# Capturing the Edge of Chaos as a Spectral Submanifold in Pipe Flows

Bálint Kaszás1  George Haller1<EMAIL_ADDRESS>1Institute for Mechanical
Systems, ETH Zürich Leonhardstrasse 21, 8092 Zürich, Switzerland

###### Abstract

An extended turbulent state can coexist with the stable laminar state in pipe
flows. We focus here on short pipes with additional discrete symmetries
imposed. In this case, the boundary between the coexisting basins of
attraction, often called the edge of chaos, is the stable manifold of an edge
state, which is a lower-branch traveling wave solution. We show that a low-
dimensional submanifold of the edge of chaos can be constructed from velocity
data using the recently developed theory of spectral submanifolds (SSMs).
These manifolds are the unique smoothest nonlinear continuations of
nonresonant spectral subspaces of the linearized system at stationary states.
Using very low dimensional SSM-based reduced-order models, we predict
transitions to turbulence or laminarization for velocity fields near the edge
of chaos.

###### keywords:

##  1 Introduction

Flow transition to turbulence has been a topic of great interest for a long
time (Eckhardt (2008)). As the laminar state remains stable for all Reynolds
numbers in plane Couette flow and pipe flow (see Schmid & Henningson (2001);
Meseguer & Trefethen (2003)), the transition to turbulence is not triggered by
a linear instability in these flows. This necessitates the use of inherently
nonlinear methods for the analysis of transitions in such flows.

Thanks to the increase in computational power, it is now possible to explore
structures directly in the phase space of the Navier-Stokes equations. In this
fashion, special invariant solutions that govern the flow behavior have been
identified as fixed points or periodic orbits. Starting with the discovery of
the upper and lower-branch fixed points of plane Couette flow by Nagata
(1990), an extensive library of Exact Coherent States (ECS) has been assembled
in various flow configurations (Waleffe (2001); Graham & Floryan (2021)), many
of which have also been observed experimentally (Hof et al. (2004)). An
important phenomenon revealed by these dynamical system approaches is that the
laminar state could coexist with the turbulent one in pipe flows (Duguet et
al. (2008)). For a discussion on the emergence of sustained turbulence, we
refer to Avila et al. (2011) and the recent review by Avila et al. (2023).

We focus here on flow in a periodic domain, where turbulence is not sustained
(Hof et al. (2008)) and hence can be characterized as a chaotic saddle (Brosa
(1989); Lai & Tél (2011)). This is also true for the dynamics of individual
puffs, whose lifetimes grow rapidly with the Reynolds number (Avila et al.
(2010)). Therefore, even if turbulence has a finite lifetime, this lifetime
can often be greater than all practically relevant time scales (Lai & Winslow
(1995)). This makes it feasible to define the boundary that separates
trajectories immediately converging to the laminar state from those exhibiting
transiently chaotic dynamics as the edge manifold or the edge of chaos (Skufca
et al. (2006); Schneider et al. (2007)). Embedded within the edge of chaos,
saddle-type edge states exist (Skufca et al. (2006); Kerswell & Tutty (2007);
De Lozar et al. (2012)) whose stable manifolds act as the boundary between the
two types of behaviors. Schneider et al. (2007) and Mellibovsky et al. (2009)
found that the dynamics within the edge may be chaotic.

Specifically, in short pipes under appropriate symmetry restrictions, the edge
of chaos is formed as the stable manifold of a traveling wave solution, known
as the lower-branch (LB) traveling wave (see Pringle & Kerswell (2007); Duguet
et al. (2008); Kerswell & Tutty (2007)). In this case, the edge manifold is a
smooth codimension-one invariant manifold that guides trajectories towards the
lower-branch traveling wave. The edge manifold and the edge state play a
crucial role in the transition to turbulence and the decay from turbulence, as
discussed by De Lozar et al. (2012).

The most reliable way to probe the edge has been the edge tracking algorithm
introduced by Itano & Toh (2001) and Skufca et al. (2006) in which one selects
a turbulent trajectory and a laminarizing trajectory and bisects them. Based
on the behavior of the resulting trajectory (whether it is turbulent or
laminarizing) one then takes a new bisection. This process is repeated until a
trajectory is found that is neither turbulent nor laminarizing for long times
(see also Beneitez et al. (2019)). The bisection method can be aided by
stabilizing the edge states, as discussed by Willis et al. (2017). They
propose a simple feedback control scheme, based on the Reynolds number, to
remove the unstable directions of the edge state. Forward integration of the
controlled system thus results in rapid convergence towards the edge. Linkmann
et al. (2020) have adapted this method to prevent the control scheme from
introducing additional unstable directions. An alternative characterization of
the edge of chaos is given by Beneitez et al. (2020), who reinterpret the edge
as a Lagrangian coherent structure (see Haller (2015, 2023)).

Reduced-order models promise an efficient way to describe the transition to
turbulence. They are typically obtained from a Galerkin projection of the
Navier-Stokes equations onto a small set of spatial modes (Eckhardt & Mersmann
(1999); Moehlis et al. (2004); Joglekar et al. (2015)). Alternatively, data-
driven methods can infer reduced-order models directly from time-resolved
simulations or experiments.

Common approaches to data-driven reduced-order modeling are linear, such as
the dynamic mode decomposition introduced by Schmid (2010) or the Koopman-mode
expansion (Rowley et al. (2009)). Such linear methods cannot capture the
characteristically nonlinear bistability of shear flows, as was demonstrated
in detail by Page & Kerswell (2019). However, one can build formal nonlinear
models as expansions based on linear models. For example, Ducimetière et al.
(2022) used the spectrum of the resolvent operator and multiple-scale
expansion to derive Stuart-Landau-type amplitude equations (Landau (1959)) for
flows exhibiting non-normality. Among other examples, the amplitude equations
were then used to predict the energy of the response observed in a plane
Poiseuille flow subjected to harmonic forcing.

In contrast to approximate linear models, invariant manifold based methods
provide a mathematically rigorous foundation for reduced-order models that
capture nonlinear features. An early demonstration of this fact was the
approximate inertial manifold approach, which argues that the dynamics evolves
on a finite, but still high-dimensional invariant manifold (Foias et al.
(1988)). Recently, deep learning methods have shown promise in learning the
dynamics within the inertial manifold (see Linot & Graham (2020)). Despite all
these advances, however, inertial manifolds have not been proven to exist in
Navier-Stokes flows.

We focus here on the recently introduced spectral submanifolds (SSMs). These
are low-dimensional invariant manifolds connected to known stationary states,
to which the dynamics of a (possibly infinite dimensional) dynamical system
can be reduced (Haller & Ponsioen (2016); Kogelbauer & Haller (2018)). SSMs
are the smoothest nonlinear continuations of spectral subspaces of the
linearized system in a neighborhood of a stationary state, such as a fixed
point or a periodic orbit. Based on earlier work by Cabre et al. (2003), SSM-
based model reduction exploits the existence and uniqueness of spectral
submanifolds, both of which are guaranteed when the spectrum of the linearized
system within the spectral subspace is not in resonance with the rest of the
spectrum outside that subspace. Restricting the dynamics to SSMs associated
with the slowest spectral subspace then provides a mathematically exact
reduced-order model. In some cases, such a slow SSM may contain the global
attractor of the system and can be considered an inertial manifold.

In its initial formulation, SSM reduction constructed the parametrization of
the invariant manifold and the reduced dynamics as a Taylor-expansion around a
steady state, which proved fruitful in reduced-order modeling for general
mechanical systems (e.g., Jain & Haller (2022); Li et al. (2023)). In addition
to its strict mathematical foundation, a noteworthy advantage of SSM-based
model reduction over projection-based methods is that the dimension of the
slowest nonresonant spectral subspace a priori determines the dimension of the
reduced-order model. Therefore, one can simply increase the order of the
Taylor expansion aprroximating the SSM and its reduced dynamics, without
increasing the dimension of the reduced model to achieve higher accuracy.

By now, Buza (2023) has also established the existence of certain classes of
SSMs for the Navier-Stokes equations. In a recent development, Haller et al.
(2023) introduced generalized families of (secondary) SSMs that can have
either a lower degree of smoothness (fractional SSMs) or can be tangent to a
spectral subspace containing stable and unstable modes at the same time
(mixed-mode SSMs). In the present study, we will invoke these results to
construct mixed-mode SSMs as a basis for reduced-order models for transitions
in a pipe flow. We rely on recent work by Cenedese et al. (2022) that combines
SSM theory with data-driven methods to construct reduced-order models directly
from experimental or numerical data. We will use an implementation of these
results in the open-source package SSMLearn (Cenedese et al. (2021)), which
has already been successfully applied to fluids problems, such as sloshing in
a horizontally forced tank (Axås et al. (2023)).

Using the same data-driven method, Kaszás et al. (2022) showed that an SSM-
based model accurately predicts the time evolution of individual trajectories
in the phase space of plane Couette flow. While indeed yielding accurate and
predictive models, those results only covered transitions between nonchaotic
states, such as fixed points and periodic orbits. In this contribution, we
show that similar results can be applied even when the phase space contains
turbulent behavior supported on a chaotic saddle.

In particular, we target the slowest two-dimensional SSM of the edge state to
characterize the dynamics of laminarization and transition to turbulence. We
use data generated by the open-source solver Openpipeflow of Willis (2017). We
do not seek to capture any turbulent dynamics because we have no accurate way
to approximate the dimension of the underlying chaotic saddle and hence cannot
guarantee that it will be contained in an SSM. Instead, we show that a data-
driven SSM-reduced model accurately captures the slowest submanifold in the
edge of chaos, thus yielding an explicit parametrization of the most
influential dynamical structures within the edge manifold. We have also made
the data and the code supporting the analysis available under the repository
Cenedese et al. (2023) as well as in the form of JFM Notebooks.

##  2 Setup

We consider the incompressible Navier-Stokes equations for the velocity field
$\mathbf{u}$ and the pressure $p$ in a domain $\Omega\subset\mathbb{R}^{3}$,
given by

$\frac{\partial\mathbf{u}}{\partial
t}+\left(\mathbf{u}\nabla\right)\mathbf{u}=-\nabla
p+\frac{1}{\text{Re}}\Delta\mathbf{u}+\mathbf{q}\quad\nabla\cdot\mathbf{u}=0,$
(1)

where $\mathbf{q}$ is a body force needed to sustain a constant mass flux. The
domain is taken as a circular pipe,

$\displaystyle\Omega_{\text{pipe}}$
$\displaystyle=\left\\{(r,\varphi,z)\in\mathbb{R}^{3}|\ 0\leq r\leq
R,\varphi\in[0,2\pi),0\leq z\leq L\right\\},$ (2)

defined in cylindrical coordinates with $x=r\cos\varphi$, $y=r\sin\varphi$.
The Reynolds number is defined as $\text{Re}=\frac{RU_{cl.}}{\nu}$, where
$U_{cl.}$ is the center line velocity of the laminar state and $\nu$ is the
kinematic viscosity. Equation (1) is nondimensionalized by the pipe radius $R$
and the center line velocity $U_{cl}$. The laminar state is the Hagen-
Poiseuille flow, given by

$\mathbf{u}_{HP}(r)=\begin{pmatrix}0\\\ 0\\\ 1-r^{2}\end{pmatrix}.$ (3)

The domain is assumed to be periodic in the $z$ direction, so that
$~{}\mathbf{u}(r,\varphi,z,t)=\mathbf{u}(r,\varphi,z+L,t)$. Following previous
studies (Duguet et al. (2008); Willis et al. (2013)), we approximate the
velocity field as a truncated Fourier expansion up to order $M$ in the
azimuthal direction and up to order $K$ in the streamwise direction as

$\mathbf{u}(r,\varphi,z,t)=\mathbf{u}_{HP}+\sum_{m=-M}^{M}\sum_{k=0}^{K}\mathbf{A}_{mk}(r,t)e^{im_{p}m\varphi}e^{ik\alpha
z},$ (4)

where $\mathbf{A}_{mk}(r,t)$ is a 3-vector of Fourier-amplitudes, $m_{p}$
determines the fundamental period in the angular direction and
$\alpha=\frac{2\pi}{L}$. Throughout this study, we fix the number of modes
used as $M=16$, $K=16$, and fix $m_{p}=2$ and $\alpha=1.25$. The latter sets
the length of the pipe as $L\approx 5.02$, which corresponds to a minimal flow
unit studied in Willis et al. (2013, 2017). Our spatial resolution also
corresponds to that of Willis et al. (2013), who have confirmed that the
dynamics are sufficiently resolved by this level of discretization.

We use Openpipeflow (Willis (2017)) to perform the discretization (4) with an
additional finite-difference approximation at $64$ points for the radial
derivative. The discretized Navier-Stokes equations are then integrated using
a pressure Poisson equation formulation to enforce incompressibility. The
time-resolved Navier-Stokes solutions are viewed as trajectories of a large
system of coupled ordinary differential equations,

$\frac{d}{dt}\mathbf{x}=\mathcal{N}(\mathbf{x}),\quad\mathbf{x}\in\mathbb{R}^{k},$
(5)

where $\mathbf{x}$ is the vector comprised of the discretized degrees of
freedom with $k\in O(10^{5})$ and $\mathcal{N}$ is the generator of the time
evolution. As expressed by (4), the solver returns the deviations with respect
to the laminar state, such that the origin $\mathbf{x}=0$ corresponds to (3).
Distinguished solutions of this dynamical system are ECSs discussed in the
Introduction. Budanur et al. (2017) has shown that some of these ECSs,
unstable (relative) periodic orbits, are indeed central organizers of
turbulence.

Figure 1: Left panel: Time evolution of the energy input rate of two
trajectories started near the lower-branch traveling wave. Below we illustrate
the flow fields corresponding to different snapshots during the time evolution
by showing isosurfaces of the streamwise velocity $u_{z}$. Middle and right
panels: Eigenvalue configuration of the laminar state and the lower-branch
traveling wave, respectively. Crosses denote the eigenvalues in the
$\mathcal{S}$-invariant subspace discussed in the text and circles denote the
full-space eigenvalues. The parameters are chosen as
$\text{Re}=2,400;\alpha=1.25$.

For low Reynolds numbers, the only stable state is the laminar state, while
for transitional flows above $\text{Re}\sim 2,000$, the phase space of (5) is
divided into two domains with characteristically different behaviors. Some
initial conditions immediately return to the vicinity of the state (3), while
others transition to turbulence.

The left panel of Fig. 1 shows two trajectories of (5) with distinct dynamic
behaviors, as seen in the time evolution of their normalized energy input rate
$I$, defined as

$I^{\prime}=\frac{1}{V}\oint_{\partial\Omega_{\text{pipe}}}d\mathbf{S}\cdot\mathbf{u}p,\quad
I=\frac{I^{\prime}}{I^{\prime}_{HP}}-1,$ (6)

with the volume of the domain $\Omega_{\text{pipe}}$ denoted as $V$. The
energy input rate $I^{\prime}$ measures the external power supplied to the
system to satisfy the constant mass flux condition. Similarly, the normalized
rate of energy dissipation is given by

$D^{\prime}=\frac{1}{\text{Re}}\left\|\text{curl }\mathbf{u}\right\|^{2},\quad
D=\frac{D^{\prime}}{D^{\prime}_{HP}}-1,$ (7)

with the norm of a vector field defined as

$\left\|\mathbf{f}\right\|:=\sqrt{\langle\mathbf{f},\mathbf{f}\rangle}=\left(\frac{1}{V}\int_{\Omega_{\text{pipe}}}dV\
\mathbf{f}\cdot\mathbf{f}\right)^{\frac{1}{2}}.$ (8)

In the above definitions we normalize the energy input and dissipation values
by their laminar values and subtract $1$ so that the laminar state corresponds
to $I_{HP}=0$, $D_{HP}=0$. An energy balance can be derived from the inner
product $\langle\mathbf{u},\frac{\partial\mathbf{u}}{\partial t}\rangle$ using
(1) (see Waleffe (2011)). The energy input rate, the dissipation rate, and the
kinetic energy, defined as $E=\|\mathbf{u}\|^{2}/2$, satisfy

$\dot{E}=I^{\prime}-D^{\prime}.$ (9)

Skufca et al. (2006) discovered that a codimension-one surface, the edge
manifold locally acts as a barrier that separates laminarizing and turbulent
trajectories. This manifold is the stable manifold of an unstable traveling
wave solution known as the lower-branch discussed by Kerswell & Tutty (2007),
Willis et al. (2013) and Budanur & Hof (2018). This solution is invariant with
respect to the shift-reflect symmetry

$\mathcal{S}\begin{pmatrix}u_{r}(r,\varphi,z)\\\ u_{\varphi}(r,\varphi,z)\\\
u_{z}(r,\varphi,z)\end{pmatrix}=\begin{pmatrix}u_{r}\left(r,\varphi,z-\frac{L}{2}\right)\\\
-u_{\varphi}\left(r,\varphi,z-\frac{L}{2}\right)\\\
u_{z}\left(r,\varphi,z-\frac{L}{2}\right)\end{pmatrix}.$ (10)

For convenience, we perform the calculations in the $S-$invariant subspace as
is customary in the literature (Willis et al. (2013); Budanur & Hof (2018)).

Spectral submanifolds (SSMs, Haller & Ponsioen (2016)) have emerged as useful
tools for reduced-order modeling of large-scale systems. They are defined as
the unique invariant manifolds of (5) that are the smoothest continuations of
the invariant spectral subspaces of the linearization of (5) at hyperbolic
fixed points or periodic orbits. Although the existence conditions for SSMs in
infinite-dimensional systems are difficult to verify (see Kogelbauer & Haller
(2018)), this analysis has been carried out by Buza (2023) for the Navier-
Stokes equations.

As any numerical solution of system (1) is inevitably finite-dimensional,
invoking the finite-dimensional results of Haller & Ponsioen (2016) is
sufficient in our present setting. These results guarantee that a stable
hyperbolic fixed point has a hierarchy of spectral submanifolds attached to it
provided that the spectral non-resonance conditions mentioned in the
Introduction are met.

A notable hyperbolic anchor point for SSM construction is the laminar Hagen-
Poiseuille flow (3). The spectrum of the linearized Navier-Stokes equation (1)
at (3) is discussed by Schmid & Henningson (2001). The slowest spectral
subspace inferred from these calculations corresponds to the least stable real
eigenvalue or complex conjugate eigenvalue pair. For our geometry with
$\alpha=1.25$ and $m_{p}=2$, these are the streamwise-independent modes with
$k=0$ in (4), which have the form

$\mathbf{v}_{nm^{\prime}}(r,\varphi)=\begin{pmatrix}0\\\ 0\\\
J_{m^{\prime}}(\sqrt{-\lambda_{n,m^{\prime}}\text{Re}}r)\end{pmatrix}e^{im^{\prime}\varphi}.$
(11)

Here, $m^{\prime}=m_{p}m$ and $J_{m}$ is the $m-$th Bessel function of the
first kind. The corresponding eigenvalue is real and is given by

$\lambda_{n,m^{\prime}}=-\frac{j^{2}_{n,m^{\prime}}}{\text{Re}},$

where $j_{n,m^{\prime}}$ denotes the $n-$th root of $J_{m^{\prime}}$. Thus,
the slowest spectral subspace is one-dimensional and is obtained for
$n=1,m^{\prime}=m_{p}=2$.

The nonlinear term $(\mathbf{u}\nabla)\mathbf{u}$ vanishes identically along
the modes (11), which means that the spectral subspace spanned by (11) is also
an invariant manifold of the nonlinear system (1). By the uniqueness of the
slowest analytic spectral submanifold, the spectral subspace spanned by (11)
is necessarily the unique, analytic spectral submanifold of system (1)
anchored at the state (3). As such, this SSM cannot carry nonlinear dynamics.
This conclusion is also supported by the results of Joseph & Hung (1971), who
show that a streamwise-independent flow such as (11) must ultimately decay.
Kogelbauer et al. (2020) also investigated slow SSMs of the laminar state in
pipe flow for a range of parameter values and found no nontrivial behavior in
Taylor approximations of those SSMs. Indeed, Haller et al. (2023) find that
heteroclinic connections among steady states generally occur along invariant
manifolds of finite smoothness. Their results indicate that typical SSMs in
the flow are only once continuously differentiable at the laminar state. This
can be deduced from the spectrum of the laminar state (see Fig. 1 middle
panel).

The other anchor point candidate for SSM-based model order reduction is the
lower-branch traveling wave, as seen from the spectrum of the linearization of
(1) at that solution in the right panel of Fig. 1.

###  2.1 Symmetry reduced phase space

The Navier-Stokes equations (1) have two continuous symmetries. Streamwise
translations $\sigma_{l}$ and rotations $\sigma_{\theta}$. This means that any
solution $\mathbf{u}(t)$ is physically equivalent to all solutions along
orbits of the group generated by the two shifts, denoted as $\sigma_{l}$ and
$\sigma_{\theta}$, that is to the set

$\Sigma(\mathbf{u}(t))=\left\\{\sigma_{l}\sigma_{\theta}\mathbf{u}(r,\varphi,z,t)=\mathbf{u}(r,\varphi-\theta,z-l,t)|\theta\in[0,2\pi),l\in[0,L)\right\\}$
(12)

Imposing the shift-reflect symmetry $\mathcal{S}$ (i.e., restricting (1) to
the $\mathcal{S}-$invariant subspace) eliminates one of these shifts, but
solutions in the invariant subspace can still be translated freely in the
streamwise direction. This translational invariance is reflected by the
appearance of an eigenvector with zero eigenvalue for the lower-branch
traveling wave.

This zero eigenvalue formally renders the lower-branch traveling wave a non-
hyperbolic fixed point and prevents us from concluding the existence of SSMs
anchored at this fixed point. However, since non-hyperbolicity arises here due
to a continuous symmetry, we can use the method of slices (Froehlich &
Cvitanović (2012)) to eliminate this symmetry. This method factorizes the
phase space of (5) by establishing an equivalence relation along group orbits.
From each group orbit (12), we select the single trajectory that is closest to
the lower-branch traveling wave in the norm (8). This construction has already
been used to study the same system by Willis et al. (2013) and to aid dynamic
mode decomposition (Marensi et al. (2023)).

The method of slices then yields a dynamical system that has a lower dimension
than the original one. The dynamics along the group orbit, which we may denote
by a phase-type variable $\psi$, can also be recovered. Thus, formally, the
dynamics is governed by

$\displaystyle\dot{\mathbf{x}}_{R}$
$\displaystyle=\mathcal{N}_{R}(\mathbf{x}_{R}),\quad\mathbf{x}_{R}\in\mathbb{R}^{k-1}$
(13) $\displaystyle\dot{\psi}$ $\displaystyle=f(\mathbf{x}_{R},\psi).$ (14)

In the following, we work with the $\mathbf{x}_{R}$ component of the dynamical
system obtained from the method of slices which makes the phase space $d=k-1$
dimensional. We illustrate the trajectories computed in the full space in the
upper left panel of Fig. 2. It can be seen that trajectories evolve in
spirals, which is a signal of their time evolution along the group orbit.

The method of slices eliminates this behavior, as can be seen in the right
panel of Fig. 2. Note also that physically inherent quantities, such as the
energy input rate and the dissipation are the same along all points of the
group orbit, which can be seen in the left panel of Fig. 2. Eliminating the
symmetry thus does not change the physically relevant observables of a
trajectory, since macroscopic quantities must be the same for flow fields
related to each other by a symmetry transformation.

##  3 Results

###  3.1 Spectral submanifolds of the lower-branch traveling wave

In the symmetry reduced phase space, the lower-branch traveling wave becomes a
hyperbolic fixed point with a single unstable eigenvalue. By the classic
unstable manifold theorem, there exists a one-dimensional unstable manifold
tangent to the unstable eigenvector and a codimension-one stable manifold. The
unstable manifold of the lower-branch traveling wave forms a heteroclinic
connection with the laminar state. Kaszás et al. (2022) demonstrated that such
heteroclinic orbits can serve as one-dimensional reduced-order models, as also
supported by the experiments of Suri et al. (2017).

To capture more of the transient dynamics near the heteroclinic orbit and
obtain an approximation for the basin boundary of the laminar state, we
extract here a two-dimensional invariant manifold that can intersect the edge.
In order to construct an attracting manifold, we need to take the slowest two-
dimensional SSM of the lower-branch traveling wave. This SSM is tangent to
both its single unstable eigenvector and its slowest stable eigenvector and
hence contains both the heteroclinic orbit and the slowest submanifold of the
edge of chaos.

Such a manifold is often called a pseudo-unstable manifold ( de la Llave &
Wayne (1995)), whose existence can also be concluded from the recent results
of Haller et al. (2023) or Buza (2023). Haller et al. (2023) also find a large
variety of generalized spectral submanifolds of (5), including ones of limited
smoothness (fractional SSMs) and others of mixed stability type (mixed-mode
SSMs). These results apply in finite dimensions and give $C^{\infty}$-smooth,
mixed-mode SSMs under the same nonresonance conditions as those of Sternberg
(1958).

As discussed by Haller et al. (2023), exact resonances among complex
eigenvalues are unlikely in a typical finite-dimensional system, such as the
numerical discretization (5). We thus expect that a unique, $C^{\infty}$,
mixed-mode SSM of the lower-branch traveling wave exists. We list the
corresponding nonresonance conditions in the Appendix and verify that they
hold for a subset of the spectrum. Alternatively, one may invoke the results
of Buza (2023), who showed that a $C^{1}$-smooth pseudo-unstable manifold
exists for the Navier-Stokes equations (1).

Based on these recent technical developments, we can apply the data-driven
methodology of Cenedese et al. (2022) to discover this mixed-mode SSM from
simulation data. We follow the approach of Kaszás et al. (2022) and introduce
the square roots of the variables $I$ and $D$ as $J:=\sqrt{|I|}$ and
$K:=\sqrt{|D|}$. This is necessary, because the velocity field $\mathbf{u}$ is
a non-differentiable function of $I$ and $D$ at the origin, due to their
quadratic dependence on the components of $\mathbf{u}$. Thus, we parametrize
the spectral submanifolds with the variables $J$ and $K$. Specifically, we
seek a two-dimensional invariant manifold in the phase space of (13) of the
form

$\mathbf{x}_{R}=\sum_{1\leq n+m\leq
M_{p}}\mathbf{c}_{nm}K^{m}J^{n},\quad\mathbf{c}_{nm}\in\mathbb{R}^{d}.$ (15)

The polynomial-type dependence up to order $M_{p}$ on the reduced coordinates
in (15) is justified because polynomials are universal approximators (Rudin
(1976)). It is also motivated by the success of local Taylor-approximations
used in the original methods (Haller & Ponsioen (2016)) for SSM-based model
order reduction.

To determine the coefficient vectors $\mathbf{c}_{nm}$ in (15) we initialize
training trajectories that lie on the mixed-mode SSM of the lower-branch
traveling wave. The parametrization of the manifold in physical coordinates
can then be recovered from (15) using the discretization (4) performed by
Openpipeflow (Willis (2017)). This results in

$\mathbf{u}(r,\varphi,z)=\mathbf{W}(J,K)=\mathbf{u}_{HP}(r)+\sum_{1\leq
n+m\leq M_{p}}\mathbf{w}_{nm}(r,\varphi,z)K^{m}J^{n},$ (16)

where the coefficient-functions $\mathbf{w}_{nm}(r,\varphi,z)$ are determined
by $\mathbf{c}_{nm}$.

To generate initial conditions, we start with the lower-branch traveling wave
discussed by Kerswell & Tutty (2007) at $Re=2,400$. The traveling wave is
found using a Newton-Krylov scheme (Viswanath (2007)). This method also
returns the leading eigenvalues and eigenvectors of the traveling wave
(computed in a co-moving frame) (see Willis (2019)).

For training, we take a total of six trajectories, four of which were randomly
initialized at a distance of $10^{-4}$ from the lower-branch traveling wave.
The distance is measured as the Euclidean distance in the phase space of (13).
In this metric, the distance between the laminar state and the lower-branch
traveling wave is 0.72. We reserve an additional randomly initialized
trajectory for validation. After the initial transients, the trajectories
approach the unstable manifold of the lower-branch traveling wave. To also
capture the dynamics in the slowest stable direction, we construct two more
training trajectories that are approximately constrained to the edge. We use
the edge-tracking algorithm of Itano & Toh (2001) started from initial guesses
lying along the subspace spanned by the slowest stable eigenvector. The energy
input and dissipation variables are computed using Openpipeflow from the
spectral representation of the flow fields.

We show the set of training trajectories in Fig. 2. The upper left panel shows
the original trajectories as a projection onto the three most dominant spatial
modes, which are displayed in the lower right panel of the figure. In all
further computations, we work with the symmetry-reduced phase space and pre-
process the trajectories using the method of slices. The symmetry-reduced
trajectories can be seen in the upper right panel of Fig. 2. We also show the
reduced coordinates, $J=\sqrt{|I|}$ and $K=\sqrt{|D|}$, which coincide for the
full-space trajectories and the symmetry-reduced trajectories.

Note that the method of slices generally can only be applied over a bounded
domain. When trajectories cross the border of the chart associated to the
template (Froehlich & Cvitanović (2012)), finite jumps are observed in the
symmetry-reduced phase space. Figure 2 indicates no such singularities in our
training trajectories, which prompts us to represent them by a single chart.
Willis et al. (2013) found that a global atlas defined using multiple template
states was necessary to construct the symmetry reduced representation of the
turbulent state. Since our training trajectories avoid the turbulent state and
remain in the neighborhood of the heteroclinic orbit, using the lower-branch
traveling wave as the only template state was sufficient.

Given the training trajectories, we identify the coefficient vectors
$\mathbf{c}_{nm}$ via a sparsity promoting ridge regression (Hastie et al.
(2009); Brunton et al. (2016)), minimizing the squared norm of the deviation
from these trajectories. Once the SSM geometry is identified, we seek the SSM-
reduced dynamics in the form

$\frac{d}{dt}\begin{pmatrix}J\\\ K\end{pmatrix}=\mathbf{f}(J,K)=\sum_{1\leq
n+m\leq M_{r}}\begin{pmatrix}R^{(J)}_{nm}K^{m}J^{n}\\\
R^{(K)}_{nm}K^{m}J^{n}\end{pmatrix}.\\\ $ (17)

Here the coefficients $R^{(J)}_{nm}$ and $R^{(K)}_{nm}$ are also determined by
ridge regression onto $\dot{J}$ and $\dot{K}$ obtained by finite differencing
along the training trajectories. We also introduce a constraint to this
optimization, forcing the lower-branch traveling wave and the laminar state to
be fixed points of (17), as in Kaszás et al. (2022).

The parameters of the regression are the polynomial orders $M_{p}$ and
$M_{r}$, as well as the weight of the ridge-type penalty term. For simplicity,
we take $M_{p}=M_{r}$. This choice is motivated by the usual Taylor-expansion
representation of SSMs, where the parametrization and the reduced dynamics are
computed up to the same order. The value of the polynomial orders and the
ridge-type penalty term are determined by cross-validation on a trajectory
initialized in the same way as the training set.

We minimize the overall mean-prediction error on the validation trajectory,
defined as

$\text{Error}=\sum_{i=0}^{n_{t}}\left\|\mathbf{u}_{true}(t_{i})-\mathbf{W}\circ
F^{t_{i}}(J_{true}(0),K_{true}(0))\right\|,$ (18)

where $F^{t}$ is the time-$t$ flow-map of the reduced dynamics (17) and
$n_{t}$ is the number of time-snapshots available along the validation
trajectory. To avoid overfitting and promote simpler models, as long as the
value of the error (18) is similar, we prioritize lower polynomial orders and
larger penalty terms. The results are reported with $M_{r}=M_{p}=5$, and the
penalty term is chosen as $10^{-5}$. We refer to the JFM Notebook accompanying
Fig. 4 for further details.

We also note that the approximation of the SSM benefits from a richer training
set. The accuracy of the reduced-order model increases if more training
trajectories are used for the regression or if the temporal resolution of the
trajectories is increased, as explained by Cenedese et al. (2022).

Figure 2: Upper left panel: Trajectories initialized near the lower-branch
traveling wave displayed in the space spanned by projections onto the three
most dominant spatial modes of the lower-branch. The coordinates are defined
as $\eta_{i}=\langle\mathbf{u},u_{i}\rangle$, where the three dominant modes
are illustrated in the lower right panel. Right panel: projections of the
symmetry reduced trajectories onto the same spatial modes. Lower left panel:
the same trajectories displayed in the space spanned by $J=\sqrt{I}$ and
$K=\sqrt{D}$. The directory, including the data and the Jupyter notebook that
generated this figure can be accessed at
https://cocalc.com/share/public_paths/2af692bffb0397d126563ec78c61bb65205d1211

The lower-branch traveling wave is a fixed point of the reduced-order model by
construction, that is $\mathbf{f}(J_{LB},K_{LB})=\mathbf{0}$. The Jacobian,
$D\mathbf{f}(J_{LB},K_{LB})$, has eigenvalues ${\lambda^{(+)}_{red.}=0.0199}$,
and ${\lambda^{(-)}_{red.}=-0.0107}$, matching the eigenvalues of the lower-
branch in the full-order model obtained by Krylov iteration in Openpipeflow,
whose values are ${\lambda^{(+)}=0.0198}$, and ${\lambda^{(-)}=-0.0010}$.

###  3.2 Predictions of the reduced-order model

To assess the predictive power of the two-dimensional, SSM-reduced model
obtained above, we also generate an ensemble of test trajectories distributed
over a sphere of radius $10^{-4}$ around the lower-branch traveling wave in
the phase space. Some of these trajectories transition to turbulence while
others laminarize, but we can make predictions based on the reduced-order
model for all of them. In Fig. 3 we compare the time evolution of the $J$
coordinate of the test trajectories to their predicted counterparts. While the
reduced-order model accurately differentiates between turbulent and laminar
trajectories, the turbulent trajectories are only reliably modeled for 200
time units, since the model was not trained on trajectories in the turbulent
state. The right panel of Fig. 3 shows the overall relative error of the
predictions (18) for the ensemble of test trajectories.

Figure 3: Predictions of the reduced-order model on an ensemble of 20 test
trajectories initialized near the lower-branch traveling wave. In the left
panel, the time evolution of their $J$ coordinate is compared to the
predictions made by the reduced-order model. For turbulent trajectories, only
the first 200 time units of the trajectories are shown. In the right panel,
the corresponding relative errors are plotted, defined as the time-dependent
quantity $\left\|\mathbf{u}_{true}(t_{i})-\mathbf{W}\circ
F^{t_{i}}(J_{true}(0),K_{true}(0))\right\|/\text{max
}\left\|\mathbf{u}_{true}\right\|$. Ensemble-averages of the relative error
are computed. The errors of turbulent and laminar trajectories are averaged
separately.

In Fig. 4 we show the SSM-reduced model vector field $\mathbf{f}$ on the $J-K$
plane. We also show the approximation of the stable manifold of the lower-
branch traveling wave, obtained by integrating initial conditions of the form

$\begin{pmatrix}J_{LB}\\\ K_{LB}\end{pmatrix}\pm 10^{-6}\mathbf{v}$ (19)

backward in time, where $\mathbf{v}\in\mathbb{R}^{2}$ is the eigenvector of
$D\mathbf{f}(J_{LB},K_{LB})$ associated with the negative eigenvalue. In Fig.
4, the predicted stable manifold matches the edge trajectory well, even
outside the domain of the training data for $J>J_{LB}$.

Having computed the parametrization $\mathbf{W}$, we can also display the
global shape of the mixed-mode SSM. In Fig. 5, we highlight the domain in the
$J-K$ plane, over which we visualize the SSM by plotting its relative kinetic
energy,

$\Delta E=\frac{1}{2}\left\|\mathbf{u}_{HP}-\mathbf{W}(J,K)\right\|^{2},$ (20)

in the right panel of Fig. 5.

Figure 4: Two-dimensional reduced-order model obtained by restricting the
dynamics to the two-dimensional mixed-mode SSM of the lower-branch traveling
wave. This manifold is parametrized by the variables $J$ and $K$ that define
the reduced dynamics. Training trajectories are shown in green. The directory,
including the data and the Jupyter notebook that generated this figure can be
accessed at
https://cocalc.com/share/public_paths/db8c0a4e84051e09f620a97c03d6c18927f47ce0
Figure 5: Visualization of the reduced-order model and the geometry of the
mixed-mode SSM carrying the model flow. The right panel shows the mixed-mode
SSM (green surface) in the $(J,K,\Delta E)$ plane obtained as the image of the
domain shown in the left panel under the parametrization $\mathbf{W}$. The
relative energy $\Delta E$ is defined as (20). The fixed points, the base
state and the lower-branch traveling wave are indicated with colored dots. The
dashed line denotes the predicted stable manifold of the lower-branch
traveling wave in the SSM-reduced model. The directory, including the data and
the Jupyter notebook that generated this figure can be accessed at
https://cocalc.com/share/public_paths/f73549a1c2a577925859b24b039047cd5a05b1cd

###  3.3 Capturing the edge of chaos in the reduced-order model

The mixed-mode SSM shown in Fig. 6 is two-dimensional, and the edge manifold (
i.e., the stable manifold of the lower-branch traveling wave) has codimension
one. Adding up their dimensions results in $d+1$, where $d$ is the dimension
of the phase space of (13). Therefore, these two manifolds generically
intersect transversely along a one-dimensional curve. This implies that the
intersection is robust under small perturbations to system (5).

The generically expected transverse intersection of the SSM with the edge
manifold is displayed in Fig. 6, where the mixed-mode SSM is the same as in
Fig. 5 and a schematic representation of the edge manifold is added. In the
three-dimensional $(J,K,\Delta E)$ space, both the edge manifold and the
mixed-mode SSM appear as two-dimensional surfaces, even though in the full
phase space the edge manifold is a much higher dimensional object.

From the spectrum of the lower-branch traveling wave in Fig. 1, we conclude
that its slowest eigenvalue with negative real part is real. Therefore, the
stable manifold of this traveling wave has a one-dimensional slow SSM tangent
to the eigenspace of that real eigenvalue. This SSM coincides with the stable
manifold of the lower-branch traveling wave in the reduced-order model, and
hence must be the line of intersection shown in dashed lines in Fig. 6.

Figure 6: Mixed-mode SSM and the edge manifold shown in the ($J,K,\Delta E$)
space. The edge manifold is sketched as a two-dimensional surface in the
three-dimensional space that intersects the edge manifold along the stable
manifold of the lower-branch traveling wave in the reduced-order model (dashed
black line).

Since the intersection of the edge manifold and the mixed-mode SSM is a curve,
it will not capture the dynamics within the whole edge manifold. Nevertheless,
this intersection identifies a clear footprint of the edge manifold in the
SSM-reduced model. To demonstrate this, we construct 10 trajectories
constrained to the edge manifold using the bisection algorithm of Itano & Toh
(2001). These trajectories converge to the lower-branch traveling wave, which
is a relative attractor within the edge. In the reduced phase space $(J,K)$,
these trajectories clearly approach the lower-branch traveling wave along the
predicted stable manifold, as shown in the left panel of Fig. 7.

The distinguishing property of the edge manifold is that it divides the phase
space locally around the lower-branch traveling wave. We show that its
footprint in the reduced-order model also has this dividing role by
initializing a set of trajectories in the reduced phase space at a fixed
distance $\delta=0.01$ from the stable manifold. Initial conditions are then
prepared using the parametrization $\mathbf{W}$ and are supplied to the full-
order model Openpipeflow. The right panel of Fig. 7 shows that the initial
conditions prepared in this way are clearly separated in both the SSM-reduced
model and the full-order model.

Figure 7: Left panel: Edge trajectories constructed using the bisection
algorithm in the full-order model approaching the lower-branch traveling wave
displayed in the reduced-phase space. A single representative member of this
ensemble of trajectories is shown with a red dashed line to guide the eye. The
stable manifold of the lower-branch traveling wave predicted from the reduced-
order model is shown in black. In the right panel, trajectories of the full-
order model are initialized by placing initial conditions on both sides of the
predicted edge in the reduced-order model.

In the left panel of Fig. 8, we show a grid of initial conditions in a
rectangle around the lower-branch traveling wave in the reduced-order model.
The initial conditions for the full-order model are prepared using the
identified parametrization $\mathbf{W}$ and are integrated forward in time.
Different colors indicate the different long-time behaviors of the grid of
initial conditions. The stable manifold of the reduced-order model clearly
separates the trajectories of the full-order model, as expected.

Another indication that this separation is indeed caused by the saddle-type
edge state is the slowdown of trajectories, which is already present in a
linear system. Consider a linear system with a saddle-type fixed point that
has a one-dimensional unstable manifold. In diagonal form, the dynamics along
the unstable subspace are

$\dot{\xi}=\lambda\xi,\text{ for }\lambda>0,$ (21)

while for the remaining coordinates we have

$\dot{\eta}_{i}=-\kappa_{i}\eta_{i},\text{ for }\kappa_{i}>0,i=1,...,d-1.$
(22)

For sufficiently long times, the distance of a trajectory from the saddle is
well approximated by its $\xi$ coordinate component, i.e. the distance
$\delta(t)$ can be written as $\delta(t)=\xi_{0}e^{\lambda t}$. Therefore, the
time $T$ it takes for a trajectory to leave the $\delta_{\max}$-neighborhood
of the saddle satisfies $\delta_{\max}=\xi_{0}e^{\lambda T}$, or, equivalently

$T=\frac{1}{\lambda}\log\delta_{\max}-\frac{1}{\lambda}\log\xi_{0},$ (23)

where $\xi_{0}$ measures the initial distance from the stable subspace.

To demonstrate the slowdown, starting from the initial conditions shown in the
left panel of Fig. 8, we record the time needed for the full-order model
trajectory to develop a distance of $\delta_{\max}=0.1$ from the lower-branch
traveling wave. The right panel of Fig. 8 shows the dependence of this time on
the initial distance from the edge manifold in the reduced model. This
relationship can be well described by a logarithmic function of the form (23),
which would hold for a linear system. Therefore, we expect that
$T=C-\frac{1}{\lambda}\log\xi_{0}$ is satisfied approximately for some
constants $C$ and $\lambda$. We obtain $\lambda=0.0175$ from a least squares
fit to the data, which reasonably matches the true unstable eigenvalue of the
lower-branch traveling wave, $\lambda^{(+)}$.

Although the one-dimensional footprint of the edge manifold cannot act as a
global barrier in the phase space, it is still an influential curve for
trajectories near the lower-branch traveling wave. Since the mixed-mode SSM is
constructed to be tangent to the slowest dynamics, it is locally attracting.
Therefore, one may project nearby trajectories onto the mixed-mode SSM to
decide their long-term dynamics: if the projection lies above the footprint of
the edge manifold within the SSM, then the trajectory is expected to become
turbulent. Otherwise, the trajectory is expected to quickly laminarize. As one
moves away from the vicinity of the lower-branch, this correspondence will
gradually break down due to the overall complicated shape of the edge
(Schneider et al. (2007); Mellibovsky et al. (2009)).

Figure 8: Left panel: Phase portrait of the SSM-reduced model with a grid of
test trajectories. Initial conditions are placed in a grid around the lower-
branch traveling wave and are integrated forward in time using the full-order
model. The initial conditions are colored according to their long-time
behavior, as seen in the insets. The predicted edge manifold footprint is
shown in black. Right panel:time spent by these trajectories near the edge as
a function of their initial distance from the edge measured in the SSM-reduced
phase space. This time is defined as the time needed to reach a distance of
$D_{\max}=0.1$ from the lower-branch traveling wave. A least-squares fit of
the form $T=C-\frac{1}{\lambda}\log\xi_{0}$ is also shown with $C=-234\pm 14$
and $\lambda=0.0175\pm 0.0005$.

###  3.4 Parameter dependent SSM-reduced models

The spectrum of the lower-branch traveling wave changes smoothly as the
Reynolds number is varied, therefore the slowest mixed-mode SSM remains two-
dimensional for a wide range of Reynolds numbers. As in Kaszás et al. (2022),
we may then seek a parametrization of the Reynolds number-dependent family of
mixed-mode SSMs. Similarly, the reduced dynamics on the family of manifolds
then also depend on the Reynolds number. Formally, this means that we seek
$\mathbf{W}$ and $\mathbf{f}$ in a Reynolds number-dependent form

$\mathbf{u}(r,\varphi,z)=\mathbf{W}(J,K,\text{Re})=\mathbf{u}_{HP}(r)+\sum_{l=0}^{N_{p}}\sum_{1\leq
n+m\leq M_{p}}\mathbf{w}_{nml}(r,\varphi,z)K^{m}J^{n}\text{Re}^{l}$ (24)

and

$\frac{d}{dt}\begin{pmatrix}J\\\
K\end{pmatrix}=\mathbf{f}_{M}(J,K,\text{Re})=\sum_{l=0}^{N_{p}}\sum_{1\leq
n+m\leq M_{r}}\begin{pmatrix}R^{(J)}_{nml}K^{m}J^{n}\text{Re}^{l}\\\
R^{(K)}_{nm}K^{m}J^{n}\text{Re}^{l}\end{pmatrix}.\\\ $ (25)

The general dependence of these expressions on the Reynolds number is
restricted by the requirement that the laminar state must be a fixed point at
$J=K=0$ for all values of the Reynolds number, therefore $\mathbf{w}_{00l}=0$
and $R_{00l}=0$ must hold for all $l$.

To construct a parameter-dependent SSM-reduced model, we select the range
${\text{Re}\in[2400,2520]}$, where the slowest eigenvalues of the lower-branch
traveling wave do not change considerably and hence the mixed-mode SSMs are
only expected to show minor variation. In this case, we can take a linear
approximation for the Reynolds-number dependence in (24) and (25), i.e.
$N_{p}=1$. We use three sets of training trajectories initialized at
$\text{Re}=2,400;2,550;2,520$. These are prepared in the same way as described
in the case of $\text{Re}=2,400$. Figure 9 shows this parametrized family of
reduced dynamics in the ($\text{Re},J,K$) space. We find that the phase
portraits of the reduced models are qualitatively the same for all Reynolds
numbers in this range.

Figure 9: Family of SSM-reduced models parametrized by the Reynolds number.
Phase portraits at the sections Re = $2,400$ and Re = $2,520$ are shown, along
with the lower-branch traveling wave and its stable and unstable manifolds
computed from the SSM-reduced models.

##  4 Conclusion

We have constructed and tested invariant manifold-based reduced-order models
for a transitional pipe flow at $Re=2,400$. Specifically, we have computed a
mixed-mode SSM, as recently defined by Haller et al. (2023), to capture the
slowest dynamics characteristic of the lower-branch traveling wave in the pipe
flow. Following the numerical setup of Willis et al. (2013), we used a
symmetry-restricted version of the dynamics, resulting in the lower-branch
traveling wave coinciding with the edge state. In addition, we used the method
of slices to factor out the physically irrelevant directions for the time
evolution of the trajectories. We believe that the present study is the first
example of applying SSM-based reduced-order models to systems exhibiting
symmetry. We have used a Python-based implementation of the data-driven SSM-
reducion method, SSMLearnPy (Cenedese et al. (2023)). To make the dataset
accessible, we provide it in a compressed format by performing linear
dimensionality reduction first (principal component analysis) on the full
dataset to reduce its size. We stress, however, that the results presented
here were all obtained using the complete dataset.

The intersection of the mixed-mode SSM with the stable manifold of the edge
state revealed a structurally stable curve in phase space, the slowest
submanifold within the edge manifold. We have demonstrated by direct numerical
simulation that the extracted curve, the one-dimensional footprint of the edge
manifold, already displays edge-like characteristics: it separates
laminarizing and turbulent trajectories in the phase space. This illustrates
that the identified structure can be used to predict whether a given initial
condition develops turbulence or simply laminarizes. We have also constructed
parameter-dependent SSM-reduced models that remain valid over a range of
Reynolds numbers.

For simplicity, we carried out the calculations following an additional step
of symmetry reduction through the method of slices. To obtain structures in
the full phase space, one simply takes the direct product of the identified
structures with the group orbit of streamwise translations. For example, the
mixed-mode SSM of Fig. 6 becomes a three-dimensional structure and its
intersection with the edge manifold becomes a cylinder in the full phase
space. Alternatively, using the non-sliced simulation data, one may look for
the mixed-mode SSM as a 3-dimensional manifold.

In this study, we utilized the small domain size and the additional shift-
reflect symmetry and the 2-fold azimuthal rotation symmetry restrictions.
These assumptions made it possible to conclude that the edge manifold is
locally of codimension one, enabling the easy characterization of its slowest
submanifold as the intersection with the mixed-mode SSM. Although these
assumptions fail for more general pipe flows, a similar approach could be
employed even in longer pipes (Avila et al. (2010, 2013)) without imposing the
symmetries. By generating trajectories restricted to the edge of chaos using
the bisection method (Itano & Toh (2001)), one could find the most influential
submanifolds within the edge along with the dynamics restricted onto them.
This would, however, require constructing invariant manifolds of considerably
higher dimensions which brings additional challenges.

## Acknowledgements

We are grateful to Florian Kogelbauer, Ashley Willis, and Mingwu Li for
several insightful conversations. The numerical simulations were performed on
the Euler cluster operated by the High Performance Computing group at ETH
Zürich. We acknowledge support from the Turbulent Superstructures Program
(SPP1881) of the German National Science Foundation (DFG).

## Data availability

The code and data (in compressed format) used in the paper is available in the
repository https://github.com/haller-group/SSMLearnPy. The complete dataset is
available from the authors upon request.

## Declaration of Interests

The authors report no conflict of interest.

## Appendix: Resonance conditions

The mixed-mode SSM of the lower-branch traveling wave, as constructed by
Haller et al. (2023), exists under the conditions of the linearization theorem
of Sternberg (1958). These require that the eigenvalues of the Jacobian of (5)
at the lower-branch traveling wave be nonresonant. Specifically, denoting the
eigenvalues as $\lambda_{i}$ and ordering them according to descending real
part, we require

$\lambda_{j}\neq\sum_{i=1}^{d}m_{i}\lambda_{i},\quad
m_{i}\in\mathbb{N}\quad\sum_{i=1}^{d}m_{i}\geq 2.$ (26)

We check that the conditions (26) hold for a finite subset of the spectrum of
the lower-branch traveling wave by constructing all admissible integer linear
combinations of the leading $23$ eigenvalues shown in the right panel of Fig.
10. To select the eigenvalues nearest to resonance, for every eigenvalue
$\lambda_{j}$, we define

$D(\lambda_{j})=\min_{m_{i}}\left|\lambda_{j}-\sum_{i=1}^{d}m_{i}\lambda_{i}\right|,$
(27)

which measures how close $\lambda_{j}$ is to being resonant. We plot the
relative measure of resonance-closeness, $D(\lambda)/|\lambda|$ for the
leading eigenvalues in the left panel of Fig. 10.

Figure 10: Left panel: Leading eigenvalues of the lower-branch traveling wave
restricted to the $\mathcal{S}-$invariant subspace. Right panel: relative
measure of resonance-closeness, $D(\lambda)/|\lambda|$ as a function of
$|\lambda|$. The eigenvalue closest to being resonant, $\lambda_{20}$, is
marked with a red point. The three eigenvalues generating this close
resonance, $\lambda_{1}$, $\lambda_{4}$, and $\lambda_{8}$ are marked with
orange points. The directory, including the data and the Jupyter notebook that
generated this figure can be accessed at
https://cocalc.com/share/public_paths/186d749169aa46fdb63d21756f1325b50964ac2e

Although no exact resonance is seen in the spectrum, near-resonances of up to
$0.1\%$ coincidence can be seen. The nearest resonance is highlighted in Fig.
10 as $\lambda_{20}$, whose value is

$\lambda_{20}=-0.06529+0.00829i.$ (28)

With a linear combination of $\lambda_{1}$, $\lambda_{4}$, and $\lambda_{8}$,
we get

$\lambda_{1}+\lambda_{4}+3\lambda_{8}=-0.06514+0.00822i,$ (29)

which yields $D(\lambda)/|\lambda|\approx 0.2\%$. Although close resonances of
similar severity may occur in the rest of the spectrum, we see no indication
of exact resonances, hence the results of Haller et al. (2023) should apply.

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