymbol{\vartheta}}=G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\theta}}\right)$
in which case we write $\boldsymbol{\vartheta}\sim\boldsymbol{\theta}$.

That this is an equivalence relation is established below.

######  Proposition 4.3.

The relation $\sim$ is an equivalence relation on
$\mathbb{T}^{N}_{\mathrm{ordered}}$.

###### Proof.

This follows immediately from the group properties of $\mathcal{G}$ [28]:
Consider any three points
$\boldsymbol{\vartheta}\sim\boldsymbol{\theta}\sim\boldsymbol{\phi}\in\mathbb{T}^{N}_{\mathrm{ordered}}$.
Then, (i) the identity map $G_{0,0}\in\mathcal{G}$ guarantees that
$\boldsymbol{\theta}\sim\boldsymbol{\theta}$, (ii) there exists a
$G_{\alpha,\psi}\in\mathcal{G}$ such that
$\rme^{\rmi\boldsymbol{\vartheta}}=G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\theta}}\right)$.
The inverse $G_{\alpha,\psi}^{-1}\in\mathcal{G}$ fulfills
$\rme^{\rmi\boldsymbol{\theta}}=G_{\alpha,\psi}^{-1}\left(\rme^{\rmi\boldsymbol{\vartheta}}\right)$
and $\boldsymbol{\theta}\sim\boldsymbol{\vartheta}$. (iii) there exists a
$G_{\beta,\chi}\in\mathcal{G}$ with
$\rme^{\rmi\boldsymbol{\theta}}=G_{\beta,\chi}\left(\rme^{\rmi\boldsymbol{\phi}}\right)$
and $\rme^{\rmi\boldsymbol{\vartheta}}=G_{\alpha,\psi}\circ
G_{\beta,\chi}\left(\rme^{\rmi\boldsymbol{\phi}}\right)$ where
$G_{\alpha,\psi}\circ G_{\beta,\chi}\in\mathcal{G}$ so that
$\boldsymbol{\vartheta}\sim\boldsymbol{\phi}$. ∎

The resulting partition of $\mathbb{T}^{N}_{\mathrm{ordered}}$ in equivalence
classes is invariant under the flow of (6) according to (7). We write

$[\boldsymbol{\theta}]:=\left\\{\boldsymbol{\vartheta}\in\mathbb{T}^{N}_{\mathrm{ordered}}\;;\,\exists
G_{\alpha,\psi}\in\mathcal{G}:\rme^{\rmi\boldsymbol{\vartheta}}=G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\theta}}\right)\right\\}$

for its equivalence classes.

The real-valued so-called _cross-ratios_

$\Lambda_{p,q,r,s}(\boldsymbol{\theta}):=\frac{\big{(}\rme^{\rmi\theta_{p}}-\rme^{\rmi\theta_{s}}\big{)}\big{(}\rme^{\rmi\theta_{q}}-\rme^{\rmi\theta_{r}}\big{)}}{\big{(}\rme^{\rmi\theta_{q}}-\rme^{\rmi\theta_{s}}\big{)}\big{(}\rme^{\rmi\theta_{p}}-\rme^{\rmi\theta_{r}}\big{)}}$

with $p,q,r,s=1,\dots,N$ are invariant under arbitrary Möbius transformations
[28] and are thus in particular conserved under the flow of (6). It was shown
in [13], that all cross-ratios can be written in terms of $N-3$ functionally
independent ones. To choose such a set, we make the following definition.

######  Definition 4.4.

Let the set $V\subset\mathbb{R}^{N-3}$ be defined as

$V:=\left\\{\boldsymbol{\lambda}\in(0,1)^{N-3}\;;\,1>\lambda_{1}>\dots>\lambda_{N-3}>0\right\\}.$

The cross-ratio function
$\boldsymbol{\Lambda}:\mathbb{T}^{N}_{\mathrm{ordered}}\to V$ is defined by

$\eqalign{\boldsymbol{\Lambda}(\boldsymbol{\theta})&:=\left(\Lambda_{1}(\boldsymbol{\theta}),\dots,\Lambda_{N-3}(\boldsymbol{\theta})\right)\\\
\Lambda_{k}(\boldsymbol{\theta})&:=\Lambda_{1,2,3,k+3}(\boldsymbol{\theta})=\frac{\big{(}\rme^{\rmi\theta_{1}}-\rme^{\rmi\theta_{k+3}}\big{)}\big{(}\rme^{\rmi\theta_{2}}-\rme^{\rmi\theta_{3}}\big{)}}{\big{(}\rme^{\rmi\theta_{2}}-\rme^{\rmi\theta_{k+3}}\big{)}\big{(}\rme^{\rmi\theta_{1}}-\rme^{\rmi\theta_{3}}\big{)}}}$
(10)

with $k=1,\dots,N-3$.

The $N-3$ components $\Lambda_{k}$ are functionally independent. We note some
important properties for the spaces $\mathbb{T}^{N}_{\mathrm{ordered}}$ and
$V$ as well as the cross-ratio function $\boldsymbol{\Lambda}$ below.

######  Lemma 4.5.

The following two assertions hold:

  1. 1.

The set $\mathbb{T}^{N}_{\mathrm{ordered}}$ is invariant under the action of
$\mathcal{G}$.

  2. 2.

The cross-ratio (10) defines a smooth function
$\boldsymbol{\Lambda}:\mathbb{T}^{N}_{\mathrm{ordered}}\to V$.

###### Proof.

The first assertion follows from the fact that all Möbius transformations
$G_{\alpha,\psi}\in\mathcal{G}$ are orientation preserving [28]. Let
$\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$ and
$\rme^{\rmi\boldsymbol{\vartheta}}=G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\theta}}\right)$.
By preservation of orientation, we have
$\boldsymbol{\vartheta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$.

The second assertion follows from the fact that all derivatives of
$\Lambda_{k}$ exist because all $\theta_{j}$ are distinct and that

$\displaystyle\mathrm{D}_{\theta_{k+3}}\Lambda_{k}(\boldsymbol{\theta})=\frac{1}{2}\frac{\sin\frac{\theta_{1}-\theta_{2}}{2}\sin\frac{\theta_{2}-\theta_{3}}{2}}{\sin\frac{\theta_{1}-\theta_{3}}{2}\sin^{2}\frac{\theta_{2}-\theta_{k+3}}{2}}<0,\qquad$
$\displaystyle k=1,\dots,N-3$

so that each $\Lambda_{k}$ is strictly monotonically decreasing in
$\theta_{k+3}$ on $\mathbb{T}^{N}_{\mathrm{ordered}}$. In particular, we find
$\Lambda_{k}(\boldsymbol{\theta})>\Lambda_{k+1}(\boldsymbol{\theta})$ on
$\mathbb{T}^{N}_{\mathrm{ordered}}$. Additionally, we find that
$\lim_{\theta_{k+3}\downarrow\theta_{3}}\Lambda_{k}(\boldsymbol{\theta})=1$
and
$\lim_{\theta_{k+3}\uparrow\theta_{1}+2\pi}\Lambda_{k}(\boldsymbol{\theta})=0$
so that $\boldsymbol{\Lambda}$ indeed maps $\mathbb{T}^{N}_{\mathrm{ordered}}$
to $V$. ∎

As a next step, we define a coordinate system on
$\mathbb{T}^{N}_{\mathrm{ordered}}$ in terms of $\alpha$, $\psi$ and the
cross-ratios $\boldsymbol{\lambda}\in V$. We start with the following lemma.

######  Lemma 4.6.

For any two
$\boldsymbol{\vartheta},\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$,
we have

$\boldsymbol{\vartheta}\sim\boldsymbol{\theta}\Leftrightarrow\boldsymbol{\Lambda}(\boldsymbol{\vartheta})=\boldsymbol{\Lambda}(\boldsymbol{\theta})$

and in particular, we have

$[\boldsymbol{\theta}]=\left\\{\boldsymbol{\vartheta}\in\mathbb{T}^{N}_{\mathrm{ordered}}\;;\,\boldsymbol{\Lambda}(\boldsymbol{\vartheta})=\boldsymbol{\Lambda}(\boldsymbol{\theta})\right\\},$

i. e., the equivalence classes $[\boldsymbol{\theta}]$ are exactly the level
sets of the cross-ratio function $\boldsymbol{\Lambda}$.

###### Proof.

($\Rightarrow$): Since $\boldsymbol{\vartheta}\sim\boldsymbol{\theta}$, there
exists a $G_{\alpha,\psi}\in\mathcal{G}$ with
$\rme^{\rmi\boldsymbol{\vartheta}}=G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\theta}}\right)$.
Then
$\boldsymbol{\Lambda}(\boldsymbol{\vartheta})=\boldsymbol{\Lambda}(\boldsymbol{\theta})$
follows the invariance of cross-ratios under arbitrary Möbius transformations.

($\Leftarrow$): Let
$\boldsymbol{\Lambda}(\boldsymbol{\vartheta})=\boldsymbol{\Lambda}(\boldsymbol{\theta})$
for some
$\boldsymbol{\vartheta},\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$.
We construct a suitable $G_{\alpha,\psi}\in\mathcal{G}$ for which
$G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\theta}}\right)=\rme^{\rmi\boldsymbol{\vartheta}}$
holds. Consider the two general Möbius transformations
$\mu_{\boldsymbol{\theta}}:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ and
$\mu_{\boldsymbol{\vartheta}}:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ with

$\displaystyle\mu_{\boldsymbol{\theta}}(z)$
$\displaystyle:=\frac{\Big{(}z-\rme^{\rmi\theta_{1}}\Big{)}\Big{(}\rme^{\rmi\theta_{2}}-\rme^{\rmi\theta_{3}}\Big{)}}{\Big{(}z-\rme^{\rmi\theta_{2}}\Big{)}\Big{(}\rme^{\rmi\theta_{1}}-\rme^{\rmi\theta_{3}}\Big{)}},\qquad\mu_{\boldsymbol{\vartheta}}(z)$
$\displaystyle:=\frac{\Big{(}z-\rme^{\rmi\vartheta_{1}}\Big{)}\Big{(}\rme^{\rmi\vartheta_{2}}-\rme^{\rmi\vartheta_{3}}\Big{)}}{\Big{(}z-\rme^{\rmi\vartheta_{2}}\Big{)}\Big{(}\rme^{\rmi\vartheta_{1}}-\rme^{\rmi\vartheta_{3}}\Big{)}}.$

These maps fulfill

$\displaystyle\mu_{\boldsymbol{\theta}}\left(\rme^{\rmi\theta_{1}}\right)=\mu_{\boldsymbol{\vartheta}}\left(\rme^{\rmi\vartheta_{1}}\right)=0,\qquad\mu_{\boldsymbol{\theta}}\left(\rme^{\rmi\theta_{2}}\right)=\mu_{\boldsymbol{\vartheta}}\left(\rme^{\rmi\vartheta_{2}}\right)=\infty,\qquad\mu_{\boldsymbol{\theta}}\left(\rme^{\rmi\theta_{3}}\right)=\mu_{\boldsymbol{\vartheta}}\left(\rme^{\rmi\vartheta_{3}}\right)=1$

so that the function
$G=\mu_{\boldsymbol{\vartheta}}^{-1}\circ\mu_{\boldsymbol{\theta}}$ maps
$\rme^{\rmi\theta_{k}}$ to $\rme^{\rmi\vartheta_{k}}$ for $k=1,2,3$. In
particular, $G$ is a bijective conformal map from $\partial\mathbb{D}$ to
$\partial\mathbb{D}$. It is also orientation preserving since
$\rme^{\rmi\theta_{1}},\rme^{\rmi\theta_{2}},\rme^{\rmi\theta_{3}}\in\partial\mathbb{D}$
and
$\rme^{\rmi\vartheta_{1}},\rme^{\rmi\vartheta_{2}},\rme^{\rmi\vartheta_{3}}\in\partial\mathbb{D}$
are in the same cyclic order by assumption. $G$ is therefore an element of
$\mathcal{G}$. Our assumption
$\Lambda_{k}(\boldsymbol{\theta})=\Lambda_{k}(\boldsymbol{\vartheta})$ implies
that

$\mu_{\boldsymbol{\theta}}\big{(}\rme^{\rmi\theta_{k+3}}\big{)}=\Lambda_{k}(\boldsymbol{\theta})=\Lambda_{k}(\boldsymbol{\vartheta})=\mu_{\boldsymbol{\vartheta}}\big{(}\rme^{\rmi\vartheta_{k+3}}\big{)}.$

and $G\left(\rme^{\rmi\theta_{k}}\right)=\rme^{\rmi\vartheta_{k}}$ holds also
for all $k=4,\dots,N$. We conclude that
$G\left(\rme^{\rmi\boldsymbol{\theta}}\right)=\rme^{\rmi\boldsymbol{\vartheta}}$
which finishes the proof. ∎

Lemma 4.6 implies that we can parameterize the partition of
$\mathbb{T}^{N}_{\mathrm{ordered}}$ in equivalence classes
$[\boldsymbol{\theta}]\subset\mathbb{T}^{N}_{\mathrm{ordered}}$ via the cross-
ratios $\boldsymbol{\lambda}\in V$ such that we may set

$[\boldsymbol{\theta}]\equiv\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda}):=\left\\{\boldsymbol{\vartheta}\in\mathbb{T}^{N}_{\mathrm{ordered}}\;;\,\boldsymbol{\Lambda}(\boldsymbol{\vartheta})=\boldsymbol{\lambda}\right\\}.$

where $\boldsymbol{\lambda}=\boldsymbol{\Lambda}(\boldsymbol{\theta})$. Each
$\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$ is an element of the
level set $\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ with
$\boldsymbol{\lambda}=\boldsymbol{\Lambda}(\boldsymbol{\theta})$. We have to
introduce a suitable coordinate system on each set
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$.

The parameters $\alpha$ and $\psi$ do not yet establish a coordinate system on
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ but only
parameterize how to transform any given point in
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ to any other point
in the same level set rather than uniquely determining where each point is
located in $\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$. This
ambiguity comes from the fact that no a priori “point-of-reference” exists in
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ relative to which
all other points can are parameterized. To introduce such a unique point-of-
reference $\boldsymbol{\Theta}(\boldsymbol{\lambda})$ for each
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$, we define
$\boldsymbol{\Theta}:V\to\mathbb{T}^{N}_{\mathrm{ordered}}$ by

$\eqalign{\boldsymbol{\Theta}(\boldsymbol{\lambda})&:=(\Theta_{1}(\boldsymbol{\lambda}),\dots,\Theta_{N}(\boldsymbol{\lambda}))\\\
\Theta_{k}(\boldsymbol{\lambda})&:=\cases{-\pi+\frac{2\pi}{N}(k-1)&for
$k=1,2,3$\\\
-\rmi\,\mathrm{Log\;}\frac{\rme^{\frac{2\pi\rmi}{N}}\left(\lambda_{k-3}+\lambda_{k-3}\,\rme^{\frac{2\pi\rmi}{N}}-1\right)}{-\lambda_{k-3}+(1-\lambda_{k-3})\,\rme^{\frac{2\pi\rmi}{N}}}&for
$k=4,\dots,N$.}}$ (11)

To see that this choice is suitable for our purposes, we need to show that it
defines for each level set
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ a unique point-of-
reference, i. e., that the image of $\boldsymbol{\Theta}$ intersects each
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ in exactly one
point. We can define for each point
$\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$ its WS-coordinates
$\left(\alpha,\psi,\boldsymbol{\lambda}\right)$ via the relation

$\rme^{\rmi\boldsymbol{\theta}}=G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}\right)$

where $\boldsymbol{\lambda}$ determines in _which_ level set
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ the point
$\boldsymbol{\theta}$ lies while $\alpha$ and $\psi$ determine _where_
$\boldsymbol{\theta}$ is located _in_
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ with respect to
$\boldsymbol{\Theta}(\boldsymbol{\lambda})$. This is the purpose of the
following lemma.

######  Lemma 4.7.

The map $\boldsymbol{\Theta}:V\to\mathbb{T}^{N}_{\mathrm{ordered}}$, defined
by (11), is smooth and a right inverse of the function $\boldsymbol{\Lambda}$,
i. e.,
$\boldsymbol{\Lambda}\circ\boldsymbol{\Theta}(\boldsymbol{\lambda})=\boldsymbol{\lambda}$
and
$\boldsymbol{\Theta}(\boldsymbol{\lambda})\in\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$.

###### Proof.

$\boldsymbol{\Theta}$ is well-defined and smooth since the the numerator in
the second line of (11) vanishes only for
$\lambda_{k}=1/\left(1+\exp(2\pi\rmi/N)\right)\notin\mathbb{R}$ while the
denominator vanishes only for
$\lambda_{k}=\exp(2\pi\rmi/N)/\left(-1+\exp(2\pi\rmi/N)\right)\notin\mathbb{R}$.
Equation (11) solves the equation
$\boldsymbol{\Lambda}(\boldsymbol{\Theta})=\boldsymbol{\lambda}$ for
$\boldsymbol{\Theta}$ and so is the right inverse of $\boldsymbol{\Lambda}$ by
construction. Finally, we show that $\boldsymbol{\Theta}$ indeed maps $V$ to
$\mathbb{T}^{N}_{\mathrm{ordered}}$. For this, we show that for every $k>3$,
$\Theta_{k}(\boldsymbol{\lambda})$ is strictly monotonically decreasing in
$\lambda_{k-3}$ and that
$\Theta_{3}(\boldsymbol{\lambda})>\Theta_{4}(\boldsymbol{\lambda})$. The
assertion follows from the fact that $1>\lambda_{1}>\dots>\lambda_{N-3}>0$ and
the limits

$\displaystyle\lim_{\lambda_{1}\uparrow
1}\Theta_{4}(\boldsymbol{\lambda})=-\pi+\frac{2\pi}{N}2\qquad\textrm{and}\qquad\lim_{\lambda_{N-3}\downarrow
0}\Theta_{N}(\boldsymbol{\lambda})=\pi.$

But since $N\geq 4$, we find

$\mathrm{D}_{\lambda_{k-3}}\Theta_{k}(\boldsymbol{\lambda})=\frac{-2\sin\frac{2\pi}{N}}{1-\underbrace{2\lambda_{k-3}(1-\lambda_{k-3})}_{\in(0,1/2]}\underbrace{\left(1+\cos\frac{2\pi}{N}\right)}_{\in[1,2)}}<0$

so that $\Theta_{k}$ is indeed strictly monotonically decreasing in
$\lambda_{k-3}$ for all $k=4,\dots,N$. ∎

######  Remark.

In what follows, particular focus lies on the point

$\eqalign{\boldsymbol{\theta^{*}}:=\left(\theta^{*}_{1},\dots,\theta^{*}_{N}\right)\qquad\textrm{with}\qquad\theta^{*}_{j}:=-\pi+\frac{2\pi}{N}(j-1)}$
(12)

for which we find

$\eqalign{\boldsymbol{\lambda^{*}}:=\left(\lambda^{*}_{1},\dots,\lambda^{*}_{N-3}\right)\qquad\textrm{with}\qquad\lambda^{*}_{k}:=\Lambda_{k}(\boldsymbol{\theta^{*}})=\frac{\sin\frac{\pi(k+2)}{N}}{2\cos\frac{\pi}{N}\sin\frac{\pi(k+1)}{N}}.}$
(13)

We state the final result of this section on the existence of the well-defined
WS-coordinate system on $\mathbb{T}^{N}_{\mathrm{ordered}}$ in the following
proposition.

######  Proposition 4.8.

The map $\boldsymbol{m}:\mathbb{D}\times\mathbb{S}^{1}\times
V\to\mathbb{T}^{N}_{\mathrm{ordered}}$ with

$\boldsymbol{m}(\alpha,\psi,\boldsymbol{\lambda}):=-\rmi\,\mathrm{Log\;}G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}\right)=-\rmi\,\mathrm{Log\;}\frac{\alpha+\rme^{\rmi\psi}\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}}{1+\bar{\alpha}\,\rme^{\rmi\psi}\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}}$

is a smooth diffeomorphism.

###### Proof.

As a composition of the smooth maps $(\alpha,\psi,\boldsymbol{\theta})\mapsto
G_{\alpha,\psi}(\rme^{\rmi\boldsymbol{\theta}})$, $\boldsymbol{\Theta}$, and
the $\mathrm{Log\;}$-function, $\boldsymbol{m}$ is smooth. We show that
$\boldsymbol{m}$ is also bijective and $\mathrm{D}\boldsymbol{m}$ has full
rank.

Injectivity of $\boldsymbol{m}$: Suppose that for
$(\alpha,\psi,\boldsymbol{\lambda}),(\alpha^{\prime},\psi^{\prime},\boldsymbol{\lambda}^{\prime})\in\mathbb{D}\times\mathbb{S}^{1}\times
V$ the equality
$\boldsymbol{m}(\alpha,\psi,\boldsymbol{\lambda})=\boldsymbol{m}(\alpha^{\prime},\psi^{\prime},\boldsymbol{\lambda}^{\prime})$
holds. We find

$\displaystyle-\rmi\,\mathrm{Log\;}G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}\right)=-\rmi\,\mathrm{Log\;}G_{\alpha^{\prime},\psi^{\prime}}\left(\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda}^{\prime})}\right)$
$\displaystyle\Rightarrow\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}=G_{\alpha,\psi}^{-1}\circ
G_{\alpha^{\prime},\psi^{\prime}}\left(\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda}^{\prime})}\right).$

By lemma 4.6, this implies

$\displaystyle\boldsymbol{\Theta}(\boldsymbol{\lambda})\in[\boldsymbol{\Theta}(\boldsymbol{\lambda}^{\prime})]=\mathcal{L}_{\boldsymbol{\lambda}^{\prime}}(\boldsymbol{\Lambda})\quad\Rightarrow\quad\boldsymbol{\lambda}=\boldsymbol{\Lambda}\circ\boldsymbol{\Theta}(\boldsymbol{\lambda})=\boldsymbol{\lambda}^{\prime}.$

Further, $\boldsymbol{\Theta}(\boldsymbol{\lambda})$ always possesses at least
three distinct coordinates (e.g. $\Theta_{1}$, $\Theta_{2}$, and
$\Theta_{3}$). But since a Möbius map is uniquely defined by the images of
three distinct points, we have
$G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}\right)=G_{\alpha^{\prime},\psi^{\prime}}\left(\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}\right)\Leftrightarrow
G_{\alpha,\psi}=G_{\alpha^{\prime},\psi^{\prime}}\Leftrightarrow(\alpha,\psi)=(\alpha^{\prime},\psi^{\prime})$
and
$(\alpha,\psi,\boldsymbol{\lambda})=(\alpha^{\prime},\psi^{\prime},\boldsymbol{\lambda}^{\prime})$.

Surjectivity of $\boldsymbol{m}$: For any
$\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$, the cross-ratios
$\boldsymbol{\lambda}=\boldsymbol{\Lambda}(\boldsymbol{\theta})\in V$ are well
defined. Since $\boldsymbol{\Theta}(\boldsymbol{\lambda})$ and
$\boldsymbol{\theta}$ are both elements of
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})=[\boldsymbol{\Theta}(\boldsymbol{\lambda})]=[\boldsymbol{\theta}]$,
there exists a $G_{\alpha,\psi}\in\mathcal{G}$ with
$G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\Theta}(\boldsymbol{\lambda})}\right)=\rme^{\rmi\boldsymbol{\theta}}$.
But this implies the existence of an
$(\alpha,\psi,\boldsymbol{\lambda})\in\mathbb{D}\times\mathbb{S}^{1}\times V$
which is mapped by $\boldsymbol{m}$ to $\boldsymbol{\theta}$.

At last, we show that the derivative $\mathrm{D}\boldsymbol{m}$ has full rank
everywhere. For each
$x\in\\{\alpha,\bar{\alpha},\psi,\lambda_{1},\dots,\lambda_{N-3}\\}$, let
$\mathrm{D}_{x}\boldsymbol{m}=(\mathrm{D}_{x}\boldsymbol{m}_{1},\dots,\mathrm{D}_{x}\boldsymbol{m}_{N})^{T}$
denote the respective column of
$\mathrm{D}\boldsymbol{m}=\left[\mathrm{D}_{\alpha}\boldsymbol{m},\mathrm{D}_{\bar{\alpha}}\boldsymbol{m},\mathrm{D}_{\psi}\boldsymbol{m},\mathrm{D}_{\lambda_{1}}\boldsymbol{m},\dots,\mathrm{D}_{\lambda_{N-3}}\boldsymbol{m}\right]$.
From the identity

$\boldsymbol{\Lambda}(\boldsymbol{m}(\alpha,\psi,\boldsymbol{\lambda}))=\boldsymbol{\lambda}$

we note first that
$\mathrm{D}_{\boldsymbol{\theta}}\boldsymbol{\Lambda}\cdot\mathrm{D}_{\boldsymbol{\lambda}}\boldsymbol{m}=\mathrm{id}_{N-3}$
or more explicitly

$\pmatrix{\mathrm{D}_{\theta_{1}}\Lambda_{1}&\dots&\mathrm{D}_{\theta_{N}}\Lambda_{1}\cr\vdots&\ddots&\vdots\cr\mathrm{D}_{\theta_{1}}\Lambda_{N-3}&\dots&\mathrm{D}_{\theta_{N}}\Lambda_{N-3}}\cdot\pmatrix{\mathrm{D}_{\lambda_{1}}m_{1}&\dots&\mathrm{D}_{\lambda_{N-3}}m_{1}\cr\vdots&\ddots&\vdots\cr\mathrm{D}_{\lambda_{1}}m_{N}&\dots&\mathrm{D}_{\lambda_{N-3}}m_{N}}=\mathrm{id}_{N-3}$
(14)

where $\mathrm{id}_{N-3}$ is the $(N-3)$-dimensional identity matrix. Thus,
$\mathrm{D}_{\boldsymbol{\lambda}}\boldsymbol{m}$ has a left inverse
$\mathrm{D}_{\boldsymbol{\theta}}\boldsymbol{\Lambda}$ and therefore has in
particular full column rank.

Secondly, we have

$\displaystyle\mathrm{D}_{\boldsymbol{\theta}}\boldsymbol{\Lambda}\cdot\mathrm{D}_{\alpha}\boldsymbol{m}=\mathrm{D}_{\boldsymbol{\theta}}\boldsymbol{\Lambda}\cdot\mathrm{D}_{\bar{\alpha}}\boldsymbol{m}=\mathrm{D}_{\boldsymbol{\theta}}\boldsymbol{\Lambda}\cdot\mathrm{D}_{\psi}\boldsymbol{m}=0$
(15)

so the column vectors $\mathrm{D}_{\alpha}\boldsymbol{m}$,
$\mathrm{D}_{\bar{\alpha}}\boldsymbol{m}$, and
$\mathrm{D}_{\psi}\boldsymbol{m}$ are orthogonal to the column vectors
$\mathrm{D}_{\theta_{j}}\boldsymbol{\Lambda}$ and thus linearly independent of
the column vectors $\mathrm{D}_{\lambda_{k}}\boldsymbol{m}$: if any linear
combination

$\boldsymbol{y}=a\,\mathrm{D}_{\alpha}\boldsymbol{m}+b\,\mathrm{D}_{\bar{\alpha}}\boldsymbol{m}+c\,\mathrm{D}_{\psi}\boldsymbol{m}$

was in the span of the $\mathrm{D}_{\lambda_{k}}\boldsymbol{m}$, i. e., if we
would have

$\boldsymbol{y}=\sum_{j=1}^{N-3}c_{j}\mathrm{D}_{\lambda_{j}}\boldsymbol{m}$

with $c_{k}\neq 0$ for some $k\in\\{1,\dots,N-3\\}$, we would find

$\displaystyle
0=\mathrm{D}_{\boldsymbol{\theta}}\Lambda_{k}\cdot\boldsymbol{y}=\sum_{j=1}^{N-3}c_{j}\mathrm{D}_{\boldsymbol{\theta}}\Lambda_{k}\cdot\mathrm{D}_{\lambda_{j}}\boldsymbol{m}=c_{k}$

where the first equality follows from (15) and the last from (14). This
contradicts the assumption that $c_{k}\neq 0$.

The three vectors $\mathrm{D}_{\alpha}\boldsymbol{m}$,
$\mathrm{D}_{\bar{\alpha}}\boldsymbol{m}$, and
$\mathrm{D}_{\psi}\boldsymbol{m}$ are linearly independent. To see this,
consider the $(N-3)\times 3$ matrix
$A=\left[\mathrm{D}_{\alpha}\boldsymbol{m},\mathrm{D}_{\bar{\alpha}}\boldsymbol{m},\mathrm{D}_{\psi}\boldsymbol{m}\right]$
with $\mathrm{rank\;}A\leq 3$. For any matrix, its rank is equal to the
largest order square submatrix with nonzero determinant [33]. Let the
submatrix $\hat{A}$ consist of the first three components of each column of
$A$. It is of the form

$\displaystyle\hat{A}=\pmatrix{\frac{\displaystyle\rmi}{\displaystyle\rme^{\rmi\psi}-\alpha}&\frac{\displaystyle\rmi\rme^{\rmi\psi}}{\displaystyle\bar{\alpha}\rme^{\rmi\psi}-1}&\frac{\displaystyle\rme^{\rmi\psi}(\left|\alpha\right|^{2}-1)}{\displaystyle(\rme^{\rmi\psi}-\alpha)(\bar{\alpha}\rme^{\rmi\psi}-1)}\cr\frac{\displaystyle\rmi}{\displaystyle\rme^{\rmi\psi}\zeta_{N}-\alpha}&\frac{\displaystyle\rmi\rme^{\rmi\psi}\zeta_{N}}{\displaystyle\bar{\alpha}\rme^{\rmi\psi}\zeta_{N}-1}&\frac{\displaystyle\rme^{\rmi\psi}\zeta_{N}(\left|\alpha\right|^{2}-1)}{\displaystyle(\rme^{\rmi\psi}\zeta_{N}-\alpha)(\bar{\alpha}\rme^{\rmi\psi}\zeta_{N}-1)}\cr\frac{\displaystyle\rmi}{\displaystyle\rme^{\rmi\psi}\zeta_{N}^{2}-\alpha}&\frac{\displaystyle\rmi\rme^{\rmi\psi}\zeta_{N}^{2}}{\displaystyle\bar{\alpha}\rme^{\rmi\psi}\zeta_{N}^{2}-1}&\frac{\displaystyle\rme^{\rmi\psi}\zeta_{N}^{2}(\left|\alpha\right|^{2}-1)}{\displaystyle(\rme^{\rmi\psi}\zeta_{N}^{2}-\alpha)(\bar{\alpha}\rme^{\rmi\psi}\zeta_{N}^{2}-1)}},$

where we set $\zeta_{N}=\exp(2\pi\rmi/N)$ and has the determinant

$\det\hat{A}=\frac{X}{Y_{1}Y_{2}}$

where

$\displaystyle
X=-(1-\zeta_{N})^{3}(1+\zeta_{N})(1-\left|\alpha\right|^{2})\rme^{3\rmi\psi}\zeta_{N}$
$\displaystyle
Y_{1}=(\alpha-\rme^{\rmi\psi})(\alpha-\rme^{\rmi\psi}\zeta_{N})(\alpha-\rme^{\rmi\psi}\zeta_{N}^{2})$
$\displaystyle
Y_{2}=(1-\bar{\alpha}\rme^{\rmi\psi})(1-\bar{\alpha}\rme^{\rmi\psi}\zeta_{N})(1-\bar{\alpha}\rme^{\rmi\psi}\zeta_{N}^{2}).$

This determinant is nonvanishing: By the triangle inequality, we have

$\displaystyle\left|Y_{1}\right|\leq(1+\left|\alpha\right|)^{3}\qquad\textrm{and}\qquad\left|Y_{2}\right|\leq(1+\left|\alpha\right|)^{3}$

while for $N\geq 3$ and with
$\left|1\pm\rme^{\rmi\theta}\right|=\sqrt{2}\sqrt{1\pm\cos\theta}$ and
$1-\cos\theta\geq\theta^{2}/8$ for all
$\theta\in\left[-\frac{2\pi}{3},\frac{2\pi}{3}\right]$, we have

$\displaystyle\left|X\right|=\left|1-\rme^{\frac{2\pi\rmi}{N}}\right|^{3}\left|1+\rme^{\frac{2\pi\rmi}{N}}\right|\left(1-\left|\alpha\right|^{2}\right)\geq\left(\frac{\pi}{N}\right)^{3}\left(1-\left|\alpha\right|^{2}\right)$

where in the third line we also used $1+\cos\theta\geq 1/2$ for all
$\theta\in\left[-\frac{2\pi}{3},\frac{2\pi}{3}\right]$. This implies

$\left|\det\hat{A}\right|\geq\left(\frac{\pi}{N}\right)^{3}\frac{1-\left|\alpha\right|^{2}}{(1+\left|\alpha\right|)^{6}}>0$

so that $\hat{A}$ has full rank 3 and thus $\mathrm{rank\;}A=3$. Hence,
$\mathrm{D}_{\alpha}\boldsymbol{m}$,
$\mathrm{D}_{\bar{\alpha}}\boldsymbol{m}$, and
$\mathrm{D}_{\psi}\boldsymbol{m}$ are linearly independent. From the inverse
function theorem for smooth functions, since $\boldsymbol{m}$ is smooth, so is
its inverse [34]. This finishes the proof. ∎

Lemma 4.8 implies that we can express the state of _any_ system of identical
phase variables $\boldsymbol{\phi}(t)$ in terms of $\alpha(t)$, $\psi(t)$, and
$\boldsymbol{\lambda}(t)$ to which we refer to in what follows as _WS-
variables_. The equations of motion can be expressed in WS-variables via
$\boldsymbol{m}^{-1}$. From now on we write the Watanabe-Strogatz equations
(9) in their full form

$\displaystyle\dot{\alpha}$
$\displaystyle=\rmi\left(f(Z)\alpha^{2}+g(Z)\alpha+\bar{f}(Z)\right)$ (16a)
$\displaystyle\dot{\psi}$
$\displaystyle=f(Z)\alpha+g(Z)+\bar{f}(Z)\bar{\alpha}$ (16b)
$\displaystyle\dot{\boldsymbol{\lambda}}$ $\displaystyle=\boldsymbol{0}.$
(16c)

Next, we discuss some properties of the level set
$\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$ which will be
useful later.

###  4.2 The level set of uniform distributions

On the level set
$\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$, we can close
(16) by truncating higher-order terms in $N$ and solve the resulting
equations. Under the right conditions, this yields a family of periodic orbits
whose union forms a NAIM. Interpreting the truncated terms as $C^{1}$-small
perturbations, this NAIM also exists for the original model. We start this
section with a discussion of the Kuramoto order parameter [9]

$Z:=\frac{1}{N}\sum_{j=1}^{N}\rme^{\rmi\phi_{j}}=\frac{1}{N}\sum_{j=1}^{N}G_{\alpha,\psi}\left(\rme^{\rmi\theta_{j}}\right).$

The key observation for systems of type (1) is that on the level set
$\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$ the order
parameter $Z$ can be approximated by $\alpha$ if $N$ is sufficiently large
[35, 14]. Since $f$ and $g$ in (1) only depend on $Z$, replacing $Z$ by
$\alpha$ decouples (16a) from $\psi$ which simplifies the problem of finding
periodic solution of (16). To use this fact within the framework of the theory
of normally attracting invariant manifolds, we need to show that the resulting
perturbation is small in $C^{1}$-norm. The main result of this section is an
estimate of $Z$ and its partial derivatives for large $N$, which is used to
estimate the error from truncating higher order terms in $N$ in (16). We start
by writing $Z$ in terms of $\alpha$, $\psi$, and $\boldsymbol{\lambda}$. We
first define the auxiliary function
$Z_{\boldsymbol{\theta}}:\mathbb{D}\times\mathbb{S}^{1}\times\mathbb{T}^{N}\to\mathbb{D}$
with

$\displaystyle Z_{\boldsymbol{\theta}}(\alpha,\psi,\boldsymbol{\theta})$
$\displaystyle:=\frac{1}{N}\sum_{j=1}^{N}\frac{\alpha+\rme^{\rmi\psi}\rme^{\rmi\theta_{j}}}{1+\bar{\alpha}\,\rme^{\rmi\psi}\rme^{\rmi\theta_{j}}}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}(\alpha+\rme^{\rmi\psi}\rme^{\rmi\theta_{j}})\sum_{k=0}^{\infty}(-\bar{\alpha}\,\rme^{\rmi\psi}\rme^{\rmi\theta_{j}})^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\alpha(-\bar{\alpha}\,\rme^{\rmi\psi})^{k}\frac{1}{N}\sum_{j=1}^{N}\rme^{\rmi
k\theta_{j}}+\rme^{\rmi\psi}(-\bar{\alpha}\,\rme^{\rmi\psi})^{k}\frac{1}{N}\sum_{j=1}^{N}\rme^{\rmi(k+1)\theta_{j}}.$

Introducing the symbols

$\left\langle\rme^{\rmi
k\boldsymbol{\theta}}\right\rangle:=\frac{1}{N}\sum_{j=1}^{N}\rme^{\rmi
k\theta_{j}},$

we arrive at the general expression

$\displaystyle Z_{\boldsymbol{\theta}}(\alpha,\psi,\boldsymbol{\theta})$
$\displaystyle=\alpha\sum_{k=0}^{\infty}(-\bar{\alpha}\,\rme^{\rmi\psi})^{k}\left\langle\rme^{\rmi
k\boldsymbol{\theta}}\right\rangle+\rme^{\rmi\psi}\sum_{k=0}^{\infty}(-\bar{\alpha}\,\rme^{\rmi\psi})^{k}\left\langle\rme^{\rmi(k+1)\boldsymbol{\theta}}\right\rangle.$
$\displaystyle=\alpha+(1-\left|\alpha\right|^{2})\,\rme^{\rmi\psi}\sum_{k=1}^{\infty}(-\bar{\alpha}\,\rme^{\rmi\psi})^{k-1}\left\langle\rme^{\rmi
k\boldsymbol{\theta}}\right\rangle.$

From this, we can define $Z:\mathbb{D}\times\mathbb{S}^{1}\times
V\to\mathbb{D}$ by setting
$\boldsymbol{\theta}=\boldsymbol{\Theta}(\boldsymbol{\lambda})$ in the
equation above so that we arrive at

$\eqalign{Z(\alpha,\psi,\boldsymbol{\lambda})&:=Z_{\boldsymbol{\theta}}(\alpha,\psi,\boldsymbol{\Theta}(\boldsymbol{\lambda}))\\\
&=\alpha+(1-\left|\alpha\right|^{2})\,\rme^{\rmi\psi}\sum_{k=1}^{\infty}(-\bar{\alpha}\,\rme^{\rmi\psi})^{k-1}\left\langle\rme^{\rmi
k\boldsymbol{\Theta}(\boldsymbol{\lambda})}\right\rangle.}$ (17)

For $\boldsymbol{\lambda}=\boldsymbol{\lambda^{*}}$, the symbol
$\left\langle\rme^{\rmi
k\boldsymbol{\Theta}(\boldsymbol{\lambda})}\right\rangle$ and $Z$ can be
written in simple form as shown next.

######  Lemma 4.9.

Recall that
$\boldsymbol{\theta^{*}}=\boldsymbol{\Theta}(\boldsymbol{\lambda^{*}})$ is
defined by $\theta^{*}_{j}=-\pi+2\pi(j-1)/N$ with $j=1,\dots,N$. For
$\boldsymbol{\theta^{*}}$, we have

$\left\langle\rme^{\rmi
k\boldsymbol{\theta^{*}}}\right\rangle=\cases{(-1)^{k}&if $k\in
N\mathbb{Z}$\cr 0&else.}$

###### Proof.

Let $k\in N\mathbb{Z}$, i. e., $k=Nl$ for some integer $l$. Then

$\rme^{\rmi k\theta^{*}_{j}}=\rme^{-\rmi\pi
k+\frac{2\pi\rmi}{N}Nl(j-1)}=\rme^{-\rmi\pi k}=(-1)^{k}$

for all $j$ and $\left\langle\rme^{\rmi
k\boldsymbol{\theta^{*}}}\right\rangle=\frac{1}{N}\sum_{j=1}^{N}(-1)^{k}=(-1)^{k}$.
On the other hand, let $k\notin N\mathbb{Z}$. We use the well known identity
$\sum_{j=1}^{N}z^{j-1}=\frac{z^{N}-1}{z-1}$ for any $z\neq 1$ which yields

$\displaystyle\left\langle\rme^{\rmi
k\boldsymbol{\theta^{*}}}\right\rangle=\frac{1}{N}\sum_{j=1}^{N}\rme^{-\rmi\pi
k+\frac{2\pi\rmi}{N}(j-1)k}=0$

since $\rme^{\frac{2\pi\rmi}{N}k}\neq 1$ for $k\notin N\mathbb{Z}$. ∎

Each level set
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})\subset\mathbb{T}^{N}_{\mathrm{ordered}}$
is diffeomorphic to the space $\mathbb{D}\times\mathbb{S}^{1}$ so that we
identify $\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ with
$\mathbb{D}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda}\\}$:

$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})\cong\mathbb{D}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda}\\}.$

Consider a smooth function

$\displaystyle\boldsymbol{F}:\mathbb{D}\times\mathbb{S}^{1}\times V$
$\displaystyle\to\mathbb{R}^{n}.$

The derivative of its restriction
$\boldsymbol{F}|_{\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})}$
to the level set $\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ is
given by

$\tilde{\mathrm{D}}\boldsymbol{F}:=\mathrm{D}\boldsymbol{F}|_{\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})}\equiv\left(\mathrm{D}_{\alpha}\boldsymbol{F},\mathrm{D}_{\bar{\alpha}}\boldsymbol{F},\mathrm{D}_{\psi}\boldsymbol{F}\right).$

Let

$\mathbb{D}_{r}:=\left\\{\alpha\in\mathbb{C}\;;\,|\alpha|<r\right\\}.$

The following lemma gives an estimate to leading order in $N$ for
$Z(\alpha,\psi,\boldsymbol{\lambda})-\alpha$ on the level set
$\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$ of uniform
distributions.

######  Lemma 4.10.

Consider $\overline{\mathbb{D}}_{r}$ for any $0<r<1$ and let

$\displaystyle\eta:(\alpha,\psi,\boldsymbol{\lambda})$ $\displaystyle\mapsto
Z(\alpha,\psi,\boldsymbol{\lambda})-\alpha\qquad\textrm{on}\qquad\mathbb{D}\times\mathbb{S}^{1}\times
V$ $\displaystyle\to\mathbb{C}$

Then,

$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|=\mathcal{O}(r^{N-1})$
(18)
$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\left|\tilde{\mathrm{D}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right|=\mathcal{O}(Nr^{N-2}).$
(19)

###### Proof.

We start with (18). From (17),
$\boldsymbol{\theta^{*}}=\boldsymbol{\Theta}(\boldsymbol{\lambda^{*}})$, and
lemma 4.9, we infer

$\displaystyle\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})$
$\displaystyle=-\left(1-\left|\alpha\right|^{2}\right)\,\rme^{\rmi\psi}\sum_{k=1}^{\infty}(\bar{\alpha}\,\rme^{\rmi\psi})^{kN-1}$
$\displaystyle=-\sum_{k=1}^{\infty}\bar{\alpha}^{kN-1}\rme^{\rmi
kN\psi}+\alpha\sum_{k=1}^{\infty}\bar{\alpha}^{kN}\rme^{\rmi kN\psi}$

yielding

$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle=\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\left(1-\left|\alpha\right|^{2}\right)\,\rme^{\rmi\psi}\sum_{k=1}^{\infty}(\bar{\alpha}\,\rme^{\rmi\psi})^{kN-1}\right|$
$\displaystyle\leq\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left(1-\left|\alpha\right|^{2}\right)\sum_{k=1}^{\infty}\left|\bar{\alpha}\,\rme^{\rmi\psi}\right|^{kN-1}$
$\displaystyle\leq\sum_{k=1}^{\infty}r^{kN-1}=\mathcal{O}(r^{N-1}).$

so that (18) holds.

To prove (19), we first compute the entries of
$\tilde{\mathrm{D}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})$ to leading
order in $N$. From this, we can expand
$\left|\left|\tilde{\mathrm{D}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right|$
to leading order. For each $x\in\\{\alpha,\bar{\alpha},\psi\\}$, we write in a
slight abuse of notation
$\mathrm{D}_{x}\eta=\mathrm{Re\;}\mathrm{D}_{x}\eta+\rmi\,\mathrm{Im\;}\mathrm{D}_{x}\eta$
for the respective column
$(\mathrm{Re\;}\mathrm{D}_{x}\eta,\mathrm{Im\;}\mathrm{D}_{x}\eta)^{T}$ of
$\mathrm{D}\eta$.

For the first column of
$\mathrm{D}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})$, we find

$\displaystyle\mathrm{D}_{\alpha}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})$
$\displaystyle=\sum_{k=1}^{\infty}(\bar{\alpha}\,\rme^{\rmi\psi})^{kN}$
$\displaystyle\left|\mathrm{D}_{\alpha}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle\leq\sum_{k=1}^{\infty}\left|\alpha\right|^{kN}\leq\sum_{k=1}^{\infty}r^{kN}$

for all $(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}$ and
thus

$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\mathrm{Re\;}\mathrm{D}_{\alpha}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle=\mathcal{O}(r^{N})$
$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\mathrm{Im\;}\mathrm{D}_{\alpha}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle=\mathcal{O}(r^{N}).$

For the second column, we find

$\displaystyle\mathrm{D}_{\bar{\alpha}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})$
$\displaystyle=-\sum_{k=1}^{\infty}(kN-1)\,\bar{\alpha}^{kN-2}\,\rme^{\rmi
kN\psi}+\alpha\sum_{k=1}^{\infty}kN\,\bar{\alpha}^{kN-1}\,\rme^{\rmi kN\psi}$
$\displaystyle\left|\mathrm{D}_{\bar{\alpha}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle\leq\;\sum_{k=1}^{\infty}\left[(kN-1)\left|\alpha\right|^{kN-2}+kN\left|\alpha\right|^{kN}\right]$
$\displaystyle\leq\;\sum_{k=1}^{\infty}\left[(kN-1)\,r^{kN-2}+kN\,r^{kN}\right]$

for all $(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}$ and

$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\mathrm{Re\;}\mathrm{D}_{\bar{\alpha}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle=\mathcal{O}(Nr^{N-2})$
$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\mathrm{Im\;}\mathrm{D}_{\bar{\alpha}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle=\mathcal{O}(Nr^{N-2}).$

Finally, we have

$\displaystyle\mathrm{D}_{\psi}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})$
$\displaystyle=\sum_{k=1}^{\infty}\rmi
kN\big{[}-\bar{\alpha}^{kN-1}+\alpha\bar{\alpha}^{kN}\big{]}\rme^{\rmi
kN\psi}$
$\displaystyle\left|\mathrm{D}_{\psi}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle\leq\sum_{k=1}^{\infty}kN\big{[}\left|\alpha\right|^{kN-1}+\left|\alpha\right|^{kN+1}\big{]}$
$\displaystyle\leq\sum_{k=1}^{\infty}kN\big{[}r^{kN-1}+r^{kN+1}\big{]}$

for all $(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}$ so
that

$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\mathrm{Re\;}\mathrm{D}_{\psi}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle=\mathcal{O}(Nr^{N-1})$
$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\mathrm{Im\;}\mathrm{D}_{\psi}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|$
$\displaystyle=\mathcal{O}(Nr^{N-1}).$

Writing

$\left|\left|A\right|\right|_{1}=\max_{1\leq j\leq
n}\sum_{i=1}^{m}\left|A_{ij}\right|$

for the induced 1-norm for any $m\times n$ matrix $A$ and using the inequality
$\left|\left|A\right|\right|\leq\sqrt{n}\left|\left|A\right|\right|_{1}$ for
its Euclidean norm, we arrive at

$\displaystyle\left|\left|\tilde{\mathrm{D}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right|\leq\sqrt{3}\max_{x\in\\{\alpha,\bar{\alpha},\psi\\}}\left(\left|\mathrm{Re\;}\tilde{\mathrm{D}}_{x}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|+\left|\mathrm{Im\;}\tilde{\mathrm{D}}_{x}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right)$

and hence

$\left|\left|\tilde{\mathrm{D}}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right|=\mathcal{O}(Nr^{N-2}),$

uniformly for all
$(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}$ which proves
(19). ∎

We are now ready to give the main result of this subsection which will be used
in the proof of theorem 4.12.

######  Lemma 4.11.

Let $0<r<1$ and consider the vector field
$\boldsymbol{\tau}:\mathbb{D}\times\mathbb{S}^{1}\times V\to\mathbb{R}^{3}$,
defined by

$\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})=\left(G(Z(\alpha,\psi,\boldsymbol{\lambda}))-G(\alpha)\right)\cdot\boldsymbol{Y}(\alpha,\psi)$
(20)

where $G:\mathbb{D}\to\mathbb{C}$ and
$\boldsymbol{Y}:\mathbb{D}\times\mathbb{S}^{1}\to\mathbb{R}^{3}$ are smooth.
Then, given $\varepsilon>0$, there exists an $N_{0}\in\mathbb{N}$ such that
for all $N\geq N_{0}$, there exists a $\delta$-neighborhood
$V_{\delta}(\boldsymbol{\lambda^{*}})$ of $\boldsymbol{\lambda^{*}}$ with

$\displaystyle\sup_{(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}}\left|\left|\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})\right|\right|+\left|\left|\tilde{\mathrm{D}}\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})\right|\right|$
$\displaystyle<\varepsilon$

for all $\boldsymbol{\lambda}\in V_{\delta}(\boldsymbol{\lambda^{*}})$.

###### Proof.

Step 1: Let $F:\mathbb{D}\to\mathbb{C}$ be defined as $F(Z):=G(Z)-G(\alpha)$.
We start by bounding the function $F$ and its partial derivatives
$\mathrm{D}_{x}F$ with $x\in\\{\alpha,\bar{\alpha},\psi\\}$. Since
$\overline{\mathbb{D}}_{r}$ is convex and $F(\alpha)=0$, we find by the mean
value theorem and with $\eta=Z-\alpha$ that

$\eqalign{F(Z)&=\int_{0}^{1}\eta\cdot\mathrm{D}_{z}F|_{z=\alpha+t\eta}+\bar{\eta}\cdot\mathrm{D}_{\bar{z}}F|_{z=\alpha+t\eta}\,\mathrm{d}t\\\
\left|F(Z)\right|&\leq\left|\eta\right|\cdot\int_{0}^{1}\left|\mathrm{D}_{z}F\right|_{z=\alpha+t\eta}+\left|\mathrm{D}_{\bar{z}}F\right|_{z=\alpha+t\eta}\,\mathrm{d}t.}$
(21)

Since $F$ is smooth and $\overline{\mathbb{D}}_{r}$ is compact, there then
exist constants $M_{1},M_{2}>0$ such that $\left|\mathrm{D}_{z}F\right|<M_{1}$
and $\left|\mathrm{D}_{\bar{z}}F\right|<M_{2}$ for all
$z\in\overline{\mathbb{D}}_{r}$ so that in particular, the integral above is
of order $\mathcal{O}(1)$ and

$\left|F(Z(\alpha,\psi,\boldsymbol{\lambda^{*}}))\right|=\mathcal{O}(\left|\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|)=\mathcal{O}(r^{N-1})$

for all $(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}$ by
lemma 4.10. On the other hand, applying the product and chain rule to (21), we
find

$\displaystyle\mathrm{D}_{x}F(Z)=$
$\displaystyle\quad\int_{0}^{1}\mathrm{D}_{z}\mathrm{D}_{z}F|_{z=\alpha+t\eta}\,\mathrm{D}_{x}(\alpha+t\eta)\,\mathrm{d}t\cdot\eta$
$\displaystyle+\int_{0}^{1}\mathrm{D}_{\bar{z}}\mathrm{D}_{z}F|_{z=\alpha+t\eta}\,\mathrm{D}_{x}(\bar{\alpha}+t\bar{\eta})\,\mathrm{d}t\cdot\eta$
$\displaystyle+\int_{0}^{1}\mathrm{D}_{z}\mathrm{D}_{\bar{z}}F|_{z=\alpha+t\eta}\,\mathrm{D}_{x}(\alpha+t\eta)\,\mathrm{d}t\cdot\bar{\eta}$
$\displaystyle+\int_{0}^{1}\mathrm{D}_{\bar{z}}\mathrm{D}_{\bar{z}}F|_{z=\alpha+t\eta}\,\mathrm{D}_{x}(\bar{\alpha}+t\bar{\eta})\,\mathrm{d}t\cdot\bar{\eta}$
$\displaystyle+\int_{0}^{1}\mathrm{D}_{z}F|_{\alpha+t\eta}\,\mathrm{d}t\cdot\mathrm{D}_{x}\eta+\int_{0}^{1}\mathrm{D}_{\bar{z}}F|_{z=\alpha+t\eta}\,\mathrm{d}t\cdot\mathrm{D}_{x}\bar{\eta}$

so that by the same argument as above and again with lemma 4.10, we find with
$\left|\mathrm{D}_{x}(\alpha+t\eta(\alpha,\psi,\boldsymbol{\lambda^{*}}))\right|=\mathcal{O}(1)+\mathcal{O}(Nr^{N-2})=\mathcal{O}(1)$

$\displaystyle\left|\mathrm{D}_{x}F(Z(\alpha,\psi,\boldsymbol{\lambda^{*}}))\right|$
$\displaystyle\leq\left|\eta\right|\cdot\int_{0}^{1}\Big{[}\left|\mathrm{D}_{z}\mathrm{D}_{z}F\right|\\!\,_{z=\alpha+t\eta}\cdot\left|\mathrm{D}_{x}(\alpha+t\eta)\right|+$
$\displaystyle\hskip
49.79231pt+\left|\mathrm{D}_{\bar{z}}\mathrm{D}_{z}F\right|\\!\,_{z=\alpha+t\eta}\cdot\left|\mathrm{D}_{x}(\bar{\alpha}+t\bar{\eta})\right|+$
$\displaystyle\hskip
49.79231pt+\left|\mathrm{D}_{z}\mathrm{D}_{\bar{z}}F\right|\\!\,_{z=\alpha+t\eta}\cdot\left|\mathrm{D}_{x}(\alpha+t\eta)\right|+$
$\displaystyle\hskip
49.79231pt+\left|\mathrm{D}_{\bar{z}}\mathrm{D}_{\bar{z}}F\right|\\!\,_{z=\alpha+t\eta}\cdot\left|\mathrm{D}_{x}(\bar{\alpha}+t\bar{\eta})\right|\Big{]}\,\mathrm{d}t\;+$
$\displaystyle\hskip
49.79231pt+\left|\mathrm{D}_{x}\eta\right|\cdot\int_{0}^{1}\left|\mathrm{D}_{z}F\right|\\!\,_{z=\alpha+t\eta}+\left|\mathrm{D}_{\bar{z}}F\right|\\!\,_{z=\alpha+t\eta}\,\mathrm{d}t$
$\displaystyle=\mathcal{O}(\left|\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|)+\mathcal{O}(\left|\mathrm{D}_{x}\eta(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|)=\mathcal{O}(Nr^{N-2}).$

Step 2: Since $\boldsymbol{Y}(\alpha,\psi)$ does not depend on $N$, we have
$\left|\left|\boldsymbol{Y}(\alpha,\psi)\right|\right|=\mathcal{O}(1)$ and
$\left|\left|\mathrm{D}_{x}\boldsymbol{Y}(\alpha,\psi)\right|\right|_{1}=\mathcal{O}(1)$
for all $(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}$ so
that

$\left|\left|\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right|=\left|F(Z(\alpha,\psi,\boldsymbol{\lambda^{*}}))\right|\cdot\left|\left|\boldsymbol{Y}(\alpha,\psi)\right|\right|=\mathcal{O}(r^{N-1})$

and, by virtue of the triangle inequality and the previous step,

$\displaystyle\left|\left|\tilde{\mathrm{D}}\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right|_{1}$
$\displaystyle=\max_{x\in\\{\alpha,\bar{\alpha},\psi\\}}\left|\left|\mathrm{D}_{x}F(Z(\alpha,\psi,\boldsymbol{\lambda^{*}}))\cdot\boldsymbol{Y}(\alpha,\psi)+F(Z(\alpha,\psi,\boldsymbol{\lambda^{*}}))\cdot\mathrm{D}_{x}\boldsymbol{Y}(\alpha,\psi)\right|\right|_{1}$
$\displaystyle=\mathcal{O}(Nr^{N-2})$

holds so that

$\left|\left|\tilde{\mathrm{D}}\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right|\leq\sqrt{3}\left|\left|\tilde{\mathrm{D}}\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda^{*}})\right|\right|_{1}=\mathcal{O}(Nr^{N-2})$

for all $(\alpha,\psi)\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}$.

Step 3: From step 2, we infer that there exists an $N_{0}$ such that
$\left|\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})\right|<\varepsilon/2$
and
$\left|\left|\tilde{\mathrm{D}}\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})\right|\right|<\varepsilon/2$,
uniformly on
$\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda^{*}}\\}$
for all $N\geq N_{0}$. By smoothness of $\boldsymbol{\tau}$, it follows that
for each
$(\alpha,\psi,\boldsymbol{\lambda^{*}})\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda^{*}}\\}$,
there exist open neighborhoods
$W(\alpha,\psi)\subseteq\mathbb{D}\times\mathbb{S}^{1}$ of $(\alpha,\psi)$ and
$V_{\delta^{\prime}}(\boldsymbol{\lambda^{*}})\subseteq V$ of
$\boldsymbol{\lambda^{*}}$ with
$\delta^{\prime}=\delta^{\prime}(\alpha,\psi)>0$ such that this inequality
also holds on $W(\alpha,\psi)\times
V_{\delta^{\prime}}(\boldsymbol{\lambda^{*}})$. Covering
$\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda^{*}}\\}$
with these open sets, there exists a finite subcover since
$\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda^{*}}\\}$
is compact. Setting $\delta:=\min\delta^{\prime}$ over this subcover, we find

$\left|\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})\right|+\left|\left|\tilde{\mathrm{D}}\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})\right|\right|<\varepsilon$

for each
$(\alpha,\psi,\boldsymbol{\lambda})\in\overline{\mathbb{D}}_{r}\times\mathbb{S}^{1}\times
V_{\delta}$ which proves the assertion. ∎

###  4.3 Existence of the NAIM

We are now ready to prove our first main result: the theorem on the existence
of a continuum of periodic orbits, whose union forms a NAIM.

######  Theorem 4.12.

Consider the system (16) where $f:\mathbb{D}\to\mathbb{C}$ and
$g:\mathbb{D}\to\mathbb{R}$ are smooth functions of the Kuramoto order
parameter $Z$. Further, let the closed equation

$\dot{\alpha}=i\left(f(\alpha)\alpha^{2}+g(\alpha)\alpha+\bar{f}(\alpha)\right)$

possess a unique stable fixed point $\alpha_{0}\in\mathbb{D}$ for which
$f(\alpha_{0})\neq 0$.

Then, there exists a $\delta$-neighborhood
$\overline{V}_{\delta}(\boldsymbol{\lambda^{*}})$ of
$\boldsymbol{\lambda^{*}}$ such that for sufficiently large $N$ and every
$\boldsymbol{\lambda}\in\overline{V}_{\delta}(\boldsymbol{\lambda^{*}})$,
there exists a unique periodic orbit
$\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$
which is exponentially stable in
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$. Additionally, the
union

$\mathcal{M}_{\delta}:=\bigcup_{\boldsymbol{\lambda}\in\overline{V}_{\delta}}\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathbb{T}^{N}_{\mathrm{ordered}}$

forms a compact NAIM of dimension $N-2$ with invariant boundary.

###### Proof.

Step 1: Substituting $Z$ by $\alpha$ in (16) yields the closed system

$\displaystyle\dot{\alpha}$
$\displaystyle=i\left(f(\alpha)\alpha^{2}+g(\alpha)\alpha+\bar{f}(\alpha)\right)$
(22a) $\displaystyle\dot{\psi}$
$\displaystyle=f(\alpha)\alpha+g(\alpha)+\bar{f}(\alpha)\bar{\alpha}$ (22b)
$\displaystyle\dot{\boldsymbol{\lambda}}$ $\displaystyle=\boldsymbol{0}.$
(22c)

By assumption, $\alpha_{0}$ is a stable fixed point of (22a). If
$\Omega:=f(\alpha_{0})\alpha_{0}+g(\alpha_{0})+\bar{f}(\alpha_{0})\bar{\alpha}_{0}\neq
0$, this gives rise to the periodic solution so
$(\alpha_{0},\Omega\,t,\boldsymbol{\lambda})$ for (22a)-(22c) with an
exponentially stable periodic orbit
$\mathcal{C}^{\mathrm{closed}}_{\boldsymbol{\lambda}}$. From the definition of
$\Omega$, we then infer

$\displaystyle\alpha_{0}\Omega$
$\displaystyle=f(\alpha_{0})\alpha_{0}^{2}+g(\alpha_{0})\alpha_{0}+\bar{f}(\alpha_{0})\left|\alpha_{0}\right|^{2}.$

Adding and subtracting $\bar{f}(\alpha_{0})$, this results in

$\alpha_{0}\Omega=\underbrace{f(\alpha_{0})\alpha_{0}^{2}+g(\alpha_{0})\alpha_{0}+\bar{f}(\alpha_{0})}_{=0\textrm{
by
(\ref{eq:WSTruncAlpha})}}+\bar{f}(\alpha_{0})\left(\left|\alpha_{0}\right|^{2}-1\right)$

so that $\Omega=0$ implies $f(\alpha_{0})=0$ and conversely,
$f(\alpha_{0})\neq 0$ implies $\Omega\neq 0$.

Step 2: We prove that for any $\delta>0$, the union

$\mathcal{M}^{\mathrm{closed}}_{\delta}:=\bigcup_{\boldsymbol{\lambda}\in\overline{V}_{\delta}}\mathcal{C}^{\mathrm{closed}}_{\boldsymbol{\lambda}}=\\{\alpha_{0}\\}\times\mathbb{S}^{1}\times\overline{V}_{\delta}$

forms a smooth compact NAIM with invariant boundary for (22) in
$\mathbb{D}\times\mathbb{S}^{1}\times V$ by checking conditions (i)-(iii) of
definition 3.1. As a product of the three smooth compact manifolds
$\\{\alpha_{0}\\}$, $\mathbb{S}^{1}$, and $\overline{V}_{\delta}$, it is
itself an $(N-2)$-dimensional smooth compact submanifold of
$\mathbb{D}\times\mathbb{S}^{1}\times V$. It is invariant by construction so
that condition (i) is fulfilled.

Let
$\boldsymbol{p}=(\alpha_{0},\psi,\boldsymbol{\lambda})\in\mathcal{M}^{\mathrm{closed}}_{\delta}$
denote any point on $\mathcal{M}$ and consider any tangential vector
$\boldsymbol{v}\in T_{\boldsymbol{p}}\mathbb{T}^{N}_{\mathrm{ordered}}$ at
$\boldsymbol{p}$ with
$\boldsymbol{v}=(v_{\alpha},v_{\psi},v_{\boldsymbol{\lambda}})^{T}$ where
$v_{\alpha}\in\mathbb{R}^{2}$, $v_{\psi}\in\mathbb{R}^{1}$, and
$v_{\boldsymbol{\lambda}}\in\mathbb{R}^{N-3}$ denote the respective tangential
components with respect to the variables $\alpha$, $\psi$, and
$\boldsymbol{\lambda}$. In particular, since the dynamics of $\alpha$
decouples from $\psi$ and $\boldsymbol{\lambda}$ in (22), the dynamics of
$v_{\alpha}$ decouples from $v_{\psi}$ and $v_{\boldsymbol{\lambda}}$ on
$\mathcal{M}_{\delta}^{\mathrm{closed}}$ so that $\mathcal{N}$ stays invariant
under the linearized flow $\mathrm{D}\Phi^{t}$. Then,
$T\mathcal{M}_{\delta}^{\mathrm{closed}}$ also stays invariant and (ii) is
fulfilled. The flow is not contracting in $v_{\psi}$ and
$v_{\boldsymbol{\lambda}}$ since $\boldsymbol{\lambda}$ stays constant and
$\psi=\Omega t$ so that we have $b=0$ while it is exponentially contracting in
normal direction $v_{\alpha}$ (with contraction rates $\lambda^{\pm}<0$ which
are the eigenvalues of the Jacobian for (22a). Let
$a=\max(\lambda^{+},\lambda^{-})<0$. Then $a<0=bm$ for $m>0$ and (iii) is also
fulfilled and $\mathcal{M}_{\delta}^{\mathrm{closed}}$ is a NAIM.

Step 3: We proceed by showing that (16) equally possesses a continuous family
of periodic orbits. By closing (16) to get equations (22), we truncated the
vector field

$\boldsymbol{\tau}=F(Z)\pmatrix{\rmi\alpha^{2}\cr\alpha\cr\boldsymbol{0}}+G(Z)\pmatrix{\rmi\alpha\cr
1\cr\boldsymbol{0}}+\bar{F}(Z)\pmatrix{\rmi\cr\bar{\alpha}\cr\boldsymbol{0}}$

with

$\displaystyle F(Z)$ $\displaystyle:=f(Z)-f(\alpha)$ $\displaystyle G(Z)$
$\displaystyle:=g(Z)-g(\alpha)$

from (16). Identifying
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})\cong\mathbb{D}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda}\\}$
with the space $\mathbb{D}\times\mathbb{S}^{1}$, we truncate

$\boldsymbol{\tau}|_{\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})}=F(Z)\pmatrix{\rmi\alpha^{2}\cr\alpha}+G(Z)\pmatrix{\rmi\alpha\cr
1}+\bar{F}(Z)\pmatrix{\rmi\cr\bar{\alpha}}$ (23)

in every level set $\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$
for any given $\boldsymbol{\lambda}\in V$ where each of the three terms on the
right hand side is of the form (20). Choose $1>r>\left|\alpha_{0}\right|$. In
$\mathbb{D}\times\mathbb{S}^{1}$, we can identify all
$\mathcal{C}_{\boldsymbol{\lambda}}^{\mathrm{closed}}$ with the stable limit
cycle
$\\{\alpha_{0}\\}\times\mathbb{S}^{1}\subset\mathbb{D}_{r}\times\mathbb{S}^{1}$,
so there exists an $\varepsilon>0$ such that for every perturbation of the
vector field that has $C^{1}$-norm smaller than $\varepsilon$ in
$\mathbb{D}_{r}\times\mathbb{S}^{1}$, the orbits
$\mathcal{C}_{\boldsymbol{\lambda}}^{\mathrm{closed}}$ persist [29]. But by
lemma 4.11, there exists an $N_{0}\in\mathbb{N}$ and a $\delta$-neighborhood
$\overline{V}_{\delta}(\boldsymbol{\lambda^{*}})$ of
$\boldsymbol{\lambda^{*}}$ such that for all $N\geq N_{0}$

$\left|\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})\right|+\left|\left|\tilde{\mathrm{D}}\boldsymbol{\tau}(\alpha,\psi,\boldsymbol{\lambda})\right|\right|<\varepsilon$

uniformly on $\mathbb{D}_{r}\times\mathbb{S}^{1}\times\overline{V}_{\delta}$
and thus, the $\mathcal{C}_{\boldsymbol{\lambda}}^{\mathrm{closed}}$ persist,
i. e., the full system (16) possesses a periodic orbit
$\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathbb{D}_{r}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda}\\}\subset\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$
for every $\boldsymbol{\lambda}\in\overline{V}_{\delta}$.

We can further choose $\delta$ such that the boundary of the union
$\mathcal{M}_{\delta}$ over these orbits is composed of orbits
$\mathcal{C}_{\boldsymbol{\lambda}}$ with
$\boldsymbol{\lambda}\in\partial\overline{V}_{\delta}$ and is invariant. We
show that $\mathcal{M}_{\delta}$ is a smooth compact manifold.

Step 4: Note that for $\boldsymbol{\lambda}\in\overline{V}_{\delta}$, there
exists a smooth immersion
$\iota_{\boldsymbol{\lambda}}:\mathcal{C}_{\boldsymbol{\lambda}}^{\mathrm{closed}}\to\mathbb{D}_{r}\times\mathbb{S}^{1}\times\\{\boldsymbol{\lambda}\\}$
whose image
$\iota_{\boldsymbol{\lambda}}(\mathcal{C}_{\boldsymbol{\lambda}}^{\mathrm{closed}})=\mathcal{C}_{\boldsymbol{\lambda}}$
lies $\varepsilon$-close to
$\mathcal{C}_{\boldsymbol{\lambda}}^{\mathrm{closed}}$ and that
$\iota_{\boldsymbol{\lambda}}$ depends smoothly on $\boldsymbol{\lambda}$
because the truncated terms (23) are smooth in $\boldsymbol{\lambda}$. Hence,
the map

$\displaystyle\iota:\mathcal{M}_{\delta}^{\mathrm{closed}}$
$\displaystyle\to\mathbb{D}\times\mathbb{S}^{1}\times V$
$\displaystyle\iota:(\alpha_{0},\psi,\boldsymbol{\lambda})$
$\displaystyle\mapsto(\iota_{\boldsymbol{\lambda}}(\alpha_{0},\psi),\boldsymbol{\lambda}),$

is (i) smooth, (ii) one-to-one on its image $\mathcal{M}_{\delta}$, and (iii)
its derivative has full rank $N-2$, in other words, $\iota$ is a smooth
immersion and $\mathcal{M}_{\delta}$ is a compact smooth invariant manifold
which is $\mathcal{O}(\varepsilon)$-close to
$\mathcal{M}_{\delta}^{\mathrm{closed}}$.

Step 5: We show that $\mathcal{M}_{\delta}$ is normally hyperbolic: For any
fixed $\boldsymbol{\lambda}\in V$, let $\Phi^{t}_{\boldsymbol{\lambda}}$
denote the flow on the level set
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$. Then, the flow on
$\mathbb{T}^{N}_{\mathrm{ordered}}$ is of the form

$\displaystyle(\alpha,\psi,\boldsymbol{\lambda})\mapsto(\Phi^{t}_{\boldsymbol{\lambda}}(\alpha,\psi),\boldsymbol{\lambda})$

and is smooth in $\boldsymbol{\lambda}$ since the vector field on
$\mathbb{T}^{N}_{\mathrm{ordered}}$ is smooth. The linearized flow at
$\boldsymbol{p}=(\alpha,\psi,\boldsymbol{\lambda})\in\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathcal{M}_{\delta}\subset\mathbb{T}^{N}_{\mathrm{ordered}}$
reads

$\mathrm{D}\Phi^{t}(\boldsymbol{p})=\pmatrix{\tilde{\mathrm{D}}\Phi_{\boldsymbol{\lambda}}^{t}(\boldsymbol{p})&\mathrm{D}_{\boldsymbol{\lambda}}\Phi_{\boldsymbol{\lambda}}^{t}(\boldsymbol{p})\cr
0&\mathrm{id}_{N-3}}.$

Let $\mu^{\pm}<0$ denote the two nonzero contraction rates of
$\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$
and $\boldsymbol{v}^{\pm}=(v^{\pm}_{\alpha},v^{\pm}_{\psi},\boldsymbol{0})$
denote the corresponding eigenvectors of $\mathrm{D}\Phi^{t}(\boldsymbol{p})$
and let $\mu^{0}=0$ denote the vanishing contraction rate in tangent direction
to $\mathcal{C}_{\boldsymbol{\lambda}}$. The remaining $N-3$ eigenvectors of
$\mathrm{D}\Phi^{t}(\boldsymbol{p})$ are also tangent vectors of
$\mathcal{M}_{\delta}$ at $\boldsymbol{p}$ and have nonvanishing components in
$\boldsymbol{\lambda}$-direction since $\mathcal{M}_{\delta}$ lies transversal
to each $\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ that it
intersects for sufficiently small $\varepsilon>0$. Since
$\boldsymbol{\lambda}$ is constant under the flow, the contraction rates in
the remaining $N-3$ tangent directions are also zero. Because $\Phi^{t}$ is
smooth in $\boldsymbol{\lambda}$, the $\boldsymbol{v}^{\pm}$ depend smoothly
on $\boldsymbol{p}$ so that condition (ii) of definition 3.1 is readily
fulfilled. For the numbers $a$, $b$, and $C$ from condition (iii), we find
$b=0$,
$a=\max_{\boldsymbol{p}\in\mathcal{M}_{\delta}}(\mu^{\pm}(\boldsymbol{p}))<0=mb$
for $m>0$ and $C=1$ so that $\mathcal{M}_{\delta}$ is a smooth $m$-normally
attracting invariant manifold of (16). ∎

We note that $\mathcal{M}_{\delta}$ is of the form

$\mathcal{M}_{\delta}=\left\\{\alpha(\psi,\boldsymbol{\lambda}),\psi,\boldsymbol{\lambda}\;;\,\psi\in\mathbb{S}^{1},\boldsymbol{\lambda}\in\overline{V}_{\delta}\right\\}$

where
$\left|\left|\alpha(\psi,\boldsymbol{\lambda})-\alpha_{0}\right|\right|_{C^{1}}\ll
1$ uniformly on $\mathbb{S}^{1}\times\overline{V}_{\delta}$.

###  4.4 Existence of splay states

Next, we study the dynamics on the periodic orbit
$\mathcal{C}_{\boldsymbol{\lambda^{*}}}\subset\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$
of (1), characterized in theorem 4.12. Recall that a _splay state_
$\boldsymbol{\phi}(t)$ is a $T$-periodic solution for which there exists a
$T$-periodic function $\varphi:\mathbb{R}\to\mathbb{S}^{1}$ such that

$\phi_{j}(t)=\varphi\left(t+j\frac{T}{N}\right)$

holds, i. e., the time series for each of the units $\phi_{j}$ are copies of
each other, shifted multiples of $T/N$ in time. The following assertion holds.

######  Proposition 4.13.

For the continuum $\mathcal{M}_{\delta}$ from theorem 4.12, the periodic
solution $(\alpha(t),\psi(t),\boldsymbol{\lambda^{*}})$ of (16) with periodic
orbit
$\mathcal{C}_{\boldsymbol{\lambda^{*}}}\subset\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$
yields a splay state $\boldsymbol{\phi}(t)$ of (1).

###### Proof.

Recall that $\theta^{*}_{j}=-\pi+2\pi(j-1)/N$. For the closed system (22), we
found the periodic solution $(\alpha_{0},\Omega\,t,\boldsymbol{\lambda^{*}})$
with period $T=2\pi/\Omega$. Its phase dynamics
$\boldsymbol{\phi}^{\mathrm{closed}}(t)$ can via the diffeomorphism
$\boldsymbol{m}$ which yields

$\displaystyle\rme^{\rmi\phi^{\mathrm{closed}}_{j}(t)}\equiv\frac{\alpha_{0}+\rme^{\rmi\Omega
t+i\theta^{*}_{j}}}{1+\bar{\alpha}_{0}\,\rme^{\rmi\Omega
t+i\theta^{*}_{j}}}=\frac{\alpha_{0}+\rme^{\rmi\Omega\left(t+j\frac{T}{N}\right)+i\theta^{*}_{N}}}{1+\bar{\alpha}_{0}\,\rme^{\rmi\Omega\left(t+j\frac{T}{N}\right)+i\theta^{*}_{N}}}\equiv\rme^{\rmi\phi^{\mathrm{closed}}_{N}\left(t+j\frac{T}{N}\right)}$

so that
$\phi^{\mathrm{closed}}_{j}(t)=\phi^{\mathrm{closed}}_{N}\left(t+jT/N\right)$
holds and $\boldsymbol{\phi}^{\mathrm{closed}}(t)$ is in fact a splay state
(with $\varphi=\phi_{N}$). We now assert that the periodic solution
$(\alpha(t),\psi(t),\boldsymbol{\lambda^{*}})$ for the true system (16) from
theorem 4.12 also yields a splay state $\boldsymbol{\phi}(t)$ of (1). For this
to be true, the following condition must hold:

$\rme^{\rmi\phi_{j}(t)}\equiv\frac{\alpha(t)+\rme^{\rmi\psi(t)+i\theta^{*}_{j}}}{1+\bar{\alpha}(t)\rme^{\rmi\psi(t)+i\theta^{*}_{j}}}=\frac{\alpha\left(t+j\frac{T}{N}\right)+\rme^{\rmi\psi\left(t+j\frac{T}{N}\right)+i\theta^{*}_{N}}}{1+\bar{\alpha}\left(t+j\frac{T}{N}\right)\rme^{\rmi\psi\left(t+j\frac{T}{N}\right)+i\theta^{*}_{N}}}\equiv\rme^{\rmi\phi_{N}\left(t+j\frac{T}{N}\right)}$

which is true if the spatio-temporal symmetry

$\displaystyle\alpha\left(t+j\frac{T}{N}\right)=\alpha(t),\qquad\psi\left(t+j\frac{T}{N}\right)=\psi(t)+j\frac{2\pi}{N}$

with $j=1,\dots,N$ holds for $(\alpha(t),\psi(t),\boldsymbol{\lambda^{*}})$.
To see that this is the case, we note first that (16a) and (16b) are
equivariant under any transformation that keeps $Z$ and $\alpha$ invariant. On
$\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$, the Kuramoto
order parameter is given by

$Z(\alpha,\psi,\boldsymbol{\lambda^{*}})=\alpha-\big{(}1-\left|\alpha\right|^{2}\big{)}\sum_{k=1}^{\infty}(\bar{\alpha})^{kN-1}\rme^{\rmi
kN\psi}$

so that on $\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$,
$Z(\alpha,\psi,\boldsymbol{\lambda^{*}})$ and $\alpha$ are invariant under the
action of the finite group $\Gamma$ of transformations

$\displaystyle(\alpha,\psi)$
$\displaystyle\mapsto\left(\alpha,\psi+j\frac{2\pi}{N}\right)$

with $j=1,\dots,N$. The splay state nature of the $\boldsymbol{\phi}(t)$ for
the full system is then a consequence of the fact that for hyperbolic orbits,
spatio-temporal symmetries are robust under perturbations that leave the
system equivariant under the finite group action of the symmetry group of the
unperturbed system [36]. ∎

In the next section, we discuss the introduction of symmetry-breaking
perturbations and how averaging theory can be employed to approximate the
dynamics on $\mathcal{M}_{\delta}$ for such perturbations.

##  5 Introducing perturbations

We consider now model (2) where we assume that $h$ is smooth and normalized:

$\left|\left|h\right|\right|_{L^{2}}:=\sqrt{\sum_{n=2}^{\infty}\left|a_{n}\right|^{2}+\left|b_{n}\right|^{2}}=1.$

###  5.1 Averaging principle for WS-integrable systems

The techniques that we employed to prove theorem 4.12 allow the construction
of a NAIM for cross-ratios near $\boldsymbol{\lambda^{*}}$. However, normally
attracting manifolds can be extended and numerical results indicate that for
the classic active rotator model (3), this extension is “large” in the sense
that periodic orbits exist for all $\boldsymbol{\lambda}\in V$ so that
$\mathcal{M}$ intersects every
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ [25, 26]. From now
on, we drop the index $\delta$ and consider the largest possible extension
$\mathcal{M}$ of the NAIM of theorem 4.12 such that $\mathcal{M}$ still
consists of periodic orbits
$\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$,
exponentially stable in
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ and neutrally
stable in normal direction to
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$. The following
results apply as well but not exclusively to $\mathcal{M}_{\delta}$ from
theorem 4.12.

In a slight abuse of notation, we denote by $\mathcal{M}_{\epsilon}$ the
perturbed NAIM for (2) with $\epsilon\neq 0$ and the original NAIM for (1) as
$\mathcal{M}$. Our main objective is to determine what happens to the
infinitely many periodic orbits when $\epsilon\neq 0$. To determine robust
periodic orbits and their stability, we develop a criterion based on averaging
theory [27].

Let
$\boldsymbol{n}=(n_{\alpha},n_{\psi},n_{\boldsymbol{\lambda}})=\boldsymbol{m}^{-1}$
denote the inverse of the smooth diffeomorphism from lemma 4.8. In particular,
we have $n_{\lambda}=\boldsymbol{\Lambda}$. For definiteness, we write

$\boldsymbol{h}(\boldsymbol{\phi}):=(h(\phi_{1}),\dots,h(\phi_{N}))$

for the diagonal action of $h$ on $\mathbb{T}^{N}$. Form the chain rule, we
conclude that for every $x\in\\{\alpha,\psi,\boldsymbol{\lambda}\\}$, we have

$\dot{x}=\mathrm{D}n_{x}\cdot\dot{\boldsymbol{\phi}}=\left(\mathrm{D}n_{x}\cdot\boldsymbol{h}\right)(\boldsymbol{\phi}):=\sum_{j=1}^{N}h(\phi_{j})\cdot\mathrm{D}n_{x}(\boldsymbol{\phi})$

at the point $\boldsymbol{\phi}$ and equation (2) can be written in WS-
variables as

$\displaystyle\dot{\alpha}$
$\displaystyle=\rmi\left(f(Z)\alpha^{2}+g(Z)\alpha+\bar{f}(Z)\right)+\epsilon\left(\mathrm{D}n_{\alpha}\cdot\boldsymbol{h}\right)\circ\boldsymbol{m}(\alpha,\psi,\boldsymbol{\lambda})$
$\displaystyle\dot{\psi}$
$\displaystyle=\left(f(Z)\alpha+g(Z)+\bar{f}(Z)\bar{\alpha}\right)+\epsilon\left(\mathrm{D}n_{\psi}\cdot\boldsymbol{h}\right)\circ\boldsymbol{m}(\alpha,\psi,\boldsymbol{\lambda})$
$\displaystyle\dot{\boldsymbol{\lambda}}$
$\displaystyle=\epsilon\left(\mathrm{D}\boldsymbol{\Lambda}\cdot\boldsymbol{h}\right)\circ\boldsymbol{m}(\alpha,\psi,\boldsymbol{\lambda}).$

We treat the $\mathcal{O}(\epsilon)$-terms above as a perturbation of the WS-
integrable system (16). The perturbation terms in the first two equations are
unproblematic in terms of WS-theory since they leave the level sets invariant.
Only the last equation makes the system nonintegrable by making the level sets
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ noninvariant.

The following theorem constitutes our second main result.

######  Theorem 5.1.

For fixed $N$, consider the system (2) for
$\boldsymbol{\phi}=(\phi_{1},\dots,\phi_{N})\in\mathbb{T}^{N}_{\mathrm{ordered}}$
where $h$ is smooth and $\left|\left|h\right|\right|_{L^{2}}=1$. For
$\epsilon=0$, let

$\mathcal{M}=\bigcup_{\boldsymbol{\lambda}\in
W}\mathcal{C}_{\boldsymbol{\lambda}}$

denote the largest extension of the NAIM from theorem 4.12 in
$\mathbb{T}^{N}_{\mathrm{ordered}}$ such that $W$ is the largest open set for
which for every $\boldsymbol{\lambda}\in W\subset V$, there exists a periodic
orbit $\mathcal{C}_{\boldsymbol{\lambda}}$ which is exponentially stable in
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ and $\mathcal{M}$
is a NAIM. For every $\boldsymbol{\lambda}\in W$, fix a
$\boldsymbol{\theta}=\boldsymbol{\theta}(\boldsymbol{\lambda})\in\mathcal{C}_{\boldsymbol{\lambda}}$
and let $\boldsymbol{\phi}_{\boldsymbol{\lambda}}(t)$ denote the solution of
(1) with initial condition
$\boldsymbol{\phi}_{\boldsymbol{\lambda}}(0)=\boldsymbol{\theta}$. Then, the
following statements hold true:

  1. 1.

The function

$\boldsymbol{F}_{h}(\boldsymbol{\lambda}):=\frac{1}{T(\boldsymbol{\lambda})}\int_{0}^{T(\boldsymbol{\lambda})}\left(\mathrm{D}\boldsymbol{\Lambda}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\boldsymbol{\lambda}}(t)\,\mathrm{d}t,$
(24)

where $T(\boldsymbol{\lambda})$ is the period of
$\boldsymbol{\phi}_{\boldsymbol{\lambda}}$, is continuously differentiable,
well-defined, and independent of the choice of
$\boldsymbol{\theta}(\boldsymbol{\lambda})\in\mathcal{C}_{\boldsymbol{\lambda}}$.

  2. 2.

For $N=4$, the function $F_{h}$ fulfills

$F_{h}(\lambda)=-F_{h}(1-\lambda)$

and in particular, we have $F_{h}(\lambda^{*})=0$ with $\lambda^{*}=1/2$.

  3. 3.

For any $N\geq 4$,

$\boldsymbol{F}_{h}(\boldsymbol{\lambda^{*}})=\boldsymbol{0}.$

We discuss the significance of the function $\boldsymbol{F}_{h}$ for studying
the asymptotic dynamics of the perturbed system (2) and the implications from
statements 2. and 3. above for the existence of robust splay states in it in
the supplementary material.

###  5.2 Proof of Theorem 5.1

####  5.2.1 Proof of statement 1.

Let $\boldsymbol{\phi}_{\boldsymbol{\lambda}}$ be the periodic solution of (2)
with period $T(\boldsymbol{\lambda})>0$ and initial condition
$\boldsymbol{\phi}_{\boldsymbol{\lambda}}(0)=\boldsymbol{\theta}=\boldsymbol{\theta}(\boldsymbol{\lambda})\in\mathcal{C}_{\boldsymbol{\lambda}}$.
Then, the average of $\mathrm{D}\boldsymbol{\Lambda}\cdot\boldsymbol{h}$ over
$\mathcal{C}_{\boldsymbol{\lambda}}$ in the right hand side of (24) exists.
For any $\boldsymbol{\theta}^{\prime}\in\mathcal{C}_{\boldsymbol{\lambda}}$,
let $(\alpha^{\prime}(t),\psi^{\prime}(t),\boldsymbol{\lambda})$ be the
solution of (16) with
$\boldsymbol{m}(\alpha^{\prime}(0),\psi^{\prime}(0),\boldsymbol{\lambda})=\boldsymbol{\theta}^{\prime}$
and set
$\boldsymbol{\phi}_{\boldsymbol{\lambda}}^{\prime}(t):=\boldsymbol{m}(\alpha^{\prime}(t),\psi^{\prime}(t),\boldsymbol{\lambda})$.
Then, there exists a $\tau=\tau(\boldsymbol{\theta}^{\prime})$ such that
$\boldsymbol{\phi}_{\boldsymbol{\lambda}}^{\prime}(t)=\boldsymbol{\phi}_{\boldsymbol{\lambda}}(t+\tau)$.
But since we average over a full period, the integral is invariant under any
shift $\tau$ and thus is independent of the choice of $\boldsymbol{\theta}$.
$\boldsymbol{F}_{h}$ is continuously differentiable because both
$\boldsymbol{\Lambda}$ and $\boldsymbol{h}$ are smooth and
$\boldsymbol{\phi}_{\boldsymbol{\lambda}}$ and $T(\boldsymbol{\lambda})$
depend smoothly on the system parameter $\boldsymbol{\lambda}$ [29].

####  5.2.2 Proof of statement 2.

###### Proof.

For $N=4$, the level sets of $\Lambda$ in $\mathbb{T}^{N}_{\mathrm{ordered}}$
are parameterized by a single cross-ratio
$\Lambda(\boldsymbol{\theta}):=\Lambda_{1}(\boldsymbol{\theta}):=\Lambda_{1,2,3,4}(\boldsymbol{\theta})$.
Consider the cyclic permutation

$\sigma=\pmatrix{1&2&3&4\cr 4&1&2&3}$

and set for any
$\boldsymbol{\theta}=(\theta^{1},\theta^{2},\theta^{3},\theta^{4})\in\mathbb{T}^{N}_{\mathrm{ordered}}$

$\displaystyle\sigma\boldsymbol{\theta}=(\theta^{\sigma(1)},\theta^{\sigma(2)},\theta^{\sigma(3)},\theta^{\sigma(4)})=(\theta^{4},\theta^{1},\theta^{2},\theta^{3})\in\mathbb{T}^{N}_{\mathrm{ordered}}.$

For fixed
$\boldsymbol{\theta}=\boldsymbol{\theta}(\lambda)\in\mathcal{C}_{\lambda}$,
let

$\boldsymbol{\phi}_{\lambda}(t,\boldsymbol{\theta})=\left(\phi_{\lambda}^{1}(t,\boldsymbol{\theta}),\phi_{\lambda}^{2}(t,\boldsymbol{\theta}),\phi_{\lambda}^{3}(t,\boldsymbol{\theta}),\phi_{\lambda}^{4}(t,\boldsymbol{\theta})\right)$

denote the solution of the unperturbed system (1) with initial condition
$\boldsymbol{\phi}_{\lambda}(0)=\boldsymbol{\theta}$. In the same spirit as
above, we set

$\displaystyle\sigma\boldsymbol{\phi}_{\lambda}(t,\sigma\boldsymbol{\theta})=\left(\phi_{\lambda}^{4}(t,\boldsymbol{\theta}),\phi_{\lambda}^{1}(t,\boldsymbol{\theta}),\phi_{\lambda}^{2}(t,\boldsymbol{\theta}),\phi_{\lambda}^{3}(t,\boldsymbol{\theta})\right)$

for the cyclic permutation of
$\boldsymbol{\phi}_{\lambda}(t,\boldsymbol{\theta})$ which again is a solution
of (1) since the functions $f$ and $g$ in it depend solely on $Z$ which is
invariant under permutations of phases. The periodic orbit of this new
solution does in general not lie in $\mathcal{L}_{\lambda}(\Lambda)$ because
permutations of the components of any $\boldsymbol{\theta}$ transform the
cross-ratios $\Lambda(\boldsymbol{\theta})$ [28]. Indeed, for
$\boldsymbol{\theta}\in\mathcal{L}_{\lambda}(\Lambda)$, we have
$\sigma\boldsymbol{\theta}\in\mathcal{L}_{1-\lambda}(\Lambda)$ because

$\eqalign{\Lambda(\sigma\boldsymbol{\theta})&=\Lambda_{1,2,3,4}(\sigma\boldsymbol{\theta})\\\
&=\frac{\big{(}\rme^{\rmi\theta_{4}}-\rme^{\rmi\theta_{3}}\big{)}\big{(}\rme^{\rmi\theta_{1}}-\rme^{\rmi\theta_{2}}\big{)}}{\big{(}\rme^{\rmi\theta_{1}}-\rme^{\rmi\theta_{3}}\big{)}\big{(}\rme^{\rmi\theta_{4}}-\rme^{\rmi\theta_{2}}\big{)}}\\\
&=\Lambda_{4,1,2,3}(\boldsymbol{\theta})\equiv
1-\Lambda_{1,2,3,4}(\boldsymbol{\theta})\\\
&=1-\Lambda(\boldsymbol{\theta}).}$ (25)

Since
$\sigma\boldsymbol{\phi}_{\lambda}(0,\boldsymbol{\theta})=\sigma\boldsymbol{\theta}=\boldsymbol{\phi}_{1-\lambda}(0,\sigma\boldsymbol{\theta})$,
we have

$\sigma\boldsymbol{\phi}_{\lambda}(t,\boldsymbol{\theta})=\boldsymbol{\phi}_{1-\lambda}(t,\sigma\boldsymbol{\theta})$
(26)

by uniqueness of solutions and it follows that the periodic orbit of
$\sigma\boldsymbol{\phi}_{\lambda}$ is $\mathcal{C}_{1-\lambda}$. Note that
from this we also read

$T(\boldsymbol{\lambda})=T(1-\boldsymbol{\lambda}),$ (27)

i. e., the orbits $\mathcal{C}_{\lambda}$ and $\mathcal{C}_{1-\lambda}$ have
the same period. From (25), we read

$\displaystyle\left(\mathrm{D}\Lambda_{1,2,3,4}\cdot\boldsymbol{h}\right)(\sigma\boldsymbol{\theta})$
$\displaystyle=\mathrm{D}_{\vartheta_{1}}\Lambda_{1,2,3,4}(\boldsymbol{\vartheta})\cdot
h(\vartheta_{1})+\mathrm{D}_{\vartheta_{2}}\Lambda_{1,2,3,4}(\boldsymbol{\vartheta})\cdot
h(\vartheta_{2})+$
$\displaystyle\quad+\mathrm{D}_{\vartheta_{3}}\Lambda_{1,2,3,4}(\boldsymbol{\vartheta})\cdot
h(\vartheta_{3})+\mathrm{D}_{\vartheta_{4}}\Lambda_{1,2,3,4}(\boldsymbol{\vartheta})\cdot
h(\vartheta_{4})\Big{|}_{\boldsymbol{\vartheta}=\sigma\boldsymbol{\theta}}$
$\displaystyle=\mathrm{D}_{\theta_{4}}\Lambda_{4,1,2,3}(\boldsymbol{\theta})\cdot
h(\theta_{4})+\mathrm{D}_{\theta_{1}}\Lambda_{4,1,2,3}(\boldsymbol{\theta})\cdot
h(\theta_{1})+$
$\displaystyle\quad+\mathrm{D}_{\theta_{2}}\Lambda_{4,1,2,3}(\boldsymbol{\theta})\cdot
h(\theta_{2})+\mathrm{D}_{\theta_{3}}\Lambda_{4,1,2,3}(\boldsymbol{\theta})\cdot
h(\theta_{3})$
$\displaystyle=\;\left(\mathrm{D}\Lambda_{4,1,2,3}\cdot\boldsymbol{h}\right)(\boldsymbol{\theta})$

and find, using (26), (27), and (25),

$\displaystyle F_{h}(1-\lambda)$
$\displaystyle=\frac{1}{T(1-\lambda)}\int_{0}^{T(1-\lambda)}\left(\mathrm{D}\Lambda_{1,2,3,4}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{1-\lambda}(t,\sigma\boldsymbol{\theta})\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\lambda)}\int_{0}^{T(\lambda)}\left(\mathrm{D}\Lambda_{1,2,3,4}\cdot\boldsymbol{h}\right)\circ\sigma\boldsymbol{\phi}_{\lambda}(t,\boldsymbol{\theta})\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\lambda)}\int_{0}^{T(\lambda)}\left(\mathrm{D}\Lambda_{4,1,2,3}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\lambda}(t,\boldsymbol{\theta})\,\mathrm{d}t$
$\displaystyle=-\frac{1}{T(\lambda)}\int_{0}^{T(\lambda)}\left(\mathrm{D}\Lambda_{1,2,3,4}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\lambda}(t,\boldsymbol{\theta})\,\mathrm{d}t$
$\displaystyle=-F_{h}(\lambda)$

which in particular yields

$F_{h}(1/2)=0.$

Since $\lambda^{*}=1/2$ by (13) for $N=4$, $F_{h}$ indeed vanishes at
$\lambda^{*}$. ∎

####  5.2.3 Proof of statement 3.

Before we come to the case of general $N$, it is instructional to show for
$N=4$ again, that $F_{h}(\lambda^{*})=\boldsymbol{0}$. This is done through
the following calculation, using the facts that the average over a full period
of $\mathcal{C}_{\lambda^{*}}$ is invariant under shifts in time, that any
splay state is of the form $\phi_{j}(t)=\varphi(t+jT/N)$ with _some_
$T$-periodic function $\varphi$ such that for a splay, a shift in time by
$T(\lambda^{*})/N$ is equivalent to a cyclic permutation of units, and
equation (25):

$\displaystyle F_{h}(\lambda^{*})$
$\displaystyle=\frac{1}{T(\lambda^{*})}\int_{0}^{T(\lambda^{*})}\left(\mathrm{D}\Lambda_{1,2,3,4}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\lambda^{*}}(t)\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\lambda^{*})}\int_{0}^{T(\lambda^{*})}\left(\mathrm{D}\Lambda_{1,2,3,4}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\lambda^{*}}\left(t-\frac{T(\lambda^{*})}{4}\right)\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\lambda^{*})}\int_{0}^{T(\lambda^{*})}\left(\mathrm{D}\Lambda_{1,2,3,4}\cdot\boldsymbol{h}\right)\circ\sigma\boldsymbol{\phi}_{\lambda^{*}}(t)\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\lambda^{*})}\int_{0}^{T(\lambda^{*})}\left(\mathrm{D}\Lambda_{4,1,2,3}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\lambda^{*}}(t)\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\lambda^{*})}\int_{0}^{T(\lambda^{*})}\left(-\mathrm{D}\Lambda_{1,2,3,4}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\lambda^{*}}(t)\,\mathrm{d}t$
$\displaystyle=-F_{h}(\lambda^{*}).$

Note that we do not use the fact that the splay state lies in
$\mathcal{L}_{\lambda^{*}}(\Lambda)$. This means that the integral above
always vanishes for _any_ splay state of $N=4$ units _regardless_ of whether
the governing equations of motion are WS-integrable or whether its orbit lies
in $\mathcal{L}_{\lambda^{*}}(\Lambda)$. However, this is not the case if one
considers $N>4$ where one has to explicitly make use of the fact that the
splay state lies in
$\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$ as we show
below.

Before we come to the proof, we need to discuss our choice (10) for the cross-
ratio coordinates $\boldsymbol{\Lambda}$ on
$\mathbb{T}^{N}_{\mathrm{ordered}}$. The purpose of these coordinates was to
parameterize the level sets
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ that partition
$\mathbb{T}^{N}_{\mathrm{ordered}}$ and $\mathcal{M}$ such that for every
$\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$,
$\boldsymbol{\lambda}=\boldsymbol{\Lambda}(\boldsymbol{\theta})$ uniquely
determines in which set
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$ the point
$\boldsymbol{\theta}$ lies. But (10) is not the only possible choice. In fact,
every set of $N-3$ functionally independent cross-ratios $\Lambda_{p,q,r,s}$
will do the job and indeed, the authors of [13] used the cross-ratios

$\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})=\frac{\left(\rme^{\rmi\theta_{k}}-\rme^{\rmi\theta_{k+3}}\right)\left(\rme^{\rmi\theta_{k+1}}-\rme^{\rmi\theta_{k+2}}\right)}{\big{(}\rme^{\rmi\theta_{k}}-\rme^{\rmi\theta_{k+2}}\big{)}\big{(}\rme^{\rmi\theta_{k+1}}-\rme^{\rmi\theta_{k+3}}\big{)}}$
(28)

with $k=1,\dots,N-3$ to parameterize the partition of
$\mathbb{T}^{N}_{\mathrm{ordered}}$ in terms of level sets. While our previous
choice (10) was more suitable for the proof of thorem 4.12, the cross-ratio
functions (28) are suited for the analysis of splay states since they capture
the spatio-temporal relation between any four consecutive phases
$(\phi_{k}(t),\phi_{k+1}(t),\phi_{k+2}(t),\phi_{k+3}(t))$ for the splay state:

$\displaystyle\phi_{k+j}(t)=\phi_{k}(t+jT(\boldsymbol{\lambda^{*}})/N),\qquad
j=1,2,3.$

Let us therefore from now define the cross-ratio functions
$\boldsymbol{\Lambda}$ as

$\eqalign{\boldsymbol{\Lambda}(\boldsymbol{\theta})&:=\left(\Lambda_{1}(\boldsymbol{\theta}),\dots,\Lambda_{N-3}(\boldsymbol{\theta})\right)\\\
\Lambda_{k}(\boldsymbol{\theta})&:=\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})=\frac{\big{(}\rme^{\rmi\theta_{k}}-\rme^{\rmi\theta_{k+3}}\big{)}\big{(}\rme^{\rmi\theta_{k+1}}-\rme^{\rmi\theta_{k+2}}\big{)}}{\big{(}\rme^{\rmi\theta_{k}}-\rme^{\rmi\theta_{k+2}}\big{)}\big{(}\rme^{\rmi\theta_{k+1}}-\rme^{\rmi\theta_{k+3}}\big{)}}}$
(29)

with $k=1,\dots,N-3$. In these new cross-ratio coordinates, the leaf of
uniform distributions
$\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$ is determined
by

$\boldsymbol{\lambda^{*}}:=\boldsymbol{\Lambda}(\boldsymbol{\theta^{*}})=\left(\lambda^{*}_{1},\dots,\lambda^{*}_{1}\right)$

where $\lambda^{*}_{1}$ is given through (13) with $k=1$. For any given
periodic orbit
$\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$,
hyperbolic fixed points of the averaged system

$\dot{\boldsymbol{\lambda}}=\boldsymbol{F}_{h}(\boldsymbol{\lambda})$

where
$\boldsymbol{F}_{h}(\boldsymbol{\lambda})=((F_{h})_{1}(\boldsymbol{\lambda}),\dots,(F_{h})_{N-3}(\boldsymbol{\lambda}))$
is defined by

$(F_{h})_{k}(\boldsymbol{\lambda}):=\frac{1}{T(\boldsymbol{\lambda})}\int_{0}^{T(\boldsymbol{\lambda})}\left(\mathrm{D}\Lambda_{k,k+1,k+2,k+3}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\boldsymbol{\lambda}}(t)\,\mathrm{d}t$

for $k=1,\dots,N-3$, correspond to periodic orbits of the original system.
With these remarks, we are ready to prove the last statement of theorem 5.1.

###### Proof.

Consider the cyclic permutation

$\sigma=\pmatrix{1&2&\dots&N\cr N&1&\dots&N-1}$

and as in the previous section, let for fixed
$\boldsymbol{\theta}\in\mathcal{C}_{\boldsymbol{\lambda^{*}}}$

$\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)=\left(\phi_{\boldsymbol{\lambda^{*}}}^{1}(t),\dots,\phi_{\boldsymbol{\lambda^{*}}}^{N}(t)\right)$

denote the the splay state solution of (1) with initial condition
$\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(0)=\boldsymbol{\theta}$ and set

$\displaystyle\sigma\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t):=\left(\phi_{\boldsymbol{\lambda^{*}}}^{\sigma(1)}(t),\phi_{\boldsymbol{\lambda^{*}}}^{\sigma(2)},\dots,\phi_{\boldsymbol{\lambda^{*}}}^{\sigma(N)}(t)\right).$

Using that for splay states
$\phi_{\boldsymbol{\lambda^{*}}}^{k+1}(t)=\phi_{\boldsymbol{\lambda^{*}}}^{k}(t+T(\boldsymbol{\lambda^{*}})/N)$
holds, we then find

$\eqalign{\sigma\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)&=\left(\phi_{\boldsymbol{\lambda^{*}}}^{\sigma(1)}(t),\phi_{\boldsymbol{\lambda^{*}}}^{\sigma(2)},\dots,\phi_{\boldsymbol{\lambda^{*}}}^{\sigma(N)}(t)\right)\\\
&=\left(\phi_{\boldsymbol{\lambda^{*}}}^{N}(t),\phi_{\boldsymbol{\lambda^{*}}}^{1}(t),\dots,\phi_{\boldsymbol{\lambda^{*}}}^{N-1}(t)\right)\\\
&=\left(\phi_{\boldsymbol{\lambda^{*}}}^{1}\left(t-\frac{T(\boldsymbol{\lambda^{*}})}{N}\right),\dots,\phi_{\boldsymbol{\lambda^{*}}}^{N}\left(t-\frac{T(\boldsymbol{\lambda^{*}})}{N}\right)\right)\\\
&=\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}\left(t-\frac{T(\boldsymbol{\lambda^{*}})}{N}\right).}$
(30)

Note that for the choice (29) for the cross-ratios, we have

$\eqalign{\Lambda_{k+1,k+2,k+3,k+4}(\sigma\boldsymbol{\theta})&=\frac{\big{(}\rme^{\rmi\theta_{\sigma(k+1)}}-\rme^{\rmi\theta_{\sigma(k+4)}}\big{)}\big{(}\rme^{\rmi\theta_{\sigma(k+2)}}-\rme^{\rmi\theta_{\sigma(k+3)}}\big{)}}{\big{(}\rme^{\rmi\theta_{\sigma(k+1)}}-\rme^{\rmi\theta_{\sigma(k+3)}}\big{)}\big{(}\rme^{\rmi\theta_{\sigma(k+2)}}-\rme^{\rmi\theta_{\sigma(k+4)}}\big{)}}\\\
&=\frac{\big{(}\rme^{\rmi\theta_{k}}-\rme^{\rmi\theta_{k+3}}\big{)}\big{(}\rme^{\rmi\theta_{k+1}}-\rme^{\rmi\theta_{k+2}}\big{)}}{\big{(}\rme^{\rmi\theta_{k}}-\rme^{\rmi\theta_{k+2}}\big{)}\big{(}\rme^{\rmi\theta_{k+1}}-\rme^{\rmi\theta_{k+3}}\big{)}}\\\
&=\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})}$ (31)

for all $\boldsymbol{\theta}\in\mathbb{T}^{N}_{\mathrm{ordered}}$. Because of
the spatio-temporal symmetry (30) of the splay state and using (31), we can
for $k=1,\dots,N$ (where we set $N+1\equiv 1$, $N+2\equiv 2$, and $N+3\equiv
3$) write

$\displaystyle(F_{h})_{k+1}(\boldsymbol{\lambda^{*}})$
$\displaystyle:=\frac{1}{T(\boldsymbol{\lambda^{*}})}\int_{0}^{T(\boldsymbol{\lambda^{*}})}\left(\mathrm{D}\Lambda_{k+1,k+2,k+3,k+4}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\boldsymbol{\lambda^{*}})}\int_{0}^{T(\boldsymbol{\lambda^{*}})}\left(\mathrm{D}\Lambda_{k+1,k+2,k+3,k+4}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}\left(t-\frac{T(\boldsymbol{\lambda^{*}})}{N}\right)\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\boldsymbol{\lambda^{*}})}\int_{0}^{T(\boldsymbol{\lambda^{*}})}\left(\mathrm{D}\Lambda_{k+1,k+2,k+3,k+4}\cdot\boldsymbol{h}\right)\circ\sigma\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)\,\mathrm{d}t$
$\displaystyle=\frac{1}{T(\boldsymbol{\lambda^{*}})}\int_{0}^{T(\boldsymbol{\lambda^{*}})}\left(\mathrm{D}\Lambda_{k,k+1,k+2,k+3}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)\,\mathrm{d}t$
$\displaystyle=(F_{h})_{k}(\boldsymbol{\lambda^{*}}).$

This means that in particular, the $N-3$ components (i. e., $k=1,\dots,N-3$)
of $\boldsymbol{F}_{h}(\boldsymbol{\lambda^{*}})$ coincide. We also have for
every $j=1,\dots,N-3$

$\displaystyle(F_{h})_{j}(\boldsymbol{\lambda^{*}})$
$\displaystyle=\frac{1}{N}\sum_{k=1}^{N}(F_{h})_{k}(\boldsymbol{\lambda^{*}})$
$\displaystyle=\frac{1}{NT(\boldsymbol{\lambda^{*}})}\int_{0}^{T(\boldsymbol{\lambda^{*}})}\left(\sum_{k=1}^{N}\mathrm{D}\Lambda_{k,k+1,k+2,k+3}\cdot\boldsymbol{h}\right)\circ\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)\,\mathrm{d}t$
$\displaystyle=\frac{1}{NT(\boldsymbol{\lambda^{*}})}\int_{0}^{T(\boldsymbol{\lambda^{*}})}\sum_{k=1}^{N}\Big{(}\mathrm{D}_{\theta_{k}}\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})\cdot
h(\theta_{k})+$ $\displaystyle\hskip
98.16191pt+\mathrm{D}_{\theta_{k+1}}\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})\cdot
h(\theta_{k+1})+$ $\displaystyle\hskip
98.16191pt+\mathrm{D}_{\theta_{k+2}}\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})\cdot
h(\theta_{k+2})+$ $\displaystyle\hskip
98.16191pt+\mathrm{D}_{\theta_{k+3}}\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})\cdot
h(\theta_{k+3})\Big{)}\Big{|}_{\boldsymbol{\theta}=\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)}\,\mathrm{d}t.$

Rearranging this sum by collecting all terms that contain derivatives with
respect to a given $\theta_{k}$ yields

$\eqalign{(\hat{F}_{h})_{j}(\boldsymbol{\lambda^{*}})&=\frac{1}{NT(\boldsymbol{\lambda^{*}})}\int_{0}^{T(\boldsymbol{\lambda^{*}})}\sum_{k=1}^{N}\mathrm{D}_{\theta_{k}}\Big{(}\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})+\\\
&\hskip 113.81102pt+\Lambda_{k-1,k,k+1,k+2}(\boldsymbol{\theta})+\\\ &\hskip
113.81102pt+\Lambda_{k-2,k-1,k,k+1}(\boldsymbol{\theta})+\\\ &\hskip
113.81102pt+\Lambda_{k-3,k-2,k-1,k}(\boldsymbol{\theta})\Big{)}\cdot
h(\theta_{k})\Big{|}_{\boldsymbol{\theta}=\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)}\,\mathrm{d}t.}$
(32)

All terms in the sum in (32) vanish so the integral yields zero. To see this,
observe that for any
$\boldsymbol{\theta}\in\mathcal{L}_{\boldsymbol{\lambda^{*}}}(\boldsymbol{\Lambda})$,
there exists $(\alpha,\psi)\in\mathbb{D}\times\mathbb{S}^{1}$ such that
$\rme^{\rmi\boldsymbol{\theta}}=G_{\alpha,\psi}\left(\rme^{\rmi\boldsymbol{\theta^{*}}}\right)$
with $\boldsymbol{\theta^{*}}$ from (12), yielding

$\displaystyle\mathrm{D}_{\theta_{k}}\Lambda_{k-3,k-2,k-1,k}(\boldsymbol{\theta})$
$\displaystyle=\rmi\rme^{\rmi\theta_{k}}\frac{\left(\rme^{\rmi\theta_{k-3}}-\rme^{\rmi\theta_{k-2}}\right)\left(\rme^{\rmi\theta_{k-2}}-\rme^{\rmi\theta_{k-1}}\right)}{\left(\rme^{\rmi\theta_{k-3}}-\rme^{\rmi\theta_{k-1}}\right)\left(\rme^{\rmi\theta_{k-2}}-\rme^{\rmi\theta_{k}}\right)^{2}}$
$\displaystyle=\frac{\rmi\rme^{\rmi\frac{2\pi(k-1)}{N}-\rmi\psi}\left(\rme^{\rmi\frac{2\pi
k}{N}+\rmi\psi}-\rme^{\rmi\frac{2\pi}{N}}\alpha\right)\left(\rme^{\rmi\frac{2\pi}{N}}-\rme^{\rmi\frac{2\pi
k}{N}+\rmi\psi}\bar{\alpha}\right)}{\left(1-\rme^{\rmi\frac{2\pi}{N}}\right)\left(1+\rme^{\rmi\frac{2\pi}{N}}\right)^{3}\left(1-\left|\alpha\right|^{2}\right)}$
$\displaystyle=-\rmi\rme^{\rmi\theta_{k}}\frac{\left(\rme^{\rmi\theta_{k+3}}-\rme^{\rmi\theta_{k+2}}\right)\left(\rme^{\rmi\theta_{k+2}}-\rme^{\rmi\theta_{k+1}}\right)}{\left(\rme^{\rmi\theta_{k+3}}-\rme^{\rmi\theta_{k+1}}\right)\left(\rme^{\rmi\theta_{k+2}}-\rme^{\rmi\theta_{k}}\right)^{2}}$
$\displaystyle=-\mathrm{D}_{\theta_{k}}\Lambda_{k,k+1,k+2,k+3}(\boldsymbol{\theta})$

so that the first and last term in each summand in (32) cancel out. We also
have

$\displaystyle\mathrm{D}_{\theta_{k}}\Lambda_{k-2,k-1,k,k+1}(\boldsymbol{\theta})$
$\displaystyle=\rmi\rme^{\rmi\theta_{k}}\frac{\left(\rme^{\rmi\theta_{k-1}}-\rme^{\rmi\theta_{k-2}}\right)\left(\rme^{\rmi\theta_{k+1}}-\rme^{\rmi\theta_{k-2}}\right)}{\left(\rme^{\rmi\theta_{k-1}}-\rme^{\rmi\theta_{k+1}}\right)\left(\rme^{\rmi\theta_{k}}-\rme^{\rmi\theta_{k-2}}\right)^{2}}$
$\displaystyle=-\frac{\rmi\left(1+\rme^{\rmi\frac{2\pi}{N}}+\rme^{\rmi\frac{4\pi}{N}}\right)\left(1-\rme^{-\rmi\frac{2\pi(k-1)}{N}-\rmi\psi}\alpha\right)\left(\rme^{\rmi\frac{2\pi}{N}}-\rme^{\rmi\frac{2\pi
k}{N}+\rmi\psi}\bar{\alpha}\right)}{\left(1-\rme^{\rmi\frac{2\pi}{N}}\right)\left(1+\rme^{\rmi\frac{2\pi}{N}}\right)^{3}\left(1-\left|\alpha\right|^{2}\right)}$
$\displaystyle=-\rmi\rme^{\rmi\theta_{k}}\frac{\left(\rme^{\rmi\theta_{k-1}}-\rme^{\rmi\theta_{k+2}}\right)\left(\rme^{\rmi\theta_{k+1}}-\rme^{\rmi\theta_{k+2}}\right)}{\left(\rme^{\rmi\theta_{k-1}}-\rme^{\rmi\theta_{k+1}}\right)\left(\rme^{\rmi\theta_{k}}-\rme^{\rmi\theta_{k+2}}\right)^{2}}$
$\displaystyle=-\mathrm{D}_{\theta_{k}}\Lambda_{k-1,k,k+1,k+2}(\boldsymbol{\theta})$

so that the second and third term in the summand also cancel and the
integrands for all components of
$\boldsymbol{F}_{h}(\boldsymbol{\lambda^{*}})$ vanish. This finishes the proof
of statement 3. of theorem 5.1. ∎

##  6 Application: repulsively coupled identical active rotators

Finally, we apply theorems 4.12 and 5.1 to our original motivational model of
identical classic and generalized active rotators (3) and (4), starting with
the classic case.

###  6.1 Classic active rotators

Recall that in order for (3) to describe a system of classic active rotators
and not a full blown oscillators, $\left|\omega\right|<1$ holds. First, we
determine the NAIM from theorem 4.12.

We start by writing down Equation (3) in terms of WS-variables. Comparing with
(1), we have $g(Z)=\omega$ and
$f(Z)=\frac{\rmi}{2}\left(1+\kappa\bar{Z}\right)$ so that in WS-variables, we
have

$\displaystyle\dot{\alpha}$
$\displaystyle=-\frac{1}{2}(1+\kappa\bar{Z})\alpha^{2}+\rmi\omega\alpha+\frac{1}{2}(1+\kappa
Z)$ $\displaystyle\dot{\psi}$
$\displaystyle=\frac{\rmi}{2}(1+\kappa\bar{Z})\alpha+\omega-\frac{\rmi}{2}(1+\kappa
Z)\bar{\alpha}$ $\displaystyle\dot{\boldsymbol{\lambda}}$
$\displaystyle=\boldsymbol{0}.$

Replacing $Z$ by $\alpha$ then yields the closed (truncated) equation

$\displaystyle\dot{\alpha}=-\frac{1}{2}(1+\kappa\bar{\alpha})\alpha^{2}+\rmi\omega\alpha+\frac{1}{2}(1+\kappa\alpha)$
(33)

for $\dot{\alpha}$. We proceed by determining the fixed point $\alpha_{0}$ for
(33) in theorem 4.12. Setting $\alpha=r\rme^{\rmi\beta}$ and subsequently
$\dot{\alpha}=\dot{r}\rme^{\rmi\beta}+\rmi\dot{\beta}r\rme^{\rmi\beta}$ yields

$\displaystyle\dot{r}+\rmi\dot{\beta}r$
$\displaystyle=\left(-\frac{1}{2}r^{2}\cos\beta-\frac{\kappa}{2}r^{3}+\frac{1}{2}\cos\beta+\frac{\kappa}{2}r\right)+\rmi\left(-\frac{1}{2}r^{2}\sin\beta+\omega
r-\frac{1}{2}\sin\beta\right).$

The fixed point condition implies

$\displaystyle(r^{2}-1)\cos\beta+(r^{2}-1)\kappa
r=0\qquad\textrm{and}\qquad\frac{1}{2}(r^{2}+1)\sin\beta=\omega r$

and thus, since $r<1$,

$\eqalign{\cos\beta=-\kappa r\qquad\textrm{and}\qquad\sin\beta=\frac{2\omega
r}{(1+r^{2})}.}$ (34)

Eliminating the trigonometric terms and setting $x=r^{2}$ yields a cubic
equation in $x$:

$0=\kappa^{2}x^{3}+(2\kappa^{2}-1)x^{2}+(\kappa^{2}+4\omega^{2}-2)x-1.$ (35)

We make the following claim:

######  Lemma 6.1.

The cubic equation (35) has for $\omega^{2}<1$ exactly one real root
$x\in(0,1)$ if $\kappa^{2}>1-\omega^{2}$ and no real roots in $(0,1)$ if
$\kappa^{2}<1-\omega^{2}$.

###### Proof.

We solve (35) for $\kappa^{2}$ and find

$\kappa^{2}=\frac{1}{x}-\frac{4\omega^{2}}{(1+x)^{2}}.$

We show that for any $\omega^{2}<1$ the map $(0,1)\ni x\mapsto
1/x-4\omega^{2}/(1+x)^{2}\in(1-\omega^{2},\infty)$ is a bijection from which
the claim follows because bijectivity implies conversely that for any
$\omega^{2}<1$ and $\kappa^{2}>1-\omega^{2}$ there exists a unique $x\in(0,1)$
that solves (35) and there is no such $x$ for $\kappa^{2}<1-\omega^{2}$.
Injectivity: The map is differentiable in $(0,1)$ so that

$\displaystyle\frac{\mathrm{d}\kappa^{2}}{\mathrm{d}x}=-\frac{1}{x^{2}}+\frac{8\omega^{2}}{(1+x)^{3}}$
$\displaystyle<0\quad\Leftrightarrow\quad\frac{(1+x)^{3}}{8x^{2}}$
$\displaystyle>\omega^{2}.$

But $\omega^{2}<1<(1+x)^{3}/8x^{2}$ holds because $(1+x)^{3}/8x^{2}$ is
strictly monotonically decreasing in $(0,1)$ since
$\,\mathrm{d}/\,\mathrm{d}x\left((1+x)^{3}/8x^{2}\right)=(x-2)(1+x)^{2}/8x^{3}<0\;\forall
x\in(0,1)$ and its infimum is $\lim_{x\to 1}(1+x)^{3}/8x^{2}=1$. Hence
$\kappa^{2}$ as a function of $x$ is strictly monotonically decreasing and
therefore injective.
Surjectivity: Because $\kappa^{2}$ is as a function of $x$ continuous and
strictly monotonically decreasing, we find that its image $(a,b)$ is given by

$\displaystyle a$ $\displaystyle=\lim_{x\to
1}\frac{1}{x}-\frac{4\omega^{2}}{(1+x)^{2}}=1-\omega^{2}\qquad\textrm{and}\qquad
b$ $\displaystyle=\lim_{x\to
0}\frac{1}{x}-\frac{4\omega^{2}}{(1+x)^{2}}=\infty$

so that the map is indeed surjective and thus bijective. ∎

Using (34), we write

$\alpha_{0}=r(\cos\beta+i\sin\beta)=-\kappa r^{2}+2\rmi\frac{\omega
r^{2}}{1+r^{2}}$

and find from lemma 6.1 that as long as $\omega\neq 0$

$\displaystyle f(\alpha_{0})$
$\displaystyle=\frac{\rmi}{2}\left(1+\kappa\bar{\alpha}_{0}\right)=\frac{\omega\kappa
r^{2}}{1+r^{2}}+\frac{\rmi}{2}\left(1-\kappa^{2}r^{2}\right)\neq 0$

since $\kappa^{2}>1-\omega^{2}>0$ and so we have

$\dot{\psi}=\omega-\mathrm{Im\;}\alpha_{0}=\omega\left(1-\frac{2r^{2}}{1+r^{2}}\right)=\omega\frac{1-r^{2}}{1+r^{2}}=:\Omega\neq
0.$

Note that for $\omega=0$, the contours $\mathcal{C}_{\boldsymbol{\lambda}}$
and therefore the manifold $\mathcal{M}_{\delta}$ consist of fixed points with
two stable and $N-2$ neutral directions.

We determine the stability of $\alpha_{0}$. Again, we treat $\alpha$ and
$\bar{\alpha}$ as independent variables in $\mathbb{D}$ so that the dynamics
in $\mathbb{D}$ is given by

$\displaystyle\dot{\alpha}$
$\displaystyle=-\frac{1}{2}(1+\kappa\bar{\alpha})\alpha^{2}+\rmi\omega\alpha+\frac{1}{2}(1+\kappa\alpha).$

Determining the Jacobian of the right hand side of this equation yields

$\frac{\partial(\dot{\alpha},\dot{\bar{\alpha}})}{\partial(\alpha,\bar{\alpha})}=\pmatrix{-(1+\kappa\bar{\alpha})\alpha+\rmi\omega+\frac{\kappa}{2}&-\frac{\kappa}{2}\alpha^{2}\cr-\frac{\kappa}{2}\bar{\alpha}^{2}&-(1+\kappa\alpha)\bar{\alpha}-\rmi\omega+\frac{\kappa}{2}}.$
(36)

At the fixed point $\alpha_{0}=r\rme^{\rmi\beta}$, using equations (34), the
eigenvalues of (36) read

$\lambda_{\pm}=\frac{\kappa}{2}\pm\sqrt{\frac{\kappa^{2}}{4}x^{2}-\omega^{2}\left(\frac{1-x}{1+x}\right)^{2}}.$
(37)

It follows that
$\mathrm{sign}\,\mathrm{Re\;}\,\lambda_{\pm}=\mathrm{sign}\,\kappa$: We have
two cases, that of $\lambda_{\pm}\in\mathbb{R}$ and that of
$\lambda_{\pm}\notin\mathbb{R}$. In the first case, the radicand in (37) is
nonnegative and we have

$\frac{\kappa^{2}}{4}>\frac{\kappa^{2}}{4}x^{2}-\omega^{2}\left(\frac{1-x}{1+x}\right)^{2}$

since $x\in(0,1)$ and the subtraction of a nonnegative term and hence
$\mathrm{sign}\,\mathrm{Re\;}\lambda_{\pm}=\mathrm{sign}\,\kappa$. In the
second case, the radicand is negative and the statement holds trivially.

To conclude, the above results on the existence and stability of $\alpha_{0}$
show that the system (3) fulfills the requirements of theorem 4.12 if
$0<\left|\omega\right|<1$ and $\kappa<-\sqrt{1-\omega^{2}}$. This establishes
the final result of this section:

######  Theorem 6.2.

Let
$\boldsymbol{\phi}(t)=(\phi_{1}(t),\dots,\phi_{N}(t))\in\mathbb{T}^{N}_{\mathrm{ordered}}$
obey (3) with parameters $0<\omega^{2}<1$ and $\kappa<-\sqrt{1-\omega^{2}}$.
Then, for every $\delta>0$, there exists an $N_{0}\in\mathbb{N}$ such that for
all $N\geq N_{0}$, there exists a closed $\delta$-neighborhood
$\overline{V}_{\delta}=\overline{V}_{\delta}(\boldsymbol{\lambda^{*}})\subset
V$ of $\boldsymbol{\lambda^{*}}$, where for every
$\boldsymbol{\lambda}\in\overline{V}_{\delta}$, there exists a unique periodic
orbit
$\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$
which is exponentially stable in
$\mathcal{L}_{\boldsymbol{\lambda}}(\boldsymbol{\Lambda})$. The union

$\mathcal{M}(\delta):=\bigcup_{\boldsymbol{\lambda}\in\overline{V}_{\delta}}\mathcal{C}_{\boldsymbol{\lambda}}\subset\mathbb{T}^{N}_{\mathrm{ordered}}$

forms a compact normally attracting manifold of dimension $N-2$. Further, the
periodic solution $\boldsymbol{\phi}_{\boldsymbol{\lambda^{*}}}(t)$ with orbit
$\mathcal{C}_{\boldsymbol{\lambda^{*}}}\subset\mathcal{M}_{\delta}$ is a splay
state.

###### Proof.

This is a corollary of theorem 4.12 and proposition 4.13. ∎

###  6.2 Generalized active rotators

Figure 1: Three examples for $F_{h}(\lambda)$ for $N=4$ Active Rotators
obeying (2) with $\omega=0.8$ and $\kappa=-0.7$. The three choices for $h$ are
$h_{1}:\phi\mapsto\sin 2\phi$, $h_{2}:\phi\mapsto-\cos 2\phi$, and
$h_{3}=(3/5)h_{1}+(4/5)h_{2}$. Note the additional zeros for the case of
$h_{3}$.

We apply the averaging method from section 5 to the generalized active rotator
model (4) to determine hyperbolic orbits. We focus on the case of $N=4$
rotators for which we are dealing with a single cross-ratio
$\lambda=\Lambda_{1,2,3,4}(\boldsymbol{\theta})$ and one-dimensional averaged
dynamics $\dot{\lambda}=F_{h}(\lambda)$.

Concerning the hyperbolicity of the splay state, figure 1 illustrates how
$F_{h}$ vanishes at $\lambda^{*}$ in accordance with statement 2. of theorem
5.1 and while $\mathrm{D}F_{h}(\lambda^{*})\neq 0$ so that $\lambda^{*}$
becomes a hyperbolic fixed point for the averaged dynamics and
$\mathcal{C}_{\epsilon,\lambda^{*}}$ a robust periodic orbit. In the figure,
we plot $F_{h}$ for three different choices of $h$, the last one being a
linear combination of the former two. The splay state persists under all three
perturbations. Note that from the numerical results, we have

$\lim_{\lambda\to 0}F_{h}(\lambda)=\lim_{\lambda\to 1}F_{h}(\lambda)=0$

which, if we include these limit cases, indicates the existence of additional
periodic orbits which must be clustered. Indeed, at least for the generalized
active rotator model (4) with $N=4$, the values $\lambda=0$ and $\lambda=1$
correspond to periodic two-cluster states [17, 26] where the ensemble splits
in two clusters of two units each. From figure 1, we read that these two-
cluster states are also exponentially (un)stable periodic solutions, depending
on the sign of $\epsilon$. Note that for the perturbation function $h_{3}$ we
find two further zeros which indicate the existence of additional periodic
orbits besides the splay state which are of broken spatio-temporal symmetry
[26]. The graph of $F_{h_{3}}$ is a linear combination of the two graphs of
from the first two Panels in figure 1 just like $h_{3}$ is a linear
combination of $h_{1}$ and $h_{2}$. This relation and its consequences on
whether splay states are robust under generic perturbations are further
elaborated in the supplementary material.

##  7 Conclusion and outlook

Families of periodic orbits are a common occurrence in Watanabe-Strogatz
integrable systems. For the case that all involved common fields only depend
on the Kuramoto order parameter, we showed that the union of these orbits
further possesses the structure of a normally attracting invariant manifold
and that one of the orbits is a splay state. To determine which of the
periodic orbits survive for the perturbed system, we developed a criterion by
applying the method of averaging to the dynamics on the NAIM.

One of the orbits that formed the invariant manifold of the unperturbed system
features splay state dynamics due to equivariance under cyclic permutations.
We showed that averaging over the splay state orbit of the unperturbed system
yields a fixed point in the averaged dynamics for the perturbed system.
Whether it also hyperbolic depends on the specific setup of the perturbed
system and in particular on the choice for the perturbation function.

Applying our results to the active rotator model by Shinomoto and Kuramoto,
our observations are two-fold: (i) We determined the critical coupling
strength below which the model possesses the family of periodic orbits. (ii)
Applying the averaging principle to the perturbed or generalized active
rotator model, numerical evidence indicates that averaging over the
unperturbed splay state indeed results in a hyperbolic fixed point for the
averaged dynamics and thus the splay state is generically hyperbolic for
ensembles of identical generalized active rotators. It also illustrates the
versatility of the proposed procedure as an easy-to-implement tool to study
perturbations to WS-integrable systems.

In restricting attention to the case of solely $Z$-dependent common fields $f$
and $g$, we excluded, e. g., systems of identically driven elements with
external forcing [38] or ensembles that consist of subpopulations [39], both
of which are WS-integrable. The question thus arises whether, and if, how the
results, presented here, may be generalized to such and other WS-integrable
systems.

We thank Zheng Bian, Jaap Eldering, and Edmilson Roque dos Santos for valuable
discussions and comments. This work has been performed within the scope of the
IRTG 1740/TRP 2015/50122-0 and funded by the DFG and FAPESP. TP was also
supported in part by FAPESP Cemeai Grant No. 2013/07375-0, by Serrapilheira
Institute (Grant No. Serra-1709-16124) and is a Newton Advanced Fellow of the
Royal Society NAF\R1\180236.

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