# Secular resonance of inner test particles in hierarchical planetary systems

Hanlun Lei1,2
1 School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China
2 Key Laboratory of Modern Astronomy and Astrophysics in Ministry of
Education, Nanjing University, Nanjing 210023, China E-mail<EMAIL_ADDRESS>
(Accepted. Received; in original form)

###### Abstract

The present work studies the secular resonance associated with the critical
argument $\sigma=\varpi$ ($\varpi$ is the longitude of pericentre) for inner
test particles moving in low-eccentricity region with inclination $i$ smaller
than $39^{\circ}$. To formulate the dynamical model, the double-averaged
Hamiltonian is formulated up to an arbitrary order in the semimajor axis
ratio, and then those high-order periodic terms are removed from the double-
averaged Hamiltonian by means of Hori–Deprit transformation technique. The
resulting Hamiltonian determines a resonant model with a single degree of
freedom. Based on the resonant model, it becomes possible to explore the
phase-space structure, resonant centre, and resonant width in an analytical
manner. In particular, an excellent correspondence is found between the
resonant width in terms of the eccentricity variation and the maximum
variation of eccentricity ($\Delta e$) for test particles initially placed on
quasi-circular orbits. It means that the secular dynamics in the low-
eccentricity space with $i<39^{\circ}$ is dominantly governed by the secular
resonance associated with $\sigma=\varpi$.

###### keywords:

celestial mechanics–planets and satellites: dynamical evolution and
stability–methods: analytical

††pagerange: Secular resonance of inner test particles in hierarchical
planetary systems–Secular resonance of inner test particles in hierarchical
planetary systems††pubyear: 2020

##  1 Introduction

The Hamiltonian of a hierarchical planetary system is composed of two
Keplerian terms plus a coupling term (also called the disturbing function)
describing the mutual interaction between the inner and outer planets
(Harrington, 1969). To study the secular dynamics on timescales much longer
than the orbital periods, it is usual to filter out the short-period effects
by averaging the planetary disturbing function over the orbital periods of the
inner and outer planets. After averaging, the Keplerian energies of the inner
and outer planets are separately conserved, and the secular evolution of
system is a dynamical outcome of their angular momentum exchange (Naoz, 2016).
The averaging approximation can be realised by means of Von Zeipel
transformation (Naoz et al., 2013).

The conventional double averaging corresponds to the secular approximation at
the first order in the magnitude of perturbation, indicating that the
approximation works well for those systems with high hierarchy (i.e., the
systems with weak perturbation). Physical hierarchy is measured by the
separability among degrees of freedom of the dynamical system or the
differences between the periods of the so-called “short-period” and “long-
period” variables (Beaugé et al., 2006), so that high hierarchy requires that
the semimajor axis ratio is small. However, if a planetary three-body system
is not highly hierarchical (in this situation the perturbation is so strong
that the separation of ‘short period’ and ‘long period’ becomes blurred), the
short-period effects within the orbital periods of the outer and/or inner
planets need to be taken into account in the secular approximation (Ćuk &
Burns, 2004; Luo et al., 2016; Lei et al., 2018; Lei, 2019; Hamers & Samsing,
2019a, b) or high-order secular Hamiltonian needs to be formulated to predict
long-term behaviours (Beaugé et al., 2006).

As for highly hierarchical planetary systems, the conventional double-
averaging approach has been widely used to investigate the secular dynamics.
Considering one of the inner objects as a test particle, both Kozai (1962) and
Lidov (1962) expanded the disturbing function as a power series of the
semimajor axis ratio ($\alpha$) between the test particle and the perturber
and considered applications when the orbit of the outer binary is circular.
Kozai (1962) investigated the secular influences of Jupiter’s perturbations
upon the main belt asteroids in our Solar system, and Lidov (1962) considered
the secular evolution of artificial satellites around the Earth perturbed by
the Sun and Moon. In both the landmark works, the double-averaged disturbing
function (or Hamiltonian) is independent on the longitude of ascending node
due to the axisymmetric potential, making the $z$-axis component of the
angular momentum be conserved (here the $z$-axis is along the vector of the
total angular momentum) and the resulting dynamical model be integrable. Under
the quadrupole-level approximation, a well-known secular resonance (also
called Kozai resonance) occurs between the longitude of pericentre and the
longitude of ascending node when the mutual inclination is higher than
$39.2^{\circ}$ (Kozai, 1962; Thomas & Morbidelli, 1996). Due to the Kozai
resonance, the eccentricity and inclination undergo a large coupled
oscillation, and such a mechanism of exciting eccentricity and inclination is
called the Kozai–Lidov (KL) mechanism. In recent years, many researchers
relaxed the assumptions made in Kozai (1962) and Lidov (1962) in terms of the
following three aspects: (a) the outer orbit is eccentric, (b) all three
objects involved are not massless, and (c) the disturbing function is relaxed
to the octupole (or higher) orders in the semimajor axis ratio (Michtchenko et
al., 2006; Libert & Henrard, 2007; Libert & Tsiganis, 2009; Migaszewski &
Goździewski, 2009; Naoz et al., 2011, 2013; Volpi et al., 2019). Under these
relaxations, the conventional KL mechanism is extended to the eccentric
Kozai–Lidov (EKL) mechanism. Both the KL and EKL mechanisms have been widely
utilised to interpret secular phenomena in practical astrophysical systems
(Soderhjelm, 1982; Thomas & Morbidelli, 1996; Holman et al., 1997; Blaes et
al., 2002; Miller & Hamilton, 2002; Wu & Murray, 2003; Michtchenko et al.,
2006; Libert & Henrard, 2007; Libert & Tsiganis, 2009; Migaszewski &
Goździewski, 2009; Lithwick & Naoz, 2011; Katz et al., 2011; Naoz et al.,
2011, 2013; Shappee & Thompson, 2013; Antognini et al., 2014; Antognini, 2015;
Petrovich, 2015; Naoz, 2016; Carvalho et al., 2016; Naoz et al., 2017;
Sidorenko, 2018; Volpi et al., 2019).

Based on the aforementioned discussions, we know that, in a hierarchical
planetary system with the disturbing body on an eccentric orbit, the secular
dynamics at mutual inclinations higher than $\sim$$39.2^{\circ}$ is dominantly
governed by Kozai resonance. A question arises: what’s the dynamical mechanism
in governing the secular dynamics in the parameter space where Kozai resonance
cannot happen (i.e., in the region with inclinations smaller than
$\sim$$39^{\circ}$)? To my knowledge, there are few publications focusing on
this topic.

Funk et al. (2011) made a parametric study in the region with $i<39^{\circ}$
for the long-term stability of inclined and quasi-circular test particles
(fictitious Earth-like planets) moving under the gravity field produced by a
central star and an eccentric giant planet. They found an interesting
dynamical region around the mutual inclination of $\sim$$35^{\circ}$ where the
low-eccentricity orbits are long-time stable, indicating that the inclined
($\sim$$35^{\circ}$) quasi-circular Earth-mass companion is possible to
survive in the habitable zone of extrasolar systems (Funk et al., 2011). Under
the double-averaged model at the octupole-level approximation, Libert &
Delsate (2012) computed the maximum variation of eccentricity as a function of
the mutual inclination (see Fig. 6 in their work) and reproduced the long-time
stable region at inclination of $\sim$$35^{\circ}$. To understand the long-
term stable dynamics in this special region, Libert & Delsate (2012) adopted
the 12th-order (double-averaged) Hamiltonian expansion in eccentricities and
inclinations (Libert & Henrard, 2007) and formulated a second-order secular
Hamiltonian by means of Lie-series transformation method to determine the
fundamental frequencies, similar to the method used in Libert & Henrard
(2008). Then, Libert & Delsate (2012) showed that this region presents an
equality of two fundamental frequencies (please refer to Fig. 8 in their
work), and thus they concluded that the long-term stable dynamics at
inclination of $\sim$$35^{\circ}$ is due to the existence of the secular
resonance associated with $\sigma=\omega-\Omega$.

Figure 1: Fundamental frequencies ($\dot{\omega}$ and $\dot{\Omega}$) as well
as their respective periods ($T_{\omega}$ and $T_{\Omega}$) as functions of
initial inclination $i_{0}$. In simulations, the double-averaged Hamiltonian
shown by equation (2) is truncated at the sixth order in the semimajor axis
ratio $\alpha$ and the secular Hamiltonian is formulated up to the second
order in $\varepsilon$ by means of Lie-series transformation (see Section 4
for the definition of parameter $\varepsilon$ and the Lie-series
transformation). The planetary system consists of a central body with mass
$m_{\rm Sun}$, a giant planet with mass $m_{\rm Jupiter}$ and a test particle
with infinitesimal mass. The same parameters as the ones in Libert & Delsate
(2012) are used in the simulations: the eccentricity of the giant planet is
assumed as $e_{p}=0.3$, the semimajor axis ratio as $\alpha=0.05$ and the
initial eccentricity of the test particle as $e=1\times 10^{-6}$.

Based on our double-averaged Hamiltonian in power series of the semimajor axis
ratio (to be presented in Section 2), we formulated the secular Hamiltonian by
using the same transformation method adopted by Libert & Delsate (2012) and
reproduced the fundamental frequencies ($\dot{\omega}$ and $\dot{\Omega}$) as
well as the corresponding periods ($T_{\omega}$ and $T_{\Omega}$) for test
particles moving in low-eccentricity regions. The fundamental frequencies as
well as their respective periods are reported in Fig. 1, where the secular
Hamiltonian is formulated up to the second order in $\varepsilon$ (see Section
4 for the definition of parameter $\varepsilon$). It is observed that the
equality between $T_{\Omega}$ and $T_{\omega}$ occurs at inclination of
$\sim$$35.9^{\circ}$. The curves of $T_{\omega}$ and $T_{\Omega}$ given in
Fig. 1 are in good agreement with the ones shown in Libert & Delsate (2012)
(see Fig. 8 in their work). However, in our simulations the fundamental
frequencies holds $\dot{\omega}>0$ and $\dot{\Omega}<0$, so that the
intersection point between the curves of $T_{\Omega}$ and $T_{\omega}$ may
lead us to a different conclusion: the secular resonance associated with
$\sigma=\omega+\Omega=\varpi$ (rather than $\sigma=\omega-\Omega$) occurs at
the inclination of the intersection point.

The purpose of this work is to understand how the secular resonance dominates
the secular dynamics in the parameter space with inclinations smaller than
$39^{\circ}$ by studying the secular resonance with critical argument
$\sigma=\omega+\Omega$. To this end, it is required to formulate the resonant
Hamiltonian and then explore phase-space structures, resonant centres and
widths. In the process of formulating resonant Hamiltonian, we adopt the
Hori–Deprit transformation technique (Hori, 1966; Deprit, 1969) to remove
high-order short-period effects from the Hamiltonian. Based on the resulting
resonant Hamiltonian, the phase-space structures, resonant centres and widths
of the resonances are analyzed. Interestingly, there is an excellent
correspondence between the resonant width in terms of the eccentricity
variation and the maximum variation of eccentricity ($\Delta e$) for particles
initially placed on quasi-circular orbits. It means that the shape of $\Delta
e$ numerically determined is sculpted by the secular resonance associated with
$\sigma=\omega+\Omega$.

The structure of the present work is organised as follows. In Section 2, the
double-averaged Hamiltonian is briefly introduced and, in Section 3, secular
dynamics in the low-eccentricity region with inclinations smaller than
$39^{\circ}$ is numerically explored. In Section 4, the perturbation theory
based on the Hori–Deprit transformation technique is introduced and the
second-order resonant Hamiltonian is formulated. Results and discussions are
provided in Section 5. Finally, Section 6 concludes this work.

##  2 Hamiltonian function

The hierarchical planetary system considered in this study is composed of a
central body with mass $m_{0}$ (e.g., a star), a terrestrial planet with mass
$m$ (e.g., an Earth-mass planet) and a faraway disturbing body with mass
$m_{p}$ (e.g., a Jupiter-mass planet). In this system, the Earth-mass planet
and the central body constitute a tight inner binary, and the disturbing body
and the barycentre of the inner binary form an outer binary (please refer to
the left panel of Fig. 2 for the relative configuration). Due to the fact that
$m$ is much smaller than $m_{p}$ and $m_{0}$, the Earth-mass planet in this
planetary system can be approximated as a test particle. In the test particle
approximation, the hierarchical planetary system reduces to an elliptic
restricted three-body problem, where the faraway disturbing body moves around
the central body on a fixed elliptic orbit.

The hierarchy of the planetary system is specified by the semimajor axis ratio
between the inner and outer planets, given by $\alpha=a/a_{p}$. For small
$\alpha$, the Hamiltonian function, describing the mutual interaction between
the orbits of the inner and outer planets, can be expanded as a power series
of $\alpha$. As a result, the complete Hamiltonian that describes the motion
of test particles can be written as (Morbidelli, 2002)

${\cal H}=-\frac{\mu}{{2a}}-\frac{{{\cal
G}{m_{p}}}}{{{a_{p}}}}\sum\limits_{n=2}^{\infty}{{\alpha^{n}}{{\left({\frac{r}{a}}\right)}^{n}}{{\left({\frac{{{a_{p}}}}{{{r_{p}}}}}\right)}^{n+1}}{P_{n}}\left({\cos\psi}\right)},$
(1)

where $\mu={\cal G}{m_{0}}$ with $\cal{G}$ as the universal gravitation
constant, $r$ and $r_{p}$ are, respectively, the magnitudes of the position
vectors $\bm{r}$ for the test particle and $\bm{r}_{p}$ for the disturbing
body, $\psi$ is the relative angle between $\bm{r}$ and $\bm{r}_{p}$, and
${P_{n}}\left({\cos\psi}\right)$ are the Legendre polynomial of $\cos\psi$
with degree $n$. Unless otherwise stated, in the entire work we use the
variables with subscript ‘$p$’ for the wide orbit of the perturber and the
ones without any subscript for the orbit of test particles.

To describe the motion of the objects involved in the planetary system, we
choose an inertial right-handed reference frame originated at the central
body, denoted by $o$-$xyz$, with the invariable plane of the system (i.e. the
orbit of the disturbing body around the central body) as the fundamental
plane, the total angular momentum vector as the direction of the $z$-axis and
the eccentricity vector of the disturbing body as the direction of the
$x$-axis (please refer to the left panel of Fig. 2 for details). Under the
coordinate system, the orbits of the test particle (perturber) are described
by the semimajor axis $a$ ($a_{p}$), eccentricity $e$ ($e_{p}$), inclination
$i$ ($i_{p}$), longitude of ascending node $\Omega$ ($\Omega_{p}$), the
argument of pericentre $\omega$ ($\omega_{p}$) and mean anomaly $M$ ($M_{p}$).
The longitude of pericentre is defined by $\varpi=\omega+\Omega$ for the test
particle and $\varpi_{p}=\omega_{p}+\Omega_{p}$ for the perturber. Due to the
specific choice of the fundamental plane about the coordinate system, it is
not difficult to get $i_{p}=0$.

The Hamiltonian function given by equation (1) can be expanded as a Fourier
series form (Kaula, 1961, 1962; Murray & Dermott, 1999; Beaugé et al., 2006;
Lei, 2019). To study the secular evolution, it is usual to perform double
averages for the Hamiltonian function over the orbital periods of the inner
and outer planets (Libert & Tsiganis, 2009; Laskar & Boué, 2010; Lithwick &
Naoz, 2011; Naoz et al., 2011, 2013; Li et al., 2014; Carvalho et al., 2016;
Naoz et al., 2017; Lei et al., 2018; Lei, 2019, 2020). The phase averaging,
also called the secular approximation (Naoz, 2016), is realised by Von Zeipel
transformation, as discussed in the Appendix of Naoz et al. (2013).
Eliminating those terms involving the mean anomalies $M$ or $M_{p}$ from the
Fourier series expression of Hamiltonian function (Lei, 2019), the double-
averaged formulation of Hamiltonian can be written as (for simplicity, we
still use ${\cal H}$ to represent the double-averaged Hamiltonian)

$\displaystyle{\cal H}=$ $\displaystyle-\frac{\mu}{{2a}}-\frac{{{\cal
G}{m_{p}}}}{{{a_{p}}}}\sum\limits_{n=2}^{N}{\sum\limits_{k=0}^{\left[{\frac{n}{2}}\right]}{\sum\limits_{q=0}^{n-2k}{\sum\limits_{{t_{1}}=0}^{n-2k-q}{\sum\limits_{{t_{2}}=0}^{q}{\sum\limits_{{t_{3}}=0}^{n-2k-q}{\sum\limits_{{t_{4}}=0}^{q}{\kappa_{1}\kappa_{2}}}}}}}}$
(2)
$\displaystyle\times{\alpha^{n}}X_{0}^{n,\left({n-2k-2{t_{1}}-2{t_{2}}}\right)}\left(e\right)X_{0}^{-\left({n+1}\right),\left({n-2k-2{t_{3}}-2{t_{4}}}\right)}\left({{e_{p}}}\right)$
$\displaystyle\times{\cos^{q}}i\cos{\theta},$

where $\left[{\frac{n}{2}}\right]$ represents the lower integer of
$\frac{n}{2}$, $X_{0}^{a,b}(e)$ are the Hansen coefficients, and the angle
$\theta$ is given by

$\displaystyle\theta$
$\displaystyle=\left({n-2k-2{t_{1}}-2{t_{2}}}\right)\varpi+\left({n-2k-2{t_{3}}-2{t_{4}}}\right){\varpi}_{p}$
$\displaystyle-2\left({n-2k-{t_{1}}-{t_{2}}-{t_{3}}-{t_{4}}}\right)\Omega,$

where the d’Alembert rule holds. Due to the specific choice of the $x$-axis
about the coordinate system (along the eccentricity vector of the perturber),
it is not difficult to get $\varpi_{p}=0$. Consequently, the angle $\theta$
becomes

$\theta=\left({n-2k-2{t_{1}}-2{t_{2}}}\right)\varpi-2\left({n-2k-{t_{1}}-{t_{2}}-{t_{3}}-{t_{4}}}\right)\Omega.$

In addition, the coefficients $\kappa_{1}$ and $\kappa_{2}$ in equation (2)
are given by

${\kappa_{1}}=\frac{{{{\left({-1}\right)}^{k+3q+{t_{2}}+{t_{4}}}}}}{{{2^{3n-4k}}}}\frac{{(2n-2k)!}}{{k!(n-k)!(n-2k)!}},$

and

${\kappa_{2}}={n-2k\choose
q}{n-2k-q\choose{t_{1}}}{q\choose{t_{2}}}{n-2k-q\choose{t_{3}}}{q\choose{t_{4}}}$

with the binomial coefficients defined by ${n\choose
m}={\frac{{n!}}{{m!(n-m)!}}}$.

When the truncated order is set as $N=2$, the Hamiltonian ${\cal H}$ given by
equation (2) stands for the quadrupole-level approximation and, when the
truncated order is assumed at $N=3$, the associated Hamiltonian ${\cal H}$
corresponds to the octupole-level approximation, and the like.

Some remarks are made here for the expansion of disturbing function. About the
coplanar and non-resonant two-planet systems, Lee & Peale (2003) expanded the
double-averaged planetary disturbing function up to the third order in terms
of the ratio of semimajor axes to develop the octupole-level secular
perturbation theory. The octupole-level approximation of disturbing function
can be found in Libert & Delsate (2012). As a generalisation of the octupole-
level secular theory, Migaszewski & Goździewski (2008) formulated the double-
averaged planetary disturbing function up to the 24th order with respect to
the semimajor axis ratio. Extending to the spatial two-planet systems, Laskar
& Boué (2010) revisited the derivation of the double-averaged planetary
disturbing function up to an arbitrary order in terms of the ratio of the
semi-major axes. In particular, when one of the two planets is massless (it
reduces to a restricted three-body problem), we believe that the expansion
given by Laskar & Boué (2010) should be equivalent to the one shown by
equation (2).

As usual, we adopt a set of canonical variables, known as modified Delaunay’s
elements, to describe the motion of test particles (Morbidelli, 2002),

$\displaystyle\Lambda=\sqrt{\mu a},\quad\lambda=M+\varpi,$ $\displaystyle
P=\sqrt{\mu a}\left(\sqrt{1-e^{2}}-1\right),\quad p=\varpi,$ $\displaystyle
Q=\sqrt{\mu a(1-e^{2})}\left(\cos{i}-1\right),\quad q=\Omega.$

It is noted that the variables $p$, $q$, $P$ and $Q$ defined here have
opposite signs in comparison to the traditional ones (please see Morbidelli
(2002) for the classical version). In the double-averaged environment, the
angular variable $\lambda$ disappears from the Hamiltonian, so that the
Keplerian energy (or the semimajor axis) of the test particle remains
stationary in the long-term evolution. Thus, the first term in the Hamiltonian
given in equation (2) can be removed.

For the sake of simplicity, we organise the double-averaged Hamiltonian in a
compact form:

${\cal H}=\sum\limits_{{k_{1}}=0}^{N}{\sum\limits_{{k_{2}(2)}=-2N}^{2N}{{{\cal
C}_{{k_{1}},{k_{2}}}}\cos\left({{k_{1}}p+{k_{2}}q}\right)}},$ (3)

where $k_{2}(2)$ means the index $k_{2}$ changes from $-2N$ to $2N$ with step
2 (i.e., $k_{2}$ should be an even number) and the coefficients ${{{\cal
C}_{{k_{1}},{k_{2}}}}}$ are functions of the action variables ($P$ and $Q$)
and their expressions can be directly derived from equation (2).

Replacing the double-averaged Hamiltonian in the canonical relations, we can
obtain the secular equations of motion as follows:

$\displaystyle\dot{p}=$ $\displaystyle\frac{{\partial{\cal H}}}{{\partial
P}},\quad\dot{P}=-\frac{{\partial{\cal H}}}{{\partial p}},$ (4)
$\displaystyle\dot{q}=$ $\displaystyle\frac{{\partial{\cal H}}}{{\partial
Q}},\quad\dot{Q}=-\frac{{\partial{\cal H}}}{{\partial q}},$

which determines a dynamical system with two degrees of freedom. In this work,
we call the model shown by equation (4) the full secular model.

To validate the full secular model truncated at different order in $\alpha$,
we introduce the full $N$-body model (i.e., the elliptic restricted three-body
problem), in which the equations of motion for test particles are given by

$\ddot{\bm{r}}=-\mu\frac{\bm{r}}{{{r^{3}}}}-{\cal
G}{m_{p}}\left({\frac{{{\bm{r}}-{{\bm{r}}_{p}}}}{{{{\left|{{\bm{r}}-{{\bm{r}}_{p}}}\right|}^{3}}}}+\frac{{{{\bm{r}}_{p}}}}{{r_{p}^{3}}}}\right).$
(5)

The Hamiltonian of the full $N$-body model is provided by equation (1).

Figure 2: Relative configuration of three objects in a hierarchical planetary
system and the definition of coordinate system adopted in this study (_left
panel_) and time histories of the eccentricity for numerically integrated
trajectories (_right panel_). In the left panel, the position vectors of the
test particle and the disturbing body are $\bm{r}$ and ${\bm{r}}_{p}$. In the
right panel, the trajectories are numerically propagated under the full
$N$-body model (i.e. the elliptic restricted three-body problem) as well as
under the double-averaged models truncated at different orders in the
semimajor axis ratio between the inner test particle and giant planet
$\alpha=a/a_{p}$. The dynamical model is characterised by
$m_{0}=m_{\rm{Sun}}$, $a_{p}=1.0$ $\rm{au}$, $m_{p}=m_{\rm{Jupiter}}$ and
$e_{p}=0.3$. The initial elements are taken as follows: $a_{0}=0.1$ $\rm{au}$,
$e_{0}=0.03$, $i_{0}=30^{\circ}$, $\Omega_{0}=90^{\circ}$ and
$\omega_{0}=270^{\circ}$.

Integrating the differential equations given by equations (4) and (5)
numerically, we can produce the trajectories in the full secular model and in
the full $N$-body model. Figure 2 shows time histories of the eccentricities
of the trajectories numerically propagated under the full secular models
truncated at different orders in $\alpha$ as well as under the full $N$-body
model. Please refer to the caption for more details about the setting of
initial conditions. The parameters specifying the dynamical model are
$m_{0}=m_{\rm Sun}$, $m_{p}=m_{\rm Jupiter}$, $a_{p}=1$ $\rm au$ and
$e_{p}=0.3$ (unless otherwise specified, in the entire work we adopt this
model to perform practical simulations).

From Fig. 2, it is observed that (a) there is an obvious deviation for the
evolution of eccentricity under the quadrupole-order approximation compared to
that produced under the full $N$-body model, meaning that the quadrupole-level
approximation (i.e., 2nd order in $\alpha$) cannot predict the long-term
behaviours in the full $N$-body model and (b) the curve of eccentricity
produced under the full secular model truncated at the 5th order in $\alpha$
is in perfect agreement with that produced under the full $N$-body model.

##  3 Secular dynamics at inclinations below $39^{\circ}$

Figure 3: The maximum variation of eccentricity as a function of the initial
inclination $i_{0}$, reached during the integration time of $1.6\times 10^{6}$
$\rm{year}$ (_left panel_) and time histories of the argument
$\sigma=\varpi(=\omega+\Omega)$ for inner test particles with
$i_{0}=30^{\circ}$ (_middle panel_) and $i_{0}=37^{\circ}$ (_right panel_).
The inner test particles are initially placed on quasi-circular orbits
($e_{0}=1\times 10^{-4}$). The initial conditions of test particles are
$a_{0}=0.1$ $\rm{au}$ (i.e., $\alpha=0.1$), $\Omega_{0}=90^{\circ}$ and
$\omega_{0}=270^{\circ}$.

To study the secular dynamics, we perform numerical integrations under the
full secular model developed in the previous section and recorded the maximum
variation of eccentricity as a function of the initial inclination ranging
from $0^{\circ}$ to $39^{\circ}$, as shown in the left panel of Fig. 3. In the
simulations, the inner test particles are placed on quasi-circular orbits
($e_{0}=1\times 10^{-4}$) and the total integration time is fixed as
$1.6\times 10^{6}$ $\rm{year}$ (it is noted that the integration period taken
here is long enough to cover several periods of secular resonances). Please
refer to the caption of Fig. 3 for more details about initial conditions. If
we change the semimajor axis ratio $\alpha$ or the eccentricity of the giant
planet $e_{p}$, similar plots could be produced (the results are not repeated
here). It is noted that the curve of eccentricity variation shown in the left
panel of Fig. 3 is in agreement with the results given by Libert & Delsate
(2012) (see Fig. 6 in their work where the octupole-order dynamical model was
used).

According to the curve of eccentricity variation, we could observe the
following dynamical behaviours: (a) a sudden decrease of $\Delta e$ occurs
from $i=35.3^{\circ}$; (b) along the curve of $\Delta e$ there is a deep well
located in the inclination interval from $35.3^{\circ}$ to $39^{\circ}$; (c)
when the mutual inclination is smaller than $30^{\circ}$, $\Delta e$ seems
insensitive to the initial inclination ($\Delta e$ remains $\sim$$0.08$, this
value will be explained in Section 5 from the viewpoint of resonant width);
(d) when the inclination is greater than $30^{\circ}$ and smaller than
$35.3^{\circ}$, $\Delta e$ is a monotonically increasing function of the
inclination (the peak of $\Delta e$ could reach $\sim$0.23); (e) when the
inclination is greater than $35.8^{\circ}$ and smaller than $38.8^{\circ}$,
$\Delta e$ is again an increasing function of inclination and, in particular,
it remains relatively small ($\Delta e<0.05$).

Observing the curve of eccentricity variation shown in the left panel of Fig.
3, we may ask: what’s the dynamical mechanism hidden in the space with
inclinations smaller than $39^{\circ}$ for sculpting the shape of $\Delta e$?
Our purpose of this work is to provide a possible dynamical explanation for
the curve of $\Delta e$.

To provide a preliminary view about this problem, two numerical examples are
reported in the middle and right panels of Fig. 3 with initial inclinations at
$30^{\circ}$ and $37^{\circ}$ and initial eccentricity at $e_{0}=1\times
10^{-4}$. It is observed that the argument $\sigma(=\omega+\Omega)$ librates
around zero with amplitude smaller than $90^{\circ}$. More numerical
simulations still show that the libration of $\sigma=\varpi$ occurs in almost
the entire interested region. These preliminary simulations provide us
reliable evidence to such a conjecture that the secular dynamics in low-
eccentricity region with inclination smaller than $39^{\circ}$ is dominated by
the secular resonance associated with $\sigma=\omega+\Omega$. In Section 4, we
will study the secular resonance in detail.

##  4 Secular resonances

In this section, we intend to investigate the secular resonance associated
with the critical argument $\sigma=\varpi$ by means of Lie-series perturbation
theory and then discuss the resonant dynamics (including the phase-space
structures, distribution of resonant centre and width of resonance) in order
to provide an explanation for the shape of $\Delta e$ shown in the left panel
of Fig. 3.

###  4.1 Perturbation theory

In the current study, the perturbation theory based on Lie-series
transformation (Hori, 1966; Deprit, 1969), also known as the Hori–Deprit
transformation method, is adopted to eliminate high-order periodic terms from
the Hamiltonian in order to formulate secular Hamiltonian.

Before performing the Hori–Deprit transformation, it needs to determine a
kernel function that gives the unperturbed frequencies in the secular model.
To this end, let us divide the double-averaged Hamiltonian into two parts,
denoted by

${\cal H}=\sum\limits_{{k_{1}}=0}^{N}{\sum\limits_{{k_{2}(2)}=-2N}^{2N}{{{\cal
C}_{{k_{1}},{k_{2}}}}\cos\left({{k_{1}}p+{k_{2}}q}\right)}}={{\cal
H}_{0}}+\varepsilon{{\cal H}_{1}},$ (6)

where the unperturbed Hamiltonian ${{\cal H}_{0}}={\cal C}_{0,0}$ is a
function of the action variables $P$ and $Q$, and the perturbed Hamiltonian
${{\cal H}_{1}}$ is the sum of all those periodic terms, given by

${{\cal
H}_{1}}=\sum\limits_{\left\\{{{k_{1}},{k_{2}}}\right\\}\backslash\left\\{{0,0}\right\\}}{{{\cal
C}_{{k_{1}},{k_{2}}}}\cos\left({{k_{1}}p+{k_{2}}q}\right)},$

where $\left\\{{{k_{1}},{k_{2}}}\right\\}\backslash\left\\{{0,0}\right\\}$
means that $k_{1}$ and $k_{2}$ are not equal to zero at the same time. Note
that the small parameter $\varepsilon$ in equation (6) is used to formally
separate ${{\cal H}_{1}}$ from ${{\cal H}_{0}}$ and it is equal to unity in
magnitude. Obviously, the unperturbed Hamiltonian ${{\cal H}_{0}}$ is totally
integrable and thus it can be taken as the kernel function (Beaugé et al.,
2006), producing the unperturbed frequencies in the following way:

${\upsilon_{p}}=\frac{{\partial{{\cal H}_{0}}}}{{\partial
P}},\quad{\upsilon_{q}}=\frac{{\partial{{\cal H}_{0}}}}{{\partial Q}}.$

The key concept of Hori–Deprit perturbation theory is searching for a
canonical transformation from an old set of variables $(p,q,P,Q)$ to a new set
of variables $(p^{*},q^{*},P^{*},Q^{*})$, realised by the Lie-series
generating function,

$S=\sum\limits_{k}{{\varepsilon^{k}}{S_{k}}\left({{p^{*}},{q^{*}},{P^{*}},{Q^{*}}}\right)},$

in order to satisfy the following identity (the Hamiltonian is conserved
before and after the transformation):

${\cal H}={{\cal H}_{0}}\left({P,Q}\right)+\varepsilon{{\cal
H}_{1}}\left({p,q,P,Q}\right)=\sum\limits_{k}{{\varepsilon^{k}}{\cal
H}_{k}^{*}\left({{p^{*}},{q^{*}},{P^{*}},{Q^{*}}}\right)},$ (7)

where the new Hamiltonian ${\cal H}_{k}^{*}$ does not involve those short-
period terms.

Applying the Lie-series expansions to the left-hand side of equation (7) and
equating those terms with equal powers of $\varepsilon$ in equation (7), it is
possible to obtain the explicit relationship between the new Hamiltonian,
generating function and the old Hamiltonian (up to the second order in
$\varepsilon$ as an example) as follows (Hori, 1966):

$\displaystyle{\rm Order\;0}:\quad{\cal H}_{0}^{*}$ $\displaystyle={{\cal
H}_{0}},$ (8) $\displaystyle{\rm Order\;1}:\quad{\cal H}_{1}^{*}$
$\displaystyle={{\cal H}_{1}}+\left\\{{{{\cal H}_{0}},{S_{1}}}\right\\},$
$\displaystyle{\rm Order\;2}:\quad{\cal H}_{2}^{*}$ $\displaystyle={{\cal
H}_{2}}+\left\\{{{{\cal H}_{0}},{S_{2}}}\right\\}+\frac{1}{2}\left\\{{{{\cal
H}_{1}}+{\cal H}_{1}^{*},{S_{1}}}\right\\},$

where both the left- and right-hand terms are evaluated at the new canonical
variables (i.e. the variables with superscript of star), and
$\left\\{*,*\right\\}$ denotes the Poisson bracket.

###  4.2 Resonant Hamiltonian

We will use equation (8) to formulate resonant Hamiltonian up to the second
order in $\varepsilon$.

If a test particle is located inside the resonant region, the critical
argument $\sigma=p(=\varpi)$ becomes the ‘long-period’ variable and the angle
$q$ is the ‘short-period’ variable. Thus, in the double-averaged Hamiltonian,
the components involving $q$ are short-period terms and the remaining
components are long-period or secular terms. In order to study the resonant
dynamics, it is usual to eliminate those short-period terms from the
Hamiltonian .

As discussed in Section 4, the elimination of high-order short-period effects
can be realised through the Hori–Deprit method by using equation (8). At the
zeroth order in $\varepsilon$, it is obvious that the new Hamiltonian is equal
to the old one,

${\cal H}_{0}^{*}={{\cal H}_{0}}.$

At the first order in $\varepsilon$, we have

${\cal H}_{1}^{*}={{\cal H}_{1}}+\left\\{{{{\cal H}_{0}},{S_{1}}}\right\\},$
(9)

where ${\cal H}_{1}^{*}$ is the long-term part of the right-hand side, given
by

${\cal H}_{1}^{*}=\sum\limits_{k=1}^{N}{{{\cal
C}_{k,0}}\cos\left({k\sigma}\right)}$ (10)

and the periodic part of equation (9) related to the generating function
$S_{1}$ holds

$-\left\\{{{{\cal H}_{0}},{S_{1}}}\right\\}={{\cal H}_{1}}-{\cal
H}_{1}^{*}=\sum\limits_{{k_{2}}\neq 0}{{{\cal
C}_{{k_{1}},{k_{2}}}}\cos\left({{k_{1}}p^{*}+{k_{2}}q^{*}}\right)},$ (11)

where $k_{1}$ changes from $0$ to $N$ and $k_{2}$ from $-2N$ to $2N$ with step
2. It should be noted that, in equation (10), $\sigma$ is evaluated at the
starred variables, i.e. $\sigma=p^{*}$. As ${\cal H}_{0}$ is the kernel
function of the problem at hand, we have

$-\left\\{{{{\cal
H}_{0}},{S_{1}}}\right\\}={\upsilon_{p}}\frac{{\partial{S_{1}}}}{{\partial
p^{*}}}+{\upsilon_{q}}\frac{{\partial{S_{1}}}}{{\partial q^{*}}},$

so that the first-order generating function can be obtained as

${S_{1}}=\sum\limits_{{k_{2}}\neq 0}{\frac{{{{\cal
C}_{{k_{1}},{k_{2}}}}}}{{{k_{1}}{\upsilon_{p}}+{k_{2}}{\upsilon_{q}}}}\sin\left({{k_{1}}p^{*}+{k_{2}}q^{*}}\right)}.$
(12)

Extending to the second order in $\varepsilon$, we have the following
equality,

${\cal H}_{2}^{*}=\left\\{{{{\cal
H}_{0}},{S_{2}}}\right\\}+\frac{1}{2}\left\\{{{{\cal H}_{1}}+{\cal
H}_{1}^{*},{S_{1}}}\right\\}.$ (13)

Similar to the previous step, it is possible to obtain the second-order
resonant Hamiltonian ${\cal H}_{2}^{*}$ which is composed of the secular terms
of the right-hand side (i.e., removing those short-period terms in the right-
hand side to obtain ${\cal H}_{2}^{*}$). For the current problem at hand, we
only perform this process up to the second order in $\varepsilon$.
Theoretically speaking, it could be formulated up to an arbitrary order.

In summary, the resonant Hamiltonian, up to the second order in $\varepsilon$,
can be expressed as

${{\cal H}^{*}}={\cal H}_{0}^{*}+\varepsilon{\cal
H}_{1}^{*}+{\varepsilon}^{2}{\cal H}_{2}^{*}.$ (14)

It is obvious that the angle variable $q$ disappears from the resonant
Hamiltonian, meaning that its conjugate variable $Q$ is a conserved quantity
in the resonant model (i.e., $Q$ is the motion integral of the resonant
model). As a result, the system specified by ${{\cal H}^{*}}$ reduces to a
dynamical model with a single degree of freedom, which is integrable. Based on
the resonant Hamiltonian given by equation (14), the phase-space structures in
the space $(\sigma,P)$ can be produced by plotting level curves of ${\cal
H}^{*}$ with a given motion integral $Q$. According to the phase-space
structures, it is possible for us to identify the location of resonant centre
and resonant width.

For convenience, we adopt the minimum value of inclination, denoted by
$i_{\min}$, to stand for $Q$ by means of $Q=\sqrt{\mu a}(\cos{i_{\min}}-1)$
(i.e., $i_{\min}$ is the magnitude of inclination when the eccentricity is
assumed at zero).

###  4.3 Resonant centre and width

As formulated in the previous subsection, the resonant Hamiltonian formulated
up to the second order in $\varepsilon$ can be organised in the Fourier-series
form:

${{\cal H}^{{*}}}\left({Q};{\sigma,P}\right)={\cal H}_{0}^{*}+\varepsilon{\cal
H}_{1}^{*}+{\varepsilon^{2}}{\cal H}_{2}^{*}=\sum\limits_{k=0}^{2N}{{{\cal
D}_{k}}\left({P,Q}\right)\cos\left({k\sigma}\right)},$ (15)

where ${\cal D}_{k}$ are coefficients related to the action variables $P$ and
$Q$ (here $Q$ is the motion integral in the resonant model). The equations of
motion for the resonant model can be written as

$\displaystyle\dot{\sigma}$ $\displaystyle=\frac{{\partial{{\cal
H}^{\rm{*}}}}}{{\partial P}}=\sum\limits_{k=0}^{2N}{\frac{{\partial{{\cal
D}_{k}}\left({P,Q}\right)}}{{\partial P}}\cos\left({k\sigma}\right)},$ (16)
$\displaystyle\dot{P}$ $\displaystyle=-\frac{{\partial{{\cal
H}^{\rm{*}}}}}{{\partial\sigma}}=\sum\limits_{k=0}^{2N}{k{{\cal
D}_{k}}\left({P,Q}\right)\sin\left({k\sigma}\right)},$

where the stationary solution (also called equilibrium point) is determined by

$\dot{\sigma}=\frac{{\partial{{\cal H}^{\rm{*}}}}}{{\partial
P}}=0,\quad\dot{P}=-\frac{{\partial{{\cal H}^{\rm{*}}}}}{{\partial\sigma}}=0,$
(17)

meaning that the resonant Hamiltonian reaches its extremum at an equilibrium
point. Putting equation (16) into the stationary condition shown by equation
(17), we can get that the equilibrium points occur at $\sigma=0$ or
$\sigma=\pi$.

Linearizing the equations of motion given by equation (16) in the vicinity of
an equilibrium point yields a linear system and the eigenvalues of its
coefficient matrix determines the stability of the equilibrium point. The
stability condition requires

${\left({\frac{{{\partial^{2}}{{\cal H}^{\rm{*}}}}}{{\partial\sigma\partial
P}}}\right)^{2}}<\frac{{{\partial^{2}}{{\cal
H}^{\rm{*}}}}}{{\partial{\sigma^{2}}}}\frac{{{\partial^{2}}{{\cal
H}^{{*}}}}}{{\partial{P^{2}}}},$

where both the left- and right-hand sides are evaluated at the considered
equilibrium point. Usually, the stable equilibrium corresponds to an elliptic
point which is the resonant centre in the resonant model and the unstable one
is called a saddle point. The level curves of Hamiltonian passing through the
saddle point correspond to the dynamical separatrices, dividing the entire
phase space into libration and circulation regions.

In the phase portrait specified by a given $i_{\min}$ (equivalently, the
motion integral $Q$ is given), let’s denote the stable solution as
$(\sigma_{s},P_{s})$ and the unstable solution as $(\sigma_{u},P_{u})$.
Usually, the level curves of resonant Hamiltonian passing through the unstable
point corresponds to dynamical separatrices. The Hamiltonian of the dynamical
separatrix is given by

${{\cal H}^{\rm{*}}}\left({{Q};{\sigma_{u}},{P_{u}}}\right)={{\cal
H}^{\rm{*}}}\left({{Q};{\sigma_{s}},{P_{1}}}\right)={{\cal
H}^{\rm{*}}}\left({{Q};{\sigma_{s}},{P_{2}}}\right)$

and the distance between the separatrices evaluated at $\sigma=\sigma_{s}$
defines the resonant width, calculated by $\Delta P=P_{2}-P_{1}$. For
convenience, the resonant width is usually measured in terms of the variation
of eccentricity, given by

$\Delta e=-\frac{{\Delta P}}{\sqrt{\mu
a}}\frac{1}{{\sqrt{{{\left({\frac{\sqrt{\mu a}}{{{P_{s}+\sqrt{\mu
a}}}}}\right)}^{2}}-1}}},$ (18)

which is used to determine the resonant width in the following simulations.
Please see Figs. 4–5 for the definition of $\Delta e$ in phase portraits.

##  5 Results

In this section, the analytical developments developed in Section 4 are
applied to secular resonances occurring in the interested parameter space
(i.e., the low-eccentricity region with inclinations smaller than
$39^{\circ}$). Initially, the phase portraits can be produced by plotting
level curves of the resonant Hamiltonian. By analyzing the structures arising
in phase portraits, it is possible to identify the libration zones and measure
the size of libration islands (i.e., width of resonance). At last, a
correspondence is made between the resonant width and the maximum variation of
eccentricity $\Delta e$, which provides an explanation for the shape of
$\Delta e$ numerically determined in section 3.

###  5.1 Phase portraits

Figure 4: Level curves of the resonant Hamiltonian ${\cal H}^{*}$ specified by
the minimum value of inclination at $i_{\min}=10^{\circ}$. In simulations, the
semimajor axis ratio is taken as $\alpha=0.1$. The resonant centres are marked
by black dots and the level curves separating the libration regions from
circulation regions are shown in red lines. The resonant width is measured by
the variation of eccentricity, denoted by $\Delta e$.

Figure 5: Similar to Fig. 4 but for the resonant Hamiltonian ${\cal H}^{*}$
specified by $i_{\min}=36^{\circ}$.

As examples, we take the minimum values of inclination at
$i_{\min}=10^{\circ}$ and $i_{\min}=36^{\circ}$ to show the phase portraits.
In this work, we adopt the polar coordinate
$(e\cos{\sigma/2},e\sin{\sigma/2})$ instead of $(e\cos{\sigma},e\sin{\sigma})$
to show phase portraits due to the consideration that the zero-eccentricity
point (i.e., the coordinate origin) is a visible saddle point in the
$(e\cos{\sigma/2},e\sin{\sigma/2})$ space.

The level curves of resonant Hamiltonian (i.e., phase portraits) specified by
$i_{\min}=10^{\circ}$ are reported in Fig. 4, in which the resonant centres
are shown by black dots and the level curves which separate the libration
regions from the circulation regions are marked in red lines. From Fig. 4, it
is observed that (a) the zero-eccentricity point (i.e., the coordinate origin
in the polar coordinate plane) is a visible saddle point in the resonant
model, (b) the resonant centre is located at $\sigma=0^{\circ}$, (c) there are
two types of motion modes, including rotation and libration, which are
separated by the level curve stemming from zero-eccentricity point (it
corresponds to the separatrix in the resonant model), (d) along the line of
separatrix the eccentricity could be excited from $\sim$0 up to $\sim$$0.08$
(remind that this value is approximately equal to the value of $\Delta e$
numerically determined in Section 3), and (e) the distance between the
separatrix in terms of the variation of eccentricity defines the resonant
width, demoted by $\Delta e$, as shown in Fig. 4.

The phase portraits of the secular resonance specified by
$i_{\min}=36^{\circ}$ are shown in Fig. 5, which shows that (a) the resonant
centre is located at $\sigma=0^{\circ}$ and (b) the level curve stemming from
zero eccentricity divides the entire phase space into regions of rotation and
libration types. The distance between the separatrices evaluated at the
resonant centre also defines the resonant width, denoted by $\Delta e$.

Figure 6: Analytical results of resonant centre and resonant width in terms of
$\Delta e$ for different semimajor axis ratios at $\alpha=0.1$, $\alpha=0.15$,
$\alpha=0.2$ and $\alpha=0.25$. The resonant width is denoted by $\Delta e$,
as shown in the first panel. For convenience, the level curves of the motion
integral $Q=\sqrt{\mu a(1-e^{2})}(\cos{i}-1)$ are also plotted (the width of
resonance should be measured along the level curve of $Q$).

###  5.2 Resonant width and explanation for the curve of $\Delta e$

According to the definition of resonant width, we can see that, the initially
quasi-circular orbits are close to the saddle point (zero-eccentricity point)
in the phase space, meaning that the real trajectories of test particles
should be close to the curve of separatrix, so that the maximum variation of
eccentricity numerically determined in Section 3 can be estimated by the
associated resonant width.

Our main results are reported in Fig. 6, where the location of resonant centre
and the resonant width in terms of the eccentricity variation $\Delta e$ are
reported for four different semimajor axis ratios at
$\alpha=0.1,0.15,0.2,0.25$. It is observed from Fig. 6 that (a) the resonant
width evaluated at a given inclination increases with the semimajor axis
ratio, (b) there is an interrupt for the curve of resonant width, occurring at
inclination of $\sim$$35.8^{\circ}$, and (c) the resonant width, denoted by
$\Delta e$, is first a decreasing function and then an increasing function in
the inclination interval smaller than $\sim$$35.8^{\circ}$, and it is again an
increasing function of inclination when the inclination changes from
$\sim$$35.8^{\circ}$ to $39^{\circ}$ .

Comparing the first panel of Fig. 6 with the curve of $\Delta e$ shown in the
left panel of Fig. 3 (both plots have the same semimajor axis ratio at
$\alpha=0.1$), we can find that the curve of $\Delta e$ is in quite good
agreement with the curve of resonant width in terms of the following points:
(a) in both plots the point of discontinuity happens at inclination
$\sim$$35.8^{\circ}$, which divides the entire interval of inclination into
two branches, and (b) the maximum variation of eccentricity determined in
Section 3 and the resonant width obtained by the analytical method have
similar behaviour as the inclination changing from zero to $39^{\circ}$. Thus,
we could conclude that the shape of $\Delta e$ numerically produced in Section
3 is mainly sculpted by the secular resonance associated with $\sigma=\varpi$.
In other words, the secular dynamics of inner test particles in the low-
eccentricity space with inclinations smaller than $39^{\circ}$ is dominantly
governed by the secular resonance with $\sigma=\varpi$.

###  5.3 Discussions on the dynamics around the critical inclination

Figure 7: Level curves of the resonant Hamiltonian ${\cal H}^{*}$ for the
parameter of minimum inclination $i_{\min}$ near the critical value. In
simulations, the semimajor axis ratio is taken as $\alpha=0.1$. The level
curves separating the libration regions from circulation regions are shown in
red lines. The size of libration island is measured by the variation of
eccentricity, denoted by $\Delta e$.

In the previous subsection, we have found that the curve of resonant width as
a function of inclination has an interrupt at the critical inclination
$\sim$$35.8^{\circ}$. What happens at such a critical inclination? To make
clear this problem, we assume the minim inclination at $35.6^{\circ}$,
$35.7^{\circ}$, $35.8^{\circ}$, $35.9^{\circ}$, $36.0^{\circ}$ and
$36.5^{\circ}$ and produce the associated phase portraits in the $(\sigma,e)$
space. The phase portraits are reported in Fig. 7.

When the minimum inclination $i_{\min}$ is at $35.6^{\circ}$, the dynamical
structures arising in the phase portrait is similar to the ones shown in Figs.
4 and 5 (there is only one island of resonance in the phase space). Similarly,
the resonant width is marked and denoted by $\Delta e$, which can be used to
estimate the maximum variation of eccentricity.

When the minimum inclination $i_{\min}$ increases up to $35.7^{\circ}$,
besides the primary island of libration entered at $\sigma=0^{\circ}$, two
small islands of libration bifurcate and their centres of resonance are at
$\sigma=\pm 180^{\circ}$. In the case of $i_{\min}=35.7^{\circ}$, the dynamics
of test particles initially placed on quasi-circular orbits are dominated by
the primary island of libration, so that the maximum variation of eccentricity
in this region can be estimated by the size of the primary island of
libration.

However, when the minimum inclination $i_{\min}$ is up to $35.8^{\circ}$
(close to the critical inclination), the dynamical structures in the phase
portrait change in terms of the following two aspects: (a) the primary island
of libration centred at $\sigma=0^{\circ}$ moves upward and this island of
libration cannot dominate the dynamics of particles initially placed on quasi-
circular orbits, (b) in the low-eccentricity region there appears a new island
of libration centred at $\sigma=\pm 180^{\circ}$. Furthermore, when the
minimum inclination $i_{\min}$ is increased up to $35.9^{\circ}$, the primary
island of libration disappears and in the low-eccentricity region there is a
new island of libration. We can see that, in both the cases of
$i_{\min}=35.8^{\circ}$ and $i_{\min}=35.9^{\circ}$, the dynamics of particles
on quasi-circular orbits are dominated by the islands of libration arising in
the low-eccentricity region. However, the size of the libration island arising
in the low-eccentricity region is very small. Thus, the maximum variation of
eccentricity at $i_{\min}=35.8^{\circ}$ and $i_{\min}=35.9^{\circ}$ should be
very small due to the small size of the libration islands.

When the minimum inclination $i_{\min}$ grows up to $36.0^{\circ}$ and
$36.5^{\circ}$, the size of the libration islands in the low-eccentricity
region increases, showing that the maximum variation of eccentricity should be
increased.

Based on the aforementioned discussions, we can conclude that the interrupt of
$\Delta e$ occurring at the critical inclination $\sim$$35.8^{\circ}$ is due
to the change of dynamical structures in phase portraits. In particular, when
the minimum inclination $i_{\min}$ becomes greater than the critical
inclination, new island of libration appears in the low-eccentricity region
and it begins to dominate the the dynamics of particles initially placed on
quasi-circular orbits.

##  6 Conclusions

In this work, we investigated the secular resonance associated with the
critical argument of $\sigma=\varpi$ for inner test particles in hierarchical
planetary systems in order to provide a dynamical explanation for the shape of
$\Delta e$ numerically determined in the inclination interval
$(0^{\circ},39^{\circ})$ under the full secular model.

The double-averaged Hamiltonian is formulated up to an arbitrary order in the
semimajor axis ratio $\alpha$ between the test particle and the disturbing
object. Under the full secular model, we reproduced the maximum variation of
eccentricity ($\Delta e$) as a function of initial inclination for those test
particles initially placed on quasi-circular orbits. We concentrated our
attention on the dynamical mechanism hidden in the shape of $\Delta e$.

To understand the secular dynamics, the perturbation theory based on Lie-
series transformation is adopted. By taking advantage of the Hori–Deprit
method, the ‘short-period’ terms in the double-averaged Hamiltonian are
removed, and the resulting resonant model reduces to a single-degree of
freedom system with the resonant angle $\sigma$ as the angular coordinate.
Based on the resonant Hamiltonian, the phase-space structures, resonant
centres, and resonant widths are identified analytically. Results show that
both the curves of resonant centre and resonant width have an interrupt at
inclination of $\sim$$35.8^{\circ}$, which is due to the change of dynamical
structures in phase portraits. In particular, we found a perfect
correspondence between the resonant width in terms of the eccentricity
variation and the maximum variation of eccentricity $\Delta e$ numerically
determined under the full secular model. It is concluded that the secular
dynamics of test particles in the interested region is dominated by the
secular resonance considered in this work.

## Acknowledgments

The author thanks Drs Cristian Beaugé and Xiyun Hou for helpful discussions
and thanks the anonymous reviewer for his/her comments. This work is
financially supported by the National Natural Science Foundation of China
(Nos. 12073011) and the National Key Research and Development Program of China
(No.2019YFA0706601).

## Data availability

The analysis and codes are available upon request.

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# Flows of 3-edge-colorable cubic signed graphs

Liangchen Li, Chong Li, Rong Luo, Cun-Quan Zhang and Hailiang Zhang  School of
Mathematical Sciences, Luoyang Normal University, Luoyang 471934, China.
Email<EMAIL_ADDRESS>Partially supported by NSFC (No. 11301254), Basic
Research Foundation of Henan Educational Committee (No. 20ZX004), and China
Scholarship Council.Department of Mathematics, West Virginia University,
Morgantown, WV 26505, USA. Email<EMAIL_ADDRESS>of Mathematics,
West Virginia University, Morgantown, WV 26505, USA. Email<EMAIL_ADDRESS>Partially supported by a grant from Simons Foundation (No. 839830)Department
of Mathematics, West Virginia University, Morgantown, WV 26505, USA. Email:
<EMAIL_ADDRESS>Partially supported by an NSF grant
DMS-1700218Department of Mathematics, Taizhou University, Taizhou 317000,
China. Email<EMAIL_ADDRESS>
###### Abstract

Bouchet conjectured in 1983 that every flow-admissible signed graph admits a
nowhere-zero 6-flow which is equivalent to the restriction to cubic signed
graphs. In this paper, we proved that every flow-admissible $3$-edge-colorable
cubic signed graph admits a nowhere-zero $10$-flow. This together with the
4-color theorem implies that every flow-admissible bridgeless planar signed
graph admits a nowhere-zero $10$-flow. As a byproduct, we also show that every
flow-admissible hamiltonian signed graph admits a nowhere-zero $8$-flow.

Keywords: Signed graph; nowhere-zero flow; 3-edge-colorable; hamiltonian
circuit.

##  1 Introduction

In 1983, Bouchet [2] proposed a flow conjecture that every flow-admissible
signed graph admits a nowhere-zero $6$-flow. Bouchet [2] himself proved that
such signed graphs admit nowhere-zero $216$-flows and Zýka [13] further
reduced to $30$-flows. Recently DeVos et al. [5] further proved that flow-
admissible signed graphs admit nowhere-zero $11$-flows.

Similar to ordinary graphs, Bouchet’s conjecture is equivalent to the
restriction to cubic signed graphs: every flow-admissible cubic signed graph
admits a nowhere-zero $6$-flow.

Schubert and Steffen [8] verified Bouchet’s Conjecture for Kotzig graphs.
Máčajová and Škoviera [6] characterized cubic signed graphs that admit a
nowhere-zero $3$-flow and that admit a nowhere-zero $4$-flow, respectively. In
this paper, we investigate integer flows in $3$-edge-colorable cubic signed
graphs and prove the following theorem.

######  Theorem 1.1.

Every flow-admissible $3$-edge-colorable cubic signed graph admits a nowhere-
zero $10$-flow.

By the $4$-color theorem, every bridgeless cubic planar graph is $3$-edge-
colorable. Therefore we have the following corollary for bridgeless signed
planar graphs.

######  Corollary 1.2.

Every flow-admissible bridgeless planar signed graph admits a nowhere-zero
$10$-flow.

Theorem 1.1 follows from the following stronger result which shows that every
connected flow-admissible $3$-edge-colorable cubic signed graph admits a
nowhere-zero $8$-flow except one case which has a nowhere-zero $10$-flow.

######  Theorem 1.3.

Let $(G,\sigma)$ be a connected $3$-edge-colorable cubic signed graph and
$E_{N}(G,\sigma)$ be the set of negative edges in $(G,\sigma)$. Let $R,B,Y$ be
the three color classes such that $|R\cap E_{N}(G,\sigma)|\equiv|B\cap
E_{N}(G,\sigma)|\pmod{2}$. If $(G,\sigma)$ is flow-admissible, then it has a
nowhere-zero $8$-flow unless $R\cup B$ contains no unbalanced circuits and the
numbers of unbalanced circuits in $R\cup Y$ and $B\cup Y$ are both odd and at
least $3$, in which case it has a nowhere-zero $10$-flow.

As a byproduct, we also prove the following $8$-flow theorem for hamiltonian
signed graphs.

######  Theorem 1.4.

If $(G,\sigma)$ is a flow-admissible hamiltonian signed graph, then
$(G,\sigma)$ admits a nowhere-zero 8-flow.

The organization of the rest of the paper is as follows: Basic notation and
terminology will be introduced in Section 2; some lemmas needed for the proofs
of the main results will be presented in Section 3; the proofs of Theorems 1.3
and Corollary 1.2 and the proof of Theorem 1.4 will be completed in Sections 4
and 5, respectively.

##  2 Notation and Terminology

Graphs considered in this paper are finite and may have multiple edges or
loops. For terminology and notations not defined here we follow [1, 3, 9].

Let $G$ be a graph. A leaf vertex is a vertex of degree $1$. For a vertex
subset $U\subseteq V(G)$, denote by $\delta_{G}(U)$, the set of edges with one
endvertex in $U$ and the other in $V(G)-U$. A path is nontrivial if it
contains at least two vertices. Let $u,v$ be two vertices in $V(G)$. A
$(u,v)$-path is a path with $u$ and $v$ as its endvertices. Let $C=v_{1}\cdots
v_{r}v_{1}$ be a circuit where $v_{1},v_{2},\dots,v_{r}$ appear in clockwise
on $C$. A segment of $C$ is the path $v_{i}v_{i+1}\cdots v_{j-1}v_{j}$
contained in $C$ and is denoted by $v_{i}Cv_{j}$, where the indices are taken
modulo $r$.

A signed graph $(G,\sigma)$ is a graph $G$ together with a signature
$\sigma\colon\ E(G)\to\\{\pm 1\\}$. An edge $e$ is positive if $\sigma(e)=1$
and negative if $\sigma(e)=-1$. Denote by $E_{N}(G,\sigma)$ the set of
negative edges in $(G,\sigma)$. Let $e=uv$ be an edge. By contracting $e$, we
mean to first identify $u$ with $v$ and then to delete the loop if
$\sigma(e)=1$ otherwise to keep the negative loop. An all-positive signed
graph is a signed graph without negative edges, which is also called an
ordinary graph. A circuit is _balanced_ if it contains an even number of
negative edges and is _unbalanced_ otherwise. A signed graph is called
balanced if it contains no unbalanced circuit. A signed graph is unbalanced if
it is not balanced. A signed circuit is defined as a signed graph of any of
the following three types:

(1) a balanced circuit;

(2) a short barbell, that is, the union of two unbalanced circuits that meet
at a single vertex;

(3) a long barbell, that is, the union of two vertex-disjoint unbalanced
circuits with a nontrivial path that meets the circuits only at its
endvertices.

We regard an edge $e=uv$ of a signed graph as two half edges $h_{e}^{u}$ and
$h_{e}^{v}$, where $h_{e}^{u}$ is incident with $u$ and $h_{e}^{v}$ is
incident with $v$. Let $H_{G}(v)$ (or simply $H(v)$ if no confusion occurs) be
the set of all half edges incident with $v$, and $H(G)$ be the set of all half
edges of $(G,\sigma)$. An orientation of $(G,\sigma)$ is a mapping
$\tau:H(G)\rightarrow\\{+1,-1\\}$ such that for each $e=uv\in E(G)$,
$\tau(h_{e}^{u})\tau(h_{e}^{v})=-\sigma(e)$. For $h_{e}^{u}\in H(G)$,
$h_{e}^{u}$ is oriented away from $u$ if $\tau(h_{e}^{u})=1$ and $h_{e}^{u}$
is oriented toward $u$ if $\tau(h_{e}^{u})=-1$. A signed graph $(G,\sigma)$
together with an orientation $\tau$ is called an oriented signed graph,
denoted by $(G,\tau)$.

######  Definition 2.1.

Let $(G,\sigma)$ be a signed graph and $\tau$ be an orientation of
$(G,\sigma)$. Let $k\geq 2$ be an integer and
$f:E(G)\to\operatorname{\mathbb{Z}}$ be a mapping such that $|f(e)|\leq k-1$.

  * (1)

For each vertex $v$, the boundary of $(\tau,f)$ at $v$ is
$\partial(\tau,f)(v)=\sum_{h\in H(v)}\tau(h)f(e_{h})$.

  * (2)

The support of $f$, denoted by $\operatorname{{\rm supp}}(f)$, is the set of
edges $e$ with $|f(e)|>0$.

  * (3)

If $\partial(\tau,f)(v)=0$ for each vertex $v$, then $(\tau,f)$ is called a
$k$-flow of $(G,\sigma)$. A $k$-flow $(\tau,f)$ is said to be nowhere-zero if
$\operatorname{{\rm supp}}(f)=E(G)$.

  * (4)

If $\partial(\tau,f)(v)\equiv 0\pmod{k}$ for each vertex $v$, then $(\tau,f)$
is called a $\operatorname{\mathbb{Z}}_{k}$-flow of $(G,\sigma)$. A
$\operatorname{\mathbb{Z}}_{k}$-flow $(\tau,f)$ is said to be nowhere-zero if
$\operatorname{{\rm supp}}(f)=E(G)$.

For a mapping $f:E(G)\to{\mathbb{Z}}$, denote $E_{f=\pm i}=\\{e\in
E(G):|f(e)|=i\\}.$

For convenience, we shorten the notation of nowhere-zero $k$-flow and nowhere-
zero $\operatorname{\mathbb{Z}}_{k}$-flow as $k$-NZF and
$\operatorname{\mathbb{Z}}_{k}$-NZF, respectively. If the orientation is
understood from the context, we use $f$ instead of $(\tau,f)$ to denote a
flow.

A signed graph is flow-admissible if it admits a nowhere-zero $k$-flow for
some integer $k$. In a signed graph, switching a vertex $u$ means reversing
the signs of all edges incident with $u$. Two signed graphs are equivalent if
one can be obtained from the other by a sequence of switches. Note that a
signed graph is balanced if and only if it is equivalent to an all-positive
graph. Note that switching a vertex does not change the parity of the number
of negative edges in a circuit and although technically it changes the flows,
it only reverses the directions of the half edges incident with the vertex,
but the directions of other half edges and the flow values of all edges remain
the same. Bouchet [2] gave a characterization for flow-admissible signed
graphs.

######  Lemma 2.2.

(Bouchet [2]) A connected signed graph $(G,\sigma)$ is flow-admissible if and
only if it is not equivalent to a signed graph with exactly one negative edge
and it has no bridge $b$ such that $(G-b,\sigma|_{G-b})$ has a balanced
component.

##  3 Lemmas

Given a signed graph $(G,\sigma)$, let $H$ be a signed subgraph of
$(G,\sigma)$ and $C$ be a balanced circuit. Define the following operation:

$\Phi_{2}\mbox{-operation}:\mbox{ add a balanced circuit $C$ into $H$ if
$|E(C)-E(H)|\leq 2$.}$

We use $\langle H\rangle_{2}$ to denote the maximal subgraph of $G$ obtained
from $H$ via $\Phi_{2}$-operations. Zýka [13] proved the following result.

######  Lemma 3.1.

(Zýka [13]) Let $(G,\sigma)$ be a signed graph and $H$ be a subgraph of $G$.
If $\langle H\rangle_{2}=G$, then $(G,\sigma)$ admits a
$\operatorname{\mathbb{Z}}_{3}$-flow $\phi$ such that
$E(G)-E(H)\subseteq\operatorname{{\rm supp}}(\phi)$.

It is clear that a signed graph admits a $\operatorname{\mathbb{Z}}_{2}$-NZF
if and only if each component of $(G,\sigma)$ is eulerian. The next lemma
gives a characterization of signed graphs admitting a $2$-NZF.

######  Lemma 3.2.

(Xu and Zhang [11]) A signed graph $(G,\sigma)$ admits a $2$-NZF if and only
if each component of $(G,\sigma)$ is eulerian and has an even number of
negative edges.

The next two lemmas show how to convert a modulo flow to an integer-valued
flow.

######  Lemma 3.3.

(Cheng et al. [4]) Let $(G,\sigma)$ be a bridgeless signed graph. If
$(G,\sigma)$ admits a $\operatorname{\mathbb{Z}}_{3}$-flow $f_{1}$, then it
admits a $4$-flow $f_{2}$ such that $\operatorname{{\rm supp}}(f_{1})\subseteq
E_{f_{2}=\pm 1}\cup E_{f_{2}=\pm 2}$.

######  Definition 3.4.

Let $f$ be a $\operatorname{\mathbb{Z}}_{2}$-flow of $(G,\sigma)$. Then
$\operatorname{{\rm supp}}(f)$ consists of vertex-disjoint eulerian subgraphs.
A component of $\operatorname{{\rm supp}}(f)$ is called even if it contains an
even number of negative edges; otherwise it is called odd.

######  Lemma 3.5.

(Cheng et al. [4]) Let $(G,\sigma)$ be a connected signed graph. If
$(G,\sigma)$ admits a $\operatorname{\mathbb{Z}}_{2}$-flow $f_{1}$ such that
$\operatorname{{\rm supp}}(f_{1})$ contains an even number of odd components,
then it admits a $3$-flow $f_{2}$ such that $\operatorname{{\rm
supp}}(f_{1})=E_{f_{2}=\pm 1}$ and $\operatorname{{\rm
supp}}(f_{2})/\operatorname{{\rm supp}}(f_{1})$ is acyclic.

Lemma 3.5 can be extended to the case when the support of a
$\operatorname{\mathbb{Z}}_{2}$-flow contains an odd number of odd components
in the following lemma.

######  Lemma 3.6.

Let $(G,\sigma)$ be a connected signed graph. If $(G,\sigma)$ admits a
$\operatorname{\mathbb{Z}}_{2}$-flow $f_{1}$ such that the number of odd
components of $\operatorname{{\rm supp}}(f_{1})$ is odd and is at least three,
then $(G,\sigma)$ has a $5$-flow $f_{2}$ satisfying

(1) $\operatorname{{\rm supp}}(f_{2})/\operatorname{{\rm supp}}(f_{1})$ is
acyclic;

(2) $\operatorname{{\rm supp}}(f_{1})\subseteq\\{e\in E(G)\colon\
1\leq|f_{2}(e)|\leq 3\\}$ and $|f_{2}(e)|\in\\{1,2\\}$ for each negative loop
$e\in\operatorname{{\rm supp}}(f_{1})$.

###### Proof.

Let $(G,\sigma)$ together with a $\operatorname{\mathbb{Z}}_{2}$-flow
$(\tau,f_{1})$ be a counterexample to Lemma 3.6 such that $|E(G)|$ is
minimized. In the following, we always assume the flows are under the
orientation $\tau$ or its restriction on according subgraphs.

Denote by $\mathcal{B}$ the set of components of $\operatorname{{\rm
supp}}(f_{1})$ and let $H=G/\operatorname{{\rm supp}}(f_{1})$. Thus $V(H)$ can
be partitioned into three parts: $X,Y$ and $W$ where $X$ and $Y$ are the sets
of vertices corresponding to even and odd components in $\mathcal{B}$
respectively and $W$ is corresponding to the vertices which are also the
vertices in $V(G)$. For $u\in X\cup Y$, let $B_{u}$ denote the corresponding
component in $\mathcal{B}$.

######  Claim 3.6.1.

$G$ contains no leaf vertices and $H$ is a tree.

###### Proof.

If $G$ contains a leaf vertex, say $x$, then $f_{1}(e)=0$ where $e$ is the
edge incident with $x$ and $G-x$ remains connected. This contradicts to the
minimality of $G$.

Clearly $H$ is connected since $G$ is connected. If $H$ is not a tree, then
there is an edge $e\in E(G)$ such that $f_{1}(e)=0$ and $G-e$ is connected, a
contradiction to the minimality of $G$ again. ∎

Let $u$ be a leaf vertex of $H$ and $v$ be its neighbor. By Claim 3.6.1, $u\in
X\cup Y$. Since $u$ is a leaf vertex of $H$, there is only one edge in $G$
with one endvertex in $B_{u}$ and the other one in $B_{v}$. Let $x_{u}x_{v}$
be the only edge in $G$ where $x_{u}\in V(B_{u})$ and $x_{v}\in V(B_{v})$.

######  Claim 3.6.2.

$u\in Y$ and $v\not\in Y$.

###### Proof.

Suppose to the contrary that either $u\in X$ or $u\in Y$ and $v\in Y$. Let
$G^{\prime}=G-V(B_{u})$. Since $B_{u}$ is a leaf block, $G^{\prime}$ is
connected.

If $u\in X$, then $B_{u}$ is an even component and thus $\mathcal{B}-B_{u}$
and $\mathcal{B}$ have the same number of odd components. Since $G^{\prime}$
is connected, by the minimality of $G$, there is an integer $5$-flow $g_{1}$
of $(G^{\prime},\sigma|_{E(G^{\prime})})$ such that $\operatorname{{\rm
supp}}(f_{1})-E(B_{u})\subseteq\operatorname{{\rm supp}}(g_{1})$ and $g_{1}$
satisfies (1) and (2). Then $g_{1}$ can be considered as a flow of
$(G,\sigma)$ under the orientation $\tau$ such that
$E(B_{u})\cap\operatorname{{\rm supp}}(g_{1})=\emptyset$. Since $B_{u}$ is an
even eulerian component, by Lemma 3.2, there is a $2$-flow $g_{2}$ of
$(G,\sigma)$ such that $\operatorname{{\rm supp}}(g_{2})=E(B_{u})$. Therefore
$g_{1}+g_{2}$ is a $5$-flow of $(G,\sigma)$ satisfying (1) and (2), a
contradiction.

Now assume that $u\in Y$ and $v\in Y$. Then $\mathcal{B}-B_{u}$ has an even
number of odd components. By Lemma 3.5, there is a $3$-flow $g_{3}$ such that
$\operatorname{{\rm supp}}(f_{1})-E(B_{u})\subseteq\operatorname{{\rm
supp}}(g_{3})$ and $g_{3}$ satisfies (1) and (2). Note that
$g_{3}(x_{u}x_{v})=0$ and $g_{3}(e)=0$ for each $e\in E(B_{u})$. By Lemma 3.5
again, there is a $3$-flow $g_{4}$ such that $\operatorname{{\rm
supp}}(g_{4})=E(B_{u})\cup E(B_{v})+x_{u}x_{v}$, $E(B_{u})\cup
E(B_{v})=E_{g_{4}=\pm 1}$, and $\\{x_{u}x_{v}\\}=E_{g_{4}=\pm 2}$. Therefore
$g_{3}+2g_{4}$ is a desired $5$-flow, a contradiction. This proves the claim.
∎

Let $(G_{1},\sigma_{1})$ be the signed graph obtained from $G-V(B_{u})$ by
adding a negative loop $e_{1}$ at $x_{v}$ where $\sigma_{1}$ is defined as
$\sigma_{1}(e)=\sigma(e)$ for each $e\in E(G_{1})-\\{e_{1}\\}$ and
$\sigma_{1}(e_{1})=-1$. The orientation $\tau_{1}$ of $(G_{1},\sigma_{1})$ is
defined as $\tau_{1}(h)=\tau(h)$ for each $h\in H(G_{1})$ and $h$ is not an
half edge of the loop $e_{1}$; for each half edge $h$ of $e_{1}$,
$\tau_{1}(h)=\tau(h_{uv}^{v})$.

Let $(G_{2},\sigma_{2})$ be the signed graph obtained from $B_{u}$ by adding a
negative loop $e_{2}$ at $x_{u}$. Its signature $\sigma_{2}$ and orientation
$\tau_{2}$ are defined similarly to $\sigma_{1}$ and $\tau_{1}$, respectively.

Denote $B_{v}^{\prime}=B_{v}\cup\\{e_{1}\\}$ and
$\mathcal{B^{\prime}}=\mathcal{B}-B_{u}-B_{v}+B_{v}^{\prime}$ if $v\in X\cup
Y$; otherwise denote $B_{v}^{\prime}=\\{e_{1}\\}$ and
$\mathcal{B^{\prime}}=\mathcal{B}-B_{u}+B_{v}^{\prime}$. Note that there is a
$\operatorname{\mathbb{Z}}_{2}$-flow of $(G_{1},\sigma_{1})$ whose support is
$\bigcup_{B\in\mathcal{B^{\prime}}}E(B)$.

By Claim 3.6.2, both $B_{u}$ and $B_{v}^{\prime}$ are odd. Thus
$\mathcal{B^{\prime}}$ and $\mathcal{B}$ have the same number of odd
components. By the minimality of $G$, there is a $5$-flow $(\tau_{1},g_{5})$
of $(G_{1},\sigma_{1})$ satisfying (1) and (2).

By Claim 3.6.2, $(G_{2},\sigma_{2})$ is a signed eulerian graph with even
number of negative edges. By Lemma 3.2, there is a $2$-flow $(\tau_{2},g_{6})$
of $(G_{2},\sigma_{2})$ such that $\operatorname{{\rm supp}}(g_{6})=E(G_{2})$.
We may assume $g_{5}(e_{1})g_{6}(e_{2})>0$ otherwise replacing $g_{6}$ with
$-g_{6}$. Let $a=g_{5}(e_{1})$. Then $|a|\in\\{1,2\\}$.

Let $(\tau,g_{7})$ be the integer flow of $(G,\sigma)$ defined as follows: for
each $e\in E(G)$,

$g_{7}(e)=\left\\{\begin{array}[]{ll}g_{5}(e)&\mbox{ if
}e\in\operatorname{{\rm supp}}(g_{5});\\\ ag_{6}(e)&\mbox{ if
}e\in\operatorname{{\rm supp}}(g_{6});\\\ 2a&\mbox{ if }e=uv;\\\ 0&\mbox{
otherwise. }\end{array}\right.$

Then $g_{7}$ is a $5$-flow of $(G,\sigma)$ satisfying (1) and (2), a
contradiction. This completes the proof of the lemma. ∎

The following lemma is due to Zaslavsky [10].

######  Lemma 3.7.

(Zaslavsky [10]) Let $T$ be a spanning tree of a signed graph $(G,\sigma)$.
For every $e\notin E(T)$, let $C_{e}$ be the unique circuit contained in
$T+e$. If the circuit $C_{e}$ is balanced for every $e\notin E(T)$, then $G$
is balanced.

The proof of the following lemma is inspired by the proof of Theorem 4.2 in
[7] due to Máčajová and Škoviera.

######  Lemma 3.8.

Let $C$ be an unbalanced circuit of a signed graph $(G,\sigma)$. If
$(G,\sigma)$ is flow-admissible and $G-E(C)$ is balanced, then $(G,\sigma)$
has a $4$-flow $f$ satisfying the following:

(1) $E(C)\subseteq\operatorname{{\rm supp}}(f)$;

(2) In $H=G[\operatorname{{\rm supp}}(f)]$ the subgraph induced by
$\operatorname{{\rm supp}}(f)$, each vertex in $V(H)-V(C)$ has degree at most
$3$ in $H$ and at most one vertex in $V(H)-V(C)$ has degree $3$.

###### Proof.

Denote by $G^{\prime}=G-E(C)$. Since $G^{\prime}$ is balanced, with some
switching operations, we may assume that all edges in $E(G^{\prime})$ are
positive and thus $E_{N}(G,\sigma)\subseteq E(C)$. Fix an orientation $\tau$
of $(G,\sigma)$ and in the following we always assume the flows are under the
orientation $\tau$ or its restriction on according subgraphs.

Let $M$ be a component of $G^{\prime}$. The circuit $C$ is divided by the
vertices of $M$ into segments whose endvertices lie in $M$ and all inner
vertices lie outside $M$. An endvertex of a segment is called an _attachment_
of $M$. A segment is called positive (negative) if it contains an even (odd)
number of negative edges. Let $S$ be a segment. Note that $M\cup S$ is
unbalanced (balanced) if and only if the segment $S$ is negative (positive).
Since $C$ is unbalanced, the number of negative segments determined by each
component $M$ is odd.

We prove the lemma by contradiction. Suppose to the contrary that $(G,\sigma)$
has no $4$-flow satisfying (1) and (2).

######  Claim 3.8.1.

Each component of $G^{\prime}$ determines exactly one negative segment.

###### Proof.

Suppose to the contrary that $M$ determines more than one negative segments.
Thus $M$ determines at least three negative segments. Let
$u_{1}Cu_{1}^{\prime}$, $u_{2}Cu_{2}^{\prime}$, $u_{3}Cu_{3}^{\prime}$ be
three consecutive negative segments (in clockwise) where $u_{i}$ and
$u_{i}^{\prime}$ are attachments for $i=1,2,3$. Then $u_{1}^{\prime}Cu_{2}$,
$u_{2}^{\prime}Cu_{3}$, $u_{3}^{\prime}Cu_{1}$ all contain even number of
negative edges. This implies that $C$ can be partitioned into three negative
segments: $u_{1}Cu_{2}$, $u_{2}Cu_{3}$, and $u_{3}Cu_{1}$.

We first show that no $(u_{1},u_{2})$-path in $M$ passes through $u_{3}$.
Otherwise let $P$ be a $(u_{1},u_{2})$-path in $M$ that passes through
$u_{3}$. Then $C_{1}=u_{1}Cu_{3}+u_{1}Pu_{3}$ and
$C_{2}=u_{3}Cu_{2}+u_{3}Pu_{2}$ both are balanced circuits. By Lemma 3.2,
there is a $2$-flow $f_{i}$ of $(G,\sigma)$ such that $\operatorname{{\rm
supp}}(f_{i})=E(C_{i})$ for each $i=1,2$. Therefore $2f_{1}+f_{2}$ is a
$4$-flow of $(G,\sigma)$ and $\operatorname{{\rm supp}}(2f_{1}+f_{2})=E(C)\cup
E(P)$, which is a desired $4$-flow, a contradiction.

By symmetry, no $(u_{i},u_{j})$-path passes through $u_{k}$ where
$\\{i,j,k\\}=\\{1,2,3\\}$. This implies that $u_{1}$ and $u_{2}$ are not
adjacent. Otherwise, a $(u_{1},u_{3})$-path together with $u_{1}u_{2}$ gives a
$(u_{2},u_{3})$-path containing $u_{1}$.

Let $P_{1}$ be a $(u_{1},u_{2})$-path. Since $M$ is connected, there is a path
$P_{2}$ from $u_{3}$ to $P_{1}$ such that $|V(P_{2})\cap V(P_{1})|=1$. Let $v$
be the only common vertex in $P_{1}$ and $P_{2}$. Then $C,P_{1}$, and $P_{2}$
form a signed graph as illustrated in Figure 1 which has a desired $4$-NZF, a
contradiction again. This completes the proof of the claim. ∎

$v$$u_{1}$$u_{2}$$u_{3}$$3$$2$$1$$3$$1$$2$ Figure 1: a 4-flow covers $C$

Let $\mathcal{M}$ denote the set of all components of $G^{\prime}$. For each
component $M$, denote by $S_{M}=uCv$ the negative segment determined by $M$
where $u$ and $v$ are two attachments of $M$ on $C$. Denote by
$S_{M}^{\prime}=vCu$ the cosegment of $S_{M}$. Then $E(S_{M})\not=\emptyset$
and $E(S_{M}^{\prime})=E(C)-E(S_{M})$.

######  Claim 3.8.2.

$\bigcap_{M\in\mathcal{M}}E(S_{M})=\emptyset$. Therefore
$\bigcup_{M\in\mathcal{M}}E(S^{\prime}_{M})=C$ and $|\mathcal{M}|\geq 2$.

###### Proof.

Suppose to the contrary $\bigcap_{M\in\mathcal{M}}E(S_{M})\not=\emptyset$. Let
$e^{*}\in\bigcap_{M\in\mathcal{M}}E(S_{M})$. Then there is a spanning tree $T$
of $G-e^{*}$ containing the path $P^{*}=C-e^{*}$. Let $e=uv\in
E(G)-e^{*}-E(T)$. Denote the unique circuit contained in $T+e$ by $C_{e}$.

If $E(C_{e})\cap E(P^{*})=\emptyset$, then $C_{e}$ contains no negative edges
and thus is balanced.

Assume that $C_{e}$ and $P^{*}$ have common edges. Since $T$ contains all the
edges in $C-e^{*}$, $E(C_{e})\cap E(C)$ is a path $P$ on $C$. Let $u^{\prime}$
and $v^{\prime}$ be the two endvertices of $P$ in clockwise order on $C$. Then
$C_{e}[(V(C_{e})-V(P))\cup\\{u^{\prime},v^{\prime}\\}]$ is a also a path and
thus it is contained in some component $M\in\mathcal{M}$. This implies that
$u^{\prime}$ and $v^{\prime}$ are two attachments of $M$ on $C$. Since $e^{*}$
belongs to the only negative segment of $C$ determined by $M$,
$u^{\prime}Cv^{\prime}$ is the union of some positive segments of $C$
determined by $M$. Therefore $C_{e}$ has an even number of negative edges and
thus is balanced. By Lemma 3.7, $G-e^{*}$ is balanced, contradicting Lemma
2.2. This proves $\bigcap_{M\in\mathcal{M}}E(S_{M})=\emptyset$.

Since $E(S_{M}^{\prime})=E(C)-E(S_{M})$ and
$\bigcap_{M\in\mathcal{M}}E(S_{M})=\emptyset$, we have
$\bigcup_{M\in\mathcal{M}}E(S_{M}^{\prime})=C$.

Since $E(S_{M})\not=\emptyset$ and
$\bigcap_{M\in\mathcal{M}}E(S_{M})=\emptyset$, we have $|\mathcal{M}|\geq 2$.
∎

Let $\mathcal{S}=\\{S^{\prime}_{1},S^{\prime}_{2},\ldots,S^{\prime}_{t}\\}$ be
a minimal cosegment cover of $C$. Then $S_{i}^{\prime}\not\subseteq
S_{j}^{\prime}$ for any $i,j$.

######  Claim 3.8.3.

(i) For each pair $i,j\in\\{1,2,\ldots,t\\}$, either $S^{\prime}_{i}\cap
S^{\prime}_{j}$ consists of some nontrivial paths or $S_{i}^{\prime}$ and
$S_{j}^{\prime}$ are vertex-disjoint;

(ii) Each edge $e\in E(C)$ is contained in at most two cosegments.

###### Proof.

(i) Note that for any two segments $S_{i}$ and $S_{j}$, their endvertices
belong to two vertex-disjoint components $M_{i}$ and $M_{j}$. Thus no
component of $S_{i}^{\prime}\cap S^{\prime}_{j}$ is an isolated vertex. This
proves (i).

(ii) Suppose to the contrary that there is an edge $e=uv$ that belongs to
three cosegments, say $S_{1}^{\prime}$, $S_{2}^{\prime}$, $S_{3}^{\prime}$.
Let $S_{i}^{\prime}=u_{i}Cv_{i}$ for each $i\in\\{1,2,3\\}$. Without loss of
generality, we may assume that $u_{1},u_{2},u_{3},u,v$ appear in this
clockwise cyclic order. Then there exists a pair $i,j$ such that $u_{i}Cv_{j}$
contains all the $u_{l},v_{l}^{\prime}s$ ($l\in\\{1,2,3\\}$) and $u,v$. Hence,
there is a $k\in\\{1,2,3\\}-\\{i,j\\}$ such that $u_{k}$ and $v_{k}$ are
properly included in $u_{i}Cv_{j}$. In this case, either
$\mathcal{S}\setminus\\{S_{k}^{\prime}\\}$ is still a cover of $C$ or
$S_{k}^{\prime}\cup S_{i}^{\prime}=C$, both in contradiction to the minimality
of $\mathcal{S}$. This completes the proof of the claim. ∎

The final step. For each $i=1,\dots,t$, denote by $S_{i}^{\prime}=x_{i}Cy_{i}$
and let $P_{i}$ be a path in $M_{i}$ connecting $x_{i}$ and $y_{i}$. Then
$C_{i}=S_{i}^{\prime}\cup P_{i}$ is a balanced eulerian subgraph. By Claim
3.8.3, we may assume that the vertices
$x_{1},y_{t},x_{2},y_{1},\dots,x_{t},y_{t-1},x_{1}$ appear on $C$ in the
acyclic order. Then $C_{i}\cap C_{j}\not=\emptyset$ if and only if
$|j-i|\equiv 1\pmod{t}$. Moreover $C_{i}\cap C_{i+1}=x_{i+1}Cy_{i}$ where the
subindices are taken modulo $t$. See Figure 2 for an illustration with $t=5$.

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$(a) a minimal cosegment cover with
$t=5$$y_{1}$$y_{2}$$y_{3}$$y_{4}$$y_{5}$$\Longrightarrow$$1$$1$$1$$1$$C_{5}$(b)
2-flow $f_{5}$ of
$C_{5}$$y_{5}$$x_{5}$$x_{1}$$y_{4}$$y_{2}$$x_{2}$$y_{5}$$x_{4}$$y_{4}$$x_{3}$$y_{1}$$x_{1}$$y_{3}$$x_{5}$$C_{2}$$C_{1}$$C_{3}$$C_{4}$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1+1$$1+1$$1+1$(c)
2-flows $f_{1},f_{2},f_{3},f_{4}$ Figure 2: Minimum cosegment cover and
$4$-flow

For each $i\in\\{1,2,\dots,t\\}$, let $(\tau,f_{i})$ be a $2$-flow of
$(G,\sigma)$ such that $\operatorname{{\rm supp}}(f_{i})=E(C_{i})$. We may
assume that for each $i=1,\dots,t-1$, $f_{i}(e)=f_{i+1}(e)$ for each $e\in
E(C_{i})\cap E(C_{i+1})$. Then $\phi=\sum_{i=1}^{t-1}f_{i}+2f_{t}$ is a
$4$-flow of $(G,\sigma)$ satisfying $E(C)\subseteq\operatorname{{\rm
supp}}(\phi)=E(C)\cup[\cup_{i=1}^{t}E(P_{i})]$. Since $P_{1},\dots,P_{t}$
belong to different components of $G^{\prime}$, they are pairwise vertex-
disjoint. Thus for each vertex $v\in V(\operatorname{{\rm supp}}(\phi))-V(C)$,
the degree of $v$ in $\operatorname{{\rm supp}}(\phi)$ is two. Therefore
$\phi$ is a $4$-flow satisfying (1) and (2), a contradiction to the assumption
that $(G,\sigma)$ is a counterexample. This contradiction completes the proof
of the lemma. ∎

The proof of the following lemma is straightforward and thus is omitted.

######  Lemma 3.9.

Let $(G,\sigma)$ be a signed graph and $C$ be a chordless circuit whose edges
are all positive. Suppose that $2\leq|\delta(V(C))|\leq 3$ and $k\geq 4$ is an
integer. If $(G/C,\sigma)$ has a $k$-NZF $f$, then $f$ can be extended to be a
$k$-NZF of $(G,\sigma)$.

##  4 Proofs of Theorem 1.3 and Corollary 1.2

Let’s first recall Theorem 1.3.

Theorem 1.3 Let $(G,\sigma)$ be a connected $3$-edge-colorable cubic signed
graph and $E_{N}(G,\sigma)$ be the set of negative edges in $(G,\sigma)$. Let
$R,B,Y$ be the three color classes such that $|R\cap
E_{N}(G,\sigma)|\equiv|B\cap E_{N}(G,\sigma)|\pmod{2}$. If $(G,\sigma)$ is
flow-admissible, then it has a nowhere-zero $8$-flow unless $R\cup B$ contains
no unbalanced circuits and the numbers of unbalanced circuits in $R\cup Y$ and
$B\cup Y$ are both odd and at least $3$, in which case it has a nowhere-zero
$10$-flow.

###### Proof.

Let $\tau$ be an orientation of $(G,\sigma)$. In the following we always
assume the flows are under the orientation $\tau$ or its restriction on
according subgraphs. Denote by $M_{1}M_{2}$ the $2$-factor induced by
$M_{1}\cup M_{2}$ for each pair $M_{1},M_{2}\in\\{R,B,Y\\}$. Since $|R\cap
E_{N}(G,\sigma)|\equiv|B\cap E_{N}(G,\sigma)|\pmod{2}$, $RB$ has an even
number of odd components.

Case 1. $RB$ contains an unbalanced circuit.

Then by Lemma 3.5, $(G,\sigma)$ has a $3$-flow $(\tau,f_{1})$ such that
$RB=E_{f_{1}=\pm 1}$ and $|f_{1}(e)|=2$ only if $e\in Y$.

Subcase 1.1. $|Y\cap E_{N}(G,\sigma)|\equiv|R\cap E_{N}(G,\sigma)|\equiv|B\cap
E_{N}(G,\sigma)|\pmod{2}$.

Then $RY$ has an even number of unbalanced circuits. By Lemma 3.5 again,
$(G,\sigma)$ has a $3$-flow $(\tau f_{2})$ such that $RY=E_{f_{2}=\pm 1}$ and
$|f_{2}(e)|=2$ only if $e\in B$.

Then $f=f_{1}+3f_{2}$ is a $9$-NZF of $(G,\sigma)$. Since $E_{f_{2}=\pm 2}\cap
E_{f_{1}=\pm 2}=\emptyset$, $|f(e)|\not=8$. Thus $f$ is indeed an $8$-NZF of
$(G,\sigma)$.

Subcase 1.2. $RY$ or $BY$ has an odd number of unbalanced circuits.

In this case, both $RY$ and $BY$ have an odd number of unbalanced circuits.

Let $C$ be an unbalanced circuit in $RB$. Let $R^{\prime}=R\bigtriangleup C$
and $B^{\prime}=B\bigtriangleup C$ with $R$ and $B$, respectively (this is
equivalent to swap colors $R$ and $B$ on $C$). This implies that $|Y\cap
E_{N}(G,\sigma)|\equiv|R^{\prime}\cap E_{N}(G,\sigma)|\equiv|B^{\prime}\cap
E_{N}(G,\sigma)|\pmod{2}$. We are back to Subcase 1.1.

Case 2. $RB$ contains no unbalanced circuit.

Then by Lemma 3.2, $(G,\sigma)$ has a $2$-flow $f_{3}$ such that
$\operatorname{{\rm supp}}(f_{3})=RB$.

Subcase 2.1. The number of unbalanced circuits in $RY$ or $BY$ is even.

Let $f_{2}$ be the $3$-flow in Subcase 1.1. Then $\operatorname{{\rm
supp}}(f_{2})\cup\operatorname{{\rm supp}}(f_{3})=E(G)$. Thus $3f_{3}+f_{2}$
is a $6$-NZF of $(G,\sigma)$.

Subcase 2.2. The number of unbalanced circuits in $RY$ or $BY$ is equal to
one.

By symmetry, assume that $RY$ has exactly one odd component, say $C_{1}$. Let
$\mathcal{C}=\\{C_{1},\dots,C_{t}\\}$ be the set of components of $RY$, where
each $C_{i}$ ($i\geq 2$) is balanced, and, with some switching operations, we
may assume that the edges of each $C_{i}$ ($i\geq 2$) are all positive. Let
$H$ be the signed graph obtained from $(G,\sigma)$ by contracting
$\mathcal{C}-C_{1}$. Then $V(H)$ can be partitioned into $K$ and
$\overline{K}$, where $K=V(C_{1})$ and $\overline{K}$ is the set of vertices
corresponding to the balanced circuits in $\mathcal{C}$. For
$u\in\overline{K}$, denote the corresponding circuit in $\mathcal{C}$ by
$C_{u}$. Since $G$ is flow-admissible, $H$ remains flow-admissible. Note that
$C_{1}$ is an unbalanced circuit in $H$.

We consider the following two cases.

Subcase 2.2.1. $H$ contains an unbalanced circuit $C^{\prime}$ that is edge-
disjoint from $C_{1}$.

Since $G$ is cubic, $C^{\prime}$ is vertex-disjoint from $C_{1}$. Thus there
is a long barbell $Q$ in $H$ with $P$ as the path connecting $C_{1}$ and
$C^{\prime}$. Let $\tau_{1}$ be the orientation of $Q$ which is a restriction
of $\tau$ on $H(Q)$. By Lemma 3.5, let $(\tau_{1},f^{\prime\prime})$ be a
$3$-NZF in $Q$. Since $d_{Q}(u)=2$ or 3 for any $u\in V(Q)-V(C_{1})$, $u$ is
corresponding to an all-positive circuit $C_{u}$ in $(G,\sigma)$ with
$|\delta_{Q}(V(C_{u}))|=2$ or $3$. Hence by Lemma 3.9 we can extend
$f^{\prime\prime}$ to a 4-flow $f^{\prime}$ $(G,\sigma)$ with $\bigcup_{u\in
V(Q)}E(C_{u})\cup E(C_{1})\subseteq\operatorname{{\rm supp}}(f^{\prime})$.
Since for each $v\in V(H)-V(Q)$, $C_{v}$ is a balanced circuit in $(G,\sigma)$
, $(G,\sigma)$ admits a $2$-flow $\phi_{v}$ with $E(C_{v})=\operatorname{{\rm
supp}}(\phi_{v})$. Thus $f_{4}=f^{\prime}+\sum_{u\in V(H)-V(Q)}\phi_{u}$ is a
$4$-flow of $(G,\sigma)$ with $RY\subseteq\operatorname{{\rm supp}}(f_{4})$.
Therefore, $f_{3}+2f_{4}$ is an $8$-NZF of $(G,\sigma)$.

Subcase 2.2.2. $H$ contains no unbalanced circuit that is edge-disjoint from
$C_{1}$.

In this case, $H-E(C_{1})$ is balanced and thus $G-E(C_{1})$ is balanced. With
some switching operations we may assume $E_{N}(G,\sigma)\subseteq
E_{G}(C_{1})$. By Lemma 3.8, $(G,\sigma)$ has a $4$-flow $f^{\prime\prime}$
such that $C_{1}\subseteq\operatorname{{\rm supp}}(f^{\prime\prime})$ and
every vertex in $\operatorname{{\rm supp}}(f^{\prime\prime})-E(C_{1})$ has
degree at most $3$ in $H$. By Lemma 3.9, we can extend $f^{\prime\prime}$ to a
$4$-flow $f_{5}$ of $(G,\sigma)$ with $RY\subseteq\operatorname{{\rm
supp}}(f_{5})$ in $(G,\sigma)$. Therefore, $f_{3}+2f_{5}$ is an $8$-NZF of
$(G,\sigma)$.

Subcase 2.3. The number of unbalanced circuits in $RY$ or $BY$ is odd and is
at least $3$.

By symmetry, assume that the number of unbalanced circuits in $RY$ is odd and
is at least $3$. By Lemma 3.6, $(G,\sigma)$ has a $5$-flow $f_{6}$ such that
$RY\subseteq\operatorname{{\rm supp}}(f_{6})$ and $E_{f_{6}=\pm 4}\subseteq
B$. Then $\operatorname{{\rm supp}}(f_{3})\cup\operatorname{{\rm
supp}}(f_{6})=E(G)$. Thus $5f_{3}+f_{6}$ is a $10$-NZF of $(G,\sigma)$. ∎

Next we will prove Corollary 1.2.

Corollary 1.2 Every flow-admissible bridgeless planar signed graph admits a
nowhere-zero $10$-flow.

###### Proof.

Let $(G,\sigma)$ be a flow-admissible bridgeless planar signed graph. Let
$\tau$ be an orientation of $(G,\sigma)$. In the following we always assume
the flows are under the orientation $\tau$ or its restriction on according
subgraphs.

We may assume that the minimum degree of $G$ is at least $3$ otherwise we can
suppress all degree $2$ vertices. We may also assume that $G$ contains no
positive loops.

If $G$ is cubic, then by Theorem 1.1 and the $4$-color theorem, $G$ admits a
nowhere-zero $10$-flow.

Suppose that $G$ is not cubic and that $G$ is already embedded in a sphere.
Let $v$ be a vertex with $d_{G}(v)\geq 4$ and $t$ be the number of negative
loops adjacent to $v$. First delete the $t$ negative loops and then blow up
$v$ into a circuit $C_{v}$ of length $d_{G}(v)-2t$ where each edge of $C_{v}$
is positive. Let $xy$ be an edge in $C_{v}$. Replace it with a subdivided edge
$u_{0}u_{1}u_{2}\cdots u_{2t+1}$ where $x=u_{0}$ and $y=u_{2t+1}$ and then
replace each $u_{i}u_{i+1}$ with an unbalanced digon for each
$i=1,3,\dots,2t-1$ (see Figure 3). Let $(G^{\prime},\sigma^{\prime})$ be the
resulting signed graph obtained from $(G,\sigma)$ by applying the above
operations on each vertex in $G$ of degree at least $4$. Then
$(G^{\prime},\sigma^{\prime})$ is cubic, planar, and flow-admissible. By
Theorem 1.1 and the $4$-color theorem, $(G^{\prime},\sigma^{\prime})$ admits a
nowhere-zero $10$-flow. Note that $(G,\sigma)$ can be obtained from
$(G^{\prime},\sigma^{\prime})$ by contracting an all-positive subgraph of
$(G^{\prime},\sigma^{\prime})$. Thus $(G,\sigma)$ admits a nowhere-zero
$10$-flow. ∎

$v$$v_{3}$$v_{2}$$v_{1}$$\Longrightarrow$

$C_{v}$$u_{4}$$u_{3}$$u_{2}$$u_{1}$$u_{0}$$u_{5}$$v_{1}$$v_{2}$$v_{3}$

Figure 3: blowing up of a vertex $v$ with $d(v)=7$ and $t=2$. Dotted lines are
negative edges.

##  5 Proof of Theorem 1.4

Let’s first recall Theorem 1.4.

Theorem 1.4 If $(G,\sigma)$ is a flow-admissible hamiltonian signed graph,
then $(G,\sigma)$ admits a nowhere-zero 8-flow.

###### Proof.

Let $\tau$ be an orientation of $(G,\sigma)$. In the following we always
assume the flows are under the orientation $\tau$ or its restriction on
according subgraphs. Let $C_{0}$ be a hamiltonian circuit of $G$. We consider
two cases according to whether $C_{0}$ is balanced or unbalanced.

Case 1.  $C_{0}$ is balanced.

We may assume that $C_{0}$ is all-positive with some switching operations. It
is known that every ordinary graph with a hamiltonian circuit admits a 4-NZF
(See Corollary 3.3.7 [12]). Thus we may further assume that $(G,\sigma)$ is
unbalanced. Hence, by Lemma 2.2, $G$ contains at least two negative edges.
Clearly, $\langle C_{0}\rangle_{2}=(G,\sigma)$. By Lemma 3.1, $(G,\sigma)$
admits a $\operatorname{\mathbb{Z}}_{3}$-flow $\phi$ such that
$E(G)-E(C_{0})\subseteq\operatorname{{\rm supp}}(\phi)$. By Lemma 3.3,
$(G,\sigma)$ admits a $4$-flow $f_{1}$ such that
$E(G)-E(C_{0})\subseteq\operatorname{{\rm
supp}}(\phi)\subseteq\operatorname{{\rm supp}}(f_{1})$.

Since $C_{0}$ is balanced, $(G,\sigma)$ has a $2$-flow $f_{2}$ such that
$E(C_{0})=\operatorname{{\rm supp}}(f_{2})$. Note that $\operatorname{{\rm
supp}}(f_{2})\cup\operatorname{{\rm supp}}(f_{1})=E(G)$. Therefore
$f=2f_{1}+f_{2}$ is an 8-NZF of $(G,\sigma)$.

Case 2. $C_{0}$ is unbalanced.

Since $C_{0}$ is unbalanced, for each edge $e\not\in E(C_{0})$, there is a
balanced circuit in $C_{0}+e$ containing $e$, denoted by $C_{e}$. Let
$H=\bigtriangleup_{e\notin E(C_{0})}C_{e}$. Then $H$ admits a
$\operatorname{\mathbb{Z}}_{2}$-NZF and has an even number of negative edges.

If $H$ doesn’t contain an unbalanced circuit, then we may assume that $E(H)$
are all positive with some switching operations. Thus $E_{N}(G)\subseteq
E(C_{0})$ and $(G,\sigma)$ has a $2$-flow $f_{3}$ such that
$\operatorname{{\rm supp}}(f_{3})=E(H)$. By Lemma 3.8, there exists a 4-flow
$f_{4}$ such that $E(C_{0})\subseteq\operatorname{{\rm supp}}(f_{4})$. Since
$E(C_{0})\cup E(H)=E(G)$, $f_{3}+2f_{4}$ is an 8-NZF of $(G,\sigma)$.

Now assume that $H$ contains an unbalanced circuit, say $C_{0}^{\prime}$.
Since $H$ admits a $\operatorname{\mathbb{Z}}_{2}$-NZF and has an even number
of negative edges, by Lemma 3.5, $(G,\sigma)$ has a $3$-flow $f_{5}$ such that
$E_{f_{5}=\pm 1}=E(H)$ and $E_{f_{5}=\pm 2}\subseteq E(C_{0})-E(H)$.

Let $H^{\prime}=C_{0}\bigtriangleup C_{0}^{\prime}$. Then $H^{\prime}$ admits
a $\operatorname{\mathbb{Z}}_{2}$-NZF and has an even number of negative
edges. Since $C_{0}\cup C_{0}^{\prime}$ is connected, by Lemma 3.5 again,
$(G,\sigma)$ has a $3$-flow $f_{6}$ such that $\operatorname{{\rm
supp}}(f_{6})\subseteq E(C_{0})\cup E(C_{0}^{\prime})$, $E_{f_{6}=\pm
1}=E(H^{\prime})$, and $E_{f_{6}=\pm 2}\subseteq E(C_{0})\cap
E(C_{0}^{\prime})$.

Therefore, $3f_{5}+f_{6}$ is a $9$-NZF of $(G,\sigma)$. Since $E_{f_{5}=\pm
2}\cap E_{f_{6}=\pm 2}=\emptyset$, $|(3f_{5}+f_{6})(e)|\not=8$ for each edge
$e\in E(G)$. Thus, $3f_{5}+f_{6}$ is indeed an $8$-NZF of $(G,\sigma)$. ∎

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