# A rapidly evolving high-amplitude $\delta$ Scuti star crossing the
Hertzsprung Gap

Jia-Shu Niu (牛家树) Institute of Theoretical Physics, Shanxi University, Taiyuan
030006, China  State Key Laboratory of Quantum Optics and Quantum Optics
Devices, Shanxi University, Taiyuan 030006, China  Collaborative Innovation
Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
Hui-Fang Xue (薛会芳) Department of Physics, Taiyuan Normal University, Jinzhong
030619, China  Institute of Computational and Applied Physics, Taiyuan Normal
University, Jinzhong 030619, China

###### Abstract

In this work, we report the discovery of the rapidly evolving high-amplitude
$\delta$ Scuti star KIC 6382916 (J19480292+4146558) which is crossing the
Hertzsprung gap. According to the analysis of the archival data, we find three
independent pulsation modes, whose amplitudes and frequencies vary distinctly
in 4 years. The linear period variation rates of the first two modes are about
3-4 times larger than the best seismic model constructed by the standard
evolution theory, while that of the third one is about 8 times larger than the
first two modes. What is more interesting is that, almost all the combinations
of the third mode have frequency peaks $0.0815\ \,\mathrm{c}\ \mathrm{d}^{-1}$
away from them in the frequency domain. A framework is proposed to interpret
the markedly large frequency and amplitude variation rates of the third mode,
in which we employ a new pulsation mode (resonating integration mode)
generated by the resonance between a radial p-mode and a nonradial mixed mode.
Moreover, a global analysis of the interactions between the three independent
pulsation modes and their harmonics/combinations is performed based on the
interaction diagrams of their amplitudes and phases, which would be a useful
tool for the future asteroseismology research.

††software: MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019), GYRE (Townsend
& Teitler, 2013; Townsend et al., 2018; Goldstein & Townsend, 2020), emcee
(Foreman-Mackey et al., 2013), seaborn (Waskom, 2021)

##  1 Introduction

People cannot clearly witness the stellar evolution process of a single star
obviously in most cases because of its extremely secular time-scale, except
for some special time nodes in it (such as the supernova explosion (Arnett et
al., 1989)). However, in some specific evolutionary phases, we have the chance
to witness such processes gradually on human timescales. When a star is
leaving from the main sequence, the hydrogen nuclei fusion in its core is
gradually transferring into the shell. In the Hertzsprung-Russell diagram, its
evolutionary phase falls into the Hertzsprung gap, which is one of the most
rapidly evolving phases in the life of a star (Kippenhahn et al., 2012).

$\delta$ Scuti stars are a class of short-period pulsating variable stars with
periods between 15 minutes and 8 hours and spectral classes of A-F, which
locate them on the main-sequence (MS) or post-MS evolutionary phases at the
bottom of the classical Cepheid instability strip and excited by the $\kappa$
mechanism (Breger, 2000; Handler, 2009; Uytterhoeven et al., 2011; Holdsworth
et al., 2014). High-amplitude $\delta$ Scuti stars (hereafter HADS) are a
subclass of $\delta$ Scuti stars that usually have larger amplitudes ($\Delta
V\geq 0_{\cdot}^{m}3$) and slower rotations (Breger, 2000).111 However, with
the accumulation of the samples, these criteria become unclear now. For
example, the HADS KIC 9408694 has an amplitude of less than 0.1 Kmag at the
highest-amplitude modes and a fast rotation ($v\sin i\approx 100\ \mathrm{km\
s}^{-1}$) (Balona et al., 2012).  Most of the HADS show single or double
radial pulsation modes, and some of them have three radial pulsation modes or
even some nonradial pulsation modes (Niu et al., 2013, 2017; Xue et al., 2018;
Wils et al., 2008; Xue & Niu, 2020). Theoretically, the period variation rates
($(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$) of HADS caused by the evolution should
roughly have a value from $10^{-10}\ \mathrm{yr}^{-1}$ on the MS to $10^{-7}\
\mathrm{yr}^{-1}$ on the post-MS phase (Breger & Pamyatnykh, 1998). According
to the seismic models, the observed period variation rates of some HADS (such
as AE UMa and VX Hya (Niu et al., 2017; Xue et al., 2018, 2022)) can be
interpreted by the evolutionary effect self-consistently, and each of these
stars should be located in the Hertzsprung gap on Hertzsprung-Russell (H-R)
diagram with a helium core and a hydrogen-burning shell. In these works, the
linear period variation rates are obtained from the ground-based time-series
photometric data accumulated over several decades, but only those of the
fundamental pulsation modes have a sufficient precision to be confirmed and
tested because of the quality of the data. The high-precision photometric data
from the Kepler space telescope lasted for about 4 years and could provide an
excellent opportunity to determine the linear period and amplitude variation
rates not only of the fundamental pulsation mode, but also of other pulsation
modes. These observed quantities would tell us more secrets about the stellar
evolution and pulsation in this special rapidly evolving phase.

KIC 6382916 (J19480292+4146558, also known as GSC 03144-595) is an HADS that
has been monitored extensively by the Kepler space telescope for its pulsation
properties, which shows three independent frequencies (pulsation modes) in its
light curves (Ulusoy et al., 2013). These three pulsation modes were at first
identified as the nonradial $l=1$ pulsation modes in Ulusoy et al. (2013), and
then identified as the fundamental, first overtone, and second overtone radial
p-modes (Mow et al., 2016). Using the long-cadence (LC) photometric data of
KIC 6382916 lasted from BJD 2454953 to 2456424 (Quarter 0-17), we extract the
first 23 frequencies with the largest amplitudes (which have a cutoff of 1.3
mmag, see Appendix A and Table 2 for more details) to study their variations
and interactions. These frequencies are also identified as the labels of the
corresponding pulsation modes in this work, from F0 to F22. Theses 23
frequencies are composed of three independent frequencies, F0, F1, and F5
(also labeled as $f_{0}$, $f_{1}$, and $f_{2}$) together with their
harmonics/combinations.

##  2 Methods and Analysis

In order to extract the variation information of the amplitudes and
frequencies over time, we use the short-time Fourier transformation to deal
with the LC photometric data. The prewhitening process in a time window of 120
days is performed when the window is moving from the start to the end time of
the LC data, with a step of 20 days. In each step, the frequencies are fixed
as that have been extracted in the complete LC data (see Table 2), while the
amplitudes and phases are obtained by the nonlinear least-square fitting.
Then, we get the amplitude and phase variations in the 23 frequencies. Using
the Fourier-phase diagram method (Paparo et al., 1998; Bowman et al., 2016;
Xue et al., 2018), the variations in the frequencies can be derived from the
variations in the phases.

When the dominating quasi-periodic signals in the variations in amplitudes and
phases have been fully handled, we obtain their nonperiodic linear and
quadratic variations of them (see Appendix B for more details).222Hereafter,
we do not consider the periodic signal when we talk about the variations in
the amplitudes and phases, unless it is specifically mentioned. The amplitude
of $f_{0}$ ($A_{0}$) is relatively stable, that of $f_{1}$ ($A_{1}$) slightly
decreases while that of $f_{2}$ ($A_{5}$) distinctly increases by about 28% in
4 years. On the other hand, the phases (subtracted by its average) of $f_{0}$,
$f_{1}$, and $f_{2}$ vary with the same trend but different levels, which
corresponds to the increasing periods of them with different linear period
variation rates: $(1/P_{0})(\,\mathrm{d}P_{0}/\,\mathrm{d}t)=(3.0\pm
1.2)\times 10^{-7}\ \mathrm{yr}^{-1}$,
$(1/P_{1})(\,\mathrm{d}P_{1}/\,\mathrm{d}t)=(3.2\pm 1.0)\times 10^{-7}\
\mathrm{yr}^{-1}$, and $(1/P_{2})(\,\mathrm{d}P_{2}/\,\mathrm{d}t)=(2.4\pm
0.6)\times 10^{-6}\ \mathrm{yr}^{-1}$. The harmonics/combinations of $f_{0}$,
$f_{1}$, and $f_{2}$ also show complex amplitude and frequency variations,
which are summarized in the appendix, Table 2.

Noting the complex amplitude variations in the F0-F22 pulsation modes, it is
quite interesting to study the amplitude interactions between them, which also
indicate the energy transfer between them. Here, we introduce the interaction
diagram (see Figure 1) to show the amplitude interactions between the
pulsation modes. In Figure 1, the color of a small square represents the
correlation coefficient333Here we use the Spearman’s rank correlation
coefficient, which is a measure of how well the relationship between two
variables can be described by a monotonic function. between the amplitudes of
the labeled pulsation modes whose columns and rows intersect at the square.
The dendrograms on the upper and left represent the agglomerative hierarchical
clustering (AHC) process (Murtagh & Contreras, 2012) in the amplitude
interaction space, which is spanned by the vectors (corresponding to each of
the pulsation modes) consisting of the correlation coefficients between a
specific pulsation mode to all of them. In general, the AHC process reorders
the pulsation modes in order to cluster those who have similar behaviors in
the interaction space together. More details can be found in Appendix C.

At first glance of Figure 1, the pulsation modes are clustered into two groups
(separated between F22 and F7), which correspond to the main features of
increase in and periodicity of the amplitudes. In the upper group, all the
$f_{2}$ related modes are closely clustered together, which have obvious
increasing amplitudes. On the other hand, it is obvious that F1, F6, F7, and
F10 are $f_{1}$ related and anticorrelated with the $f_{2}$ related modes,
which indicates that they have decreasing amplitudes. What is more interesting
is that F12, F21, F19, F8, and F22 are not $f_{2}$ related, but clustered
together with the $f_{2}$ related modes. After a closer look on these
pulsation modes in the frequency domain, we find an interesting discovery:
almost all these modes have partners $\Delta\omega=0.0815\ \,\mathrm{c}\
\mathrm{d}^{-1}$ away from them.444For F21, we cannot find any significant
peaks (larger than 0.4 mmag) in both sides. Inspired by this phenomenon, we
carefully review all the 23 pulsation modes in the frequency domain carefully,
and find that all the $f_{2}$ related modes have partners
$\Delta\omega=0.0815\ \,\mathrm{c}\ \mathrm{d}^{-1}$ away from them.555The
zoom-in of all the 23 pulsation modes in the frequency domain is shown in
Figure 6, 7, 8, and 9. It indicates that $f_{2}$ could be a nonradial modes
with $l=1$666Here, all the $1\leq l\leq 3$ modes are possible. However, if we
consider the relative large amplitude and all the doublets in the frequency
domain, $l=1$ is the most likely scenario. and all these $f_{2}$ related
pulsation modes and their partners are locked by the star’s rotation (with a
rotation frequency of $\Delta\omega=0.0815\ \,\mathrm{c}\ \mathrm{d}^{-1}$),
which are interacting with each other. Additionally, all the $f_{2}$ related
modes (F5, F11, F13, F14, F15, and F18) have partners $\Delta\omega=0.0815\
\,\mathrm{c}\ \mathrm{d}^{-1}$ smaller than them, and the $-f_{2}$ related
modes (F16) have partners $\Delta\omega=0.0815\ \,\mathrm{c}\ \mathrm{d}^{-1}$
larger than them, which implies that $f_{2}$ has an azimuthal order of $m=1$.
If this is true, the linear period variation rates of all $f_{2}$ (including
$-f_{2}$) related modes with significant nonzero values should be dominated by
the slowed-down rotation of the star, which gives rise to the linear period
variation rates that are about an order of magnitude greater than those of the
purely $f_{0}$ and $f_{1}$ related modes without partners.

Figure 1: Interaction diagram of amplitudes of the 23 pulsation modes.

Based on the above inferences, it is easy to understand the significant
nonzero $(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$ values reaching up to an order of
magnitude from $10^{-6}$ to $10^{-5}$, which are seriously influenced by the
slowed-down rotation and the interactions with their partners. In Figure 6, 7,
8, and 9, we show the pulsation modes without significant partners; if their
$(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$ have significant nonzero values (F0, F1,
F2, F4, and F7), these values fall into a range of about $2.4\times 10^{-7}\
\,\mathrm{yr}^{-1}$ to $4.0\times 10^{-7}\ \,\mathrm{yr}^{-1}$, which should
be the values of global linear period variations that are unaffected (or
minimally affected) by the rotation and interactions of the modes. Considering
that $f_{0}$ and $f_{1}$ have almost the same period variation and have a
period ratio ($P_{1}/P_{0}=f_{0}/f_{1}$) of 0.763 (Petersen & Christensen-
Dalsgaard, 1996; Poretti et al., 2005), they could be considered as the
fundamental and first overtone radial pulsation modes, respectively.

Single-star nonrotating evolutionary models are constructed by using different
initial masses with three groups of [Fe/H] ($0.040$ (Mathur et al., 2017),
$0.111$ (Xiang et al., 2019), and $0.266$ (Luo et al., 2019)) from
observations (which are labeled by $Z=0.014$, $Z=0.016$, and $Z=0.022$; see
more details in Appendix D), from the pre-main sequence to the start of the
red giant branch. At each of the steps on the evolutionary tracks, the
pulsation frequencies are calculated based on the corresponding stellar
structure. Figure 2 shows the best-fit seismic models of the observed
independent frequencies ($f_{0}$ and $f_{1}$) together with the corresponding
evolutionary tracks for different $Z$ values. The detailed information of the
best-fit seismic models and the observations are collected in Table 1.

The results of the seismic models show that KIC 6382916 is located in the
post-MS phase with the mass from 1.81-1.87 $M_{\odot}$. The most incredible
thing is that the observed linear period variation rates of $f_{0}$ and
$f_{1}$ are 3-4 times larger than the theoretical predicted ones. If we
ascribe these observed linear period variation rates to the stellar evolution,
we find that the star is evolving more rapidly than theoretical prediction and
the standard stellar evolution theory cannot precisely describe the star’s
evolution process in this rapidly evolving phase. Another possible scenario is
that the large linear period variations might be caused by mass transfer,
which could be induced by the stellar wind or an undiscovered binary companion
of the star. These possibilities should be studied and confirmed in the
future.

Figure 2: Hertzsprung-Russell diagram with all the evolutionary tracks and the best-fit seismic models of KIC 6382916 with different $Z$ values. The colored solid evolutionary tracks show from the zero-age MS to the post-MS evolutionary stage, with initial masses from 1.50 $M_{\odot}$ to 2.50 $M_{\odot}$. The regions surrounded by the black rectangular boxes are selected to zoom in the best-fit seismic models, which are presented by the red stars. The colored dashed lines present the evolutionary tracks with initial mass steps of 0.01 $M_{\odot}$.  Table 1: Information of the best-fit seismic models and the observations. | $Z=0.014$ | $Z=0.016$ | $Z=0.022$ | Observed
---|---|---|---|---
$f_{0}\ (\,\mathrm{c}\ \mathrm{d}^{-1})$ $(l=0,n_{p}=0)$ | 4.909573 | 4.909774 | 4.910073 | $4.909842\pm 0.000001$
$f_{1}\ (\,\mathrm{c}\ \mathrm{d}^{-1})$ $(l=0,n_{p}=1)$ | 6.432067 | 6.431822 | 6.431710 | $6.431890\pm 0.000001$
$(1/P_{0})(\,\mathrm{d}P_{0}/\,\mathrm{d}t)\ (\,\mathrm{yr}^{-1})$ | $7.78\times 10^{-8}$ | $8.83\times 10^{-8}$ | $1.197\times 10^{-7}$ | $(3.0\pm 1.2)\times 10^{-7}$
$(1/P_{1})(\,\mathrm{d}P_{1}/\,\mathrm{d}t)\ (\,\mathrm{yr}^{-1})$ | $7.49\times 10^{-8}$ | $8.50\times 10^{-8}$ | $1.150\times 10^{-7}$ | $(3.2\pm 1.0)\times 10^{-7}$
$f_{2}^{+}\ (\,\mathrm{c}\ \mathrm{d}^{-1})$ $(l=0,n_{p}=2)$ | 8.061727 | 8.067854 | 8.085706 | $8.035410\pm 0.000002$
$f_{2}^{-}\ (\,\mathrm{c}\ \mathrm{d}^{-1})$ $(l=1,n_{p}=2)$ | 7.759899 | 7.993722 | 7.670764 | $7.953927\pm 0.000002$
$(1/P_{2}^{+})(\,\mathrm{d}P_{2}^{+}/\,\mathrm{d}t)\ (\,\mathrm{yr}^{-1})$ | $7.39\times 10^{-8}$ | $8.39\times 10^{-8}$ | $1.123\times 10^{-7}$ | $(2.4\pm 0.6)\times 10^{-6}$
$(1/P_{2}^{-})(\,\mathrm{d}P_{2}^{-}/\,\mathrm{d}t)\ (\,\mathrm{yr}^{-1})$ | $-7.98\times 10^{-8}$ | $-8.41\times 10^{-8}$ | $-1.349\times 10^{-7}$ | $(2.4\pm 0.6)\times 10^{-6}$
$\log\,T_{\mathrm{eff}}$ | 3.809 | 3.809 | 3.806 | [3.796, 3.867]777The observed $\log\,T_{\mathrm{eff}}$ and $\log g$ values are collected from Bai et al. (2019); Ulusoy et al. (2013); Luo et al. (2019); Xiang et al. (2019); Mathur et al. (2017); Berger et al. (2018); Anders et al. (2019, 2022).
$\,\log{g}$ | 3.478 | 3.480 | 3.487 | [3.46, 4.31]
$\log L/L_{\odot}$ | 1.408 | 1.411 | 1.404 | —
Mass ($M_{\odot}$) | 1.81 | 1.83 | 1.87 | —
Age (Gyr) | 1.39 | 1.37 | 1.36 | —
$\log R/R_{\odot}$ | 0.609 | 0.610 | 0.612 | —

Based on the above best-fit seismic models from the constraint of $f_{0}$ and
$f_{1}$, we calculate the frequencies of the second overtone mode, which are
systematically greater than the observed $f_{2}$ and labeled as $f_{2}^{+}$
(see in Table 1). The best-fit results of the nonradial ($l=1$, $m=0$) modes
to the observed $f_{2}-\Delta\omega$ have $n_{p}=2,n_{g}=23$ (for $Z=0.014$),
$n_{p}=2,n_{g}=21$ (for $Z=0.016$), and $n_{p}=2,n_{g}=19$ (for $Z=0.022$),
which are mixed modes of the p-mode and g-mode types (see in Table 1) and
labeled as $f_{2}^{-}$. Although in the $Z=0.016$ case the observed $f_{2}$
can be roughly fitted by $f_{2}^{-}+\Delta\omega$ and its large linear period
variation rate can be ascribed to the slowed-down rotation of the star, the
distinctly increasing amplitude of the $f_{2}$ mode remains an unsolved
problem in this framework.

Here, we propose a framework to interpret all the important characteristics of
$f_{2}$. Firstly, a resonance happens between a radial p-mode and a nonradial
mixed mode, i.e. $f_{2}^{+}$ and $f_{2}^{-}$, and generates a resonating
integration mode (RI mode) $f_{2}^{\pm}$. The requirements of the resonance
can be met by (i) $f_{2}^{+}$ and $f_{2}^{-}$ having close frequency values
and (ii) their radial displacements having close nodes and similar phases
along with the radius in the outer layers (see in Figure 3). Secondly, the RI
mode $f_{2}^{\pm}$ should have the properties of radial and nonradial
pulsation modes, which splits into two frequencies ($f_{2}^{\pm 1}$ and
$f_{2}^{\pm 2}$, $f_{2}^{\pm 2}-f_{2}^{\pm 1}=\Delta\omega$) because of the
star’s rotation. $f_{2}^{\pm 1}$ resonates and superimposes with the
combination of $f_{0}$/$f_{1}$ (i.e. $-f_{0}+2f_{1}$) to form the observed
partner of $f_{2}$ (i.e. F8), while $f_{2}^{\pm 2}$ corresponds to the
observed $f_{2}$.

In the above framework, the frequency of $f_{2}^{\pm}$ should be modulated
between $f_{2}^{+}$ and $f_{2}^{-}$ (similar to the situation in Goupil et al.
(1998)), and then $f_{2}$ and its partner ($f_{2}^{\pm 2}$ and $f_{2}^{\pm
1}$, respectively) should represent the modulations, which are covered by
current long-time scale data sets and could be shown out in a short-time scale
data set. Because the linear period variation of $f_{2}$ is dominantly
affected by the slowed-down rotation of the star, it shows a rate about 8
times larger than the global ones (that of $f_{0}$ and $f_{1}$). Taking it a
step further, we note that the theoretical linear period variations in
$f_{2}^{+}$ and $f_{2}^{-}$ have opposite directions in the stellar evolution.
The distance between $f_{2}^{+}$ and $f_{2}^{-}$ decreases with the evolution
of the star, and the resonance between them will be strengthened and the
amplitude of $f_{2}$ will increase.888Here, we suppose a mechanism that
increases the global linear period variation rates of $f_{0}$ and $f_{1}$ and
also works for $f_{2}^{+}$ and $f_{2}^{-}$ to some extent, but keeps the signs
of them unchanged.

About the interactions between the $f_{2}$ related modes and their
rotationally locked partners, on one hand, all of them have components
inherited from the RI mode, which gives rise to increasing amplitudes. On the
other hand, each of the observed partners is the resonance result of the
$f_{0}$/$f_{1}$ related mode and the radial component of the RI mode, which
will also cause energy transfer between them and then amplitude modulation.
All these factors together produce the interaction relationships in Figure 1.
A more detailed quantitative theory about the framework should be constructed
in future works.

Figure 3: Radial displacement (represented by $r\rho^{1/2}\delta r$) against
the fractional radius ($r/R$) for the best-fit seismic models with different
$Z$ values. $r$ is the radial distance to the center; $\rho$ is the density at
$r$; $\delta r$ is the radial displacement eigenfunction at $r$; $R$ is the
radius of the star.

For KIC 6382916, following the current amplitude variation rates, the
amplitude of $f_{2}$ will exceed that of $f_{0}$ at about BJD 2460098 (2023
June), and exceed that of $f_{1}$ at about BJD 2460196 (2023 September). This
prediction can be tested in the near future by the following photometric
observations, which also could also provide us with an opportunity to witness
the stellar evolution process of a single star gradually in this special
evolutionary phase.

##  3 Discussions and Conclusions

The period variation rates of HADS have been studied for a long time as their
large amplitudes are always dominated by one pulsation mode, and the classical
O-C method can be put into practice based on the accumulation of times of
maximum light of about several decades (Breger & Pamyatnykh, 1998). In recent
years, based on the continuous high-precision time-series photometric data
from space telescopes like Kepler, the linear period variation rates of HADS
can be extracted directly from the light curves, not only for the dominating
pulsation modes, but also for other nondominating pulsation modes and the
harmonics/combinations of these modes (see, e.g., Breger & Montgomery (2014);
Bowman et al. (2016, 2021)) . These linear period variation rates
($(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$) generally have the values from
$\pm(10^{-9}-10^{-6})\,\mathrm{yr}^{-1}$, while some of them are not that
reliable and would be updated by additional high-quality photometric data.
Theoretically, the period variation of an HADS can be caused by stellar
evolution (Niu et al., 2017; Xue et al., 2018, 2022), mass transfer (Xue &
Niu, 2020), light traveling time effect in a multiple system (Xue & Niu,
2020), and pulsational nonlinearity (Breger & Montgomery, 2014; Zong et al.,
2016; Bowman et al., 2016). In these possible origins, the first three will
affect all pulsation modes globally999In more detail, the stellar evolution
will also lead to different pulsation modes (p-mode and g-mode) to linear
period variation rates with different signs but similar absolute values.,
while the last one will affect on the pulsation modes involved in the
nonlinear interactions individually. In general, the observed period variation
of an HADS is the result of a combination of several or all factors above.
Especially, as a potential marker of stellar evolution, it is quite
interesting and important whether the observed linear period variation of an
HADS can be ascribed to the stellar evolution (Niu et al., 2017; Xue et al.,
2018, 2022).

Harmonics/combinations of independent pulsation modes are common amongst
pulsating stars, such as $\delta$ Scuti stars (Breger & Montgomery, 2014),
$\beta$ Cep stars (Degroote et al., 2009), SPB stars (Pápics et al., 2017),
$\gamma$ Dor stars (Kurtz et al., 2015), and pulsating white dwarfs (Wu,
2001). Mathematically, the harmonics/combinations come from the nonsinusoidal
properties of light curves, which indicate the nonlinearity of the star’s
pulsation physically.101010Such as the nonlinear transformation from the
temperature variation to the flux variation ($F=\sigma T^{4}$), and the
nonlinear response of the stellar medium to the pulsation waves. In general,
the harmonics/combinations are not considered as intrinsic stellar pulsation
modes (Brickhill, 1992; Wu, 2001) and should mimic the behaviors of the
parent’s pulsation modes. As a result, in practice, the harmonics/combinations
are always removed in the prewhitening process and not taken as the key
information for the asteroseismology. Although the period variation rates of
HADS for different pulsation modes have been studied by some pioneering works
(see, e.g., Breger & Montgomery (2014); Bowman et al. (2016)), those of
harmonics/combinations and the global interactions between
harmonics/combinations have received little attention, which could provide us
with abundant information about the nonlinear characteristics of both the
harmonics/combinations themselves and the interactions between them.

For KIC 6382916, each of the period variations in the 23 pulsation modes are
dominated by a quasi-periodic plus a linear signal (for phase variation,
quadratic signals). The quasi-periodic signals are not the focus of in this
work (which might have instrumental origin, or due to the light-time effect in
a binary system), and will be studied in future works. About the linear period
variation, unlike in AI Vel (Walraven et al., 1992), the fundamental and first
overtone modes ($f_{0}$ and $f_{1}$) have similar period variation rates,
which are 3-4 times larger than the theoretical predictions. Considering the
similar values on both the $f_{0}$ and $f_{1}$ modes (and their
harmonics/combinations without significant partners), the most possible
origins of this result is a faster stellar evolution than that expected in
this specific phase for the mass loss of the star, which needs further studies
based on observational evidences. $f_{2}$ has a
$(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$ value 8 times larger than those of
$f_{0}$ and $f_{1}$, which could come from the slowed-down rotation of the
star and indicates that $f_{2}$ is a nonradial mode. Meanwhile, some of the
harmonics/combinations show different period variations from their parent’s
modes (such as F22), which could be ascribed to the nonlinear interactions to
the partners.

Beside the large linear period variation rates, the $f_{2}$ mode also has a
distinctly increasing amplitude (see in Figure 11). Moreover, in the
interaction diagram of amplitudes (see in Figure 1), the $f_{2}$ related modes
are closely clustered together, which also represents the particularity of
$f_{2}$. Inspired by the results of theoretical calculations, we propose a
framework to explain the strange behaviors of $f_{2}$ qualitatively. In this
framework, the observed $f_{2}$ is the nonradial component of an RI mode
(split by the rotation of the star), while the RI mode comes from a resonance
between a radial p-mode and a nonradial mixed mode. In this case, the linear
period variation rate of $f_{2}$ is dominated by the slowed-down rotation and
the increasing amplitude comes from the gradually enhanced resonance along
with the stellar evolution. In this sense, we can say that we witness the
stellar evolution process of a single star gradually in this special
evolutionary phase.

In the above framework, the variation rate of the rotation
($\,\mathrm{d}\Delta\omega/\,\mathrm{d}t$) can be obtained either directly by
the observed frequency variations in the $f_{2}$ related modes and their
partners (the differences between them), or indirectly by the observed
frequency variations in the three independent frequencies ($f_{0}$, $f_{1}$,
and $f_{2}$) based on the frequency decomposition relation (see Appendix E for
more details). Moreover, if we assume the conservation of angular momentum in
this evolutionary phase and a uniform rotation of the star, the variation in
the structure along with the stellar evolution will also cause a slowed-down
rotation. The variation rates of the rotation frequency
($\,\mathrm{d}\Delta\omega/\,\mathrm{d}t$) obtained from these estimations are
shown in Figure 4, more details can be found in Appendix E. In this work, five
pairs can be used to calculate $\,\mathrm{d}\Delta\omega/\,\mathrm{d}t$:
F3-F14, F5-F8, F6-F11, F9-F16, and F15-F22. In Figure 4, all the five pairs
indicate a slowed-down rotation, although pairs F3-F14 and F6-F11 show large
uncertainties and have no definite answer. Pairs F5-F8 and F9-F16 have
consistent results with that obtained from $f_{0}$, $f_{1}$, and $f_{2}$
(F0-F1-F5). The different results between the pair F15-F22 and F0-F1-F5 might
indicate the nonlinear interaction between F15 and F22 or the influence from a
hidden g-mode on pair F15-F22 in this low-frequency domain. On the other hand,
it is obvious that the slowed-down rotation frequency
($|\,\mathrm{d}\Delta\omega/\,\mathrm{d}t|$) rates from theoretical models are
significantly smaller than those from observations, which can be ascribed to a
more effective angular momentum transfer based on the differential rotation of
the star or an extra angular momentum loss induced by the stellar wind or its
undiscovered binary companion. It needs more precise theoretical models and
observational data for further study.

Figure 4: Variation rates of the rotation frequency from different estimates.
The red dots with error bars represent those obtained from the five pairs. The
blue region represents those obtained from the variation of $f_{0}$, $f_{1}$,
and $f_{2}$. The three dashed lines represent those from the best-fit seismic
models of different $Z$ values.

It is interesting to take a closer look on F22. On the one hand, the frequency
variation rate of F22 can be obtained directly from Table 4
($\dot{f}_{22}=(19.12\pm 10.05)\times 10^{-8}\ \mathrm{c\
d}^{-2}$).111111Here, the method of uncertainty estimation is the same as that
in Appendix E. On the other hand, we have identified $f_{22}=-2f_{0}+2f_{1}$,
and then we should have $\dot{f}_{22}=-2\dot{f}_{0}+2\dot{f}_{1}$. Based on
$\dot{f}_{0}=(-4.0\pm 1.5)\times 10^{-9}\ \mathrm{c\ d}^{-2}$ and
$\dot{f}_{1}=(-5.7\pm 1.8)\times 10^{-9}\ \mathrm{c\ d}^{-2}$, we can get
$-2\dot{f}_{0}+2\dot{f}_{1}=(-3.4\pm 4.6)\times 10^{-9}\ \mathrm{c\ d}^{-2}$.
Even taking the large uncertainties into account, it shows that
$\dot{f}_{22}\neq-2\dot{f}_{0}+2\dot{f}_{1}$, which indicates that the
observed F22 mode is not a simple superposition of $f_{0}$ and $f_{1}$. The
nonlinear interactions from its partner or a hidden g-mode might be the origin
of its significantly large frequency variation.

This is not the only case. For $\delta$ Scuti stars KIC 9700322 and KIC
5950759, one can also find partners similar to the case in this work (equally
spaced frequency multiplets caused by the rotation of the star) beside the
dominant radial pulsation modes (Breger et al., 2011; Yang et al., 2018). We
believe that this phenomenon is common in slowly rotating HADS, which evolve
into a special phase and meet the resonant conditions, and the above framework
of the RI mode will be refined in future research.

About KIC 6382916, there are still many interesting questions to be addressed,
such as the origin of the quasi-periodic signals in the amplitudes and phases;
the origin of the large linear period variation rates of $f_{0}$ and $f_{1}$
compared with the theoretical prediction; more quantitative details of the
large amplitude increases in $f_{2}$ (28% in 4 years); whether the amplitude
of $f_{2}$ will exceed those of $f_{0}$ and $f_{1}$ in the future; the origin
of the observed large slowed-down rotation rate compared with the theoretical
predictions; the inconsistency of the frequency variation rates of the
harmonics/combinations and their parent’s pulsation modes (such as the case in
F22). More detailed research and observations are needed.

## Acknowledgments

We would like to thank the anonymous reviewer for his/her professional and
detailed suggestions and Jue-Ran Niu for providing us with an efficient
working environment. J.S.N. acknowledges support from the National Natural
Science Foundation of China (NSFC) (No. 12005124 and No. 12147215). H.F.X.
acknowledges support from the Scientific and Technological Innovation Programs
of Higher Education Institutions in Shanxi (STIP) (No. 2020L0528) and the
Applied Basic Research Programs of Natural Science Foundation of Shanxi
Province (No. 202103021223320).

The MESA work folder for reproducing the best-fit models with different $Z$
values is available on Zenodo under an open-source Creative Commons
Attribution license: https://doi.org/10.5281/zenodo.7133342 (catalog
doi:10.5281/zenodo.7133342).

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##  Appendix A Photometric Data Reduction

The long-cadence (LC) photometric data of KIC 6382916 from the Kepler space
telescope were used in this work, which covers from BJD 2454953 to 2456424
(Quarter 0-17) (publicly available PDC data (Stumpe et al., 2012; Smith et
al., 2012)). We downloaded the light curves (in the format of reduced BJD and
magnitudes) of KIC 6382916 from the Mikulski Archive for Space Telescope
(MAST) 121212http://archive.stsci.edu/kepler, which were then normalized to be
zero in the mean for each quarter. An overview of all the normalized LC data
in the time domain and frequency domain are shown in Figure 5.

Figure 5: Overview of the normalized LC data in the time domain and frequency
domain of KIC 6382916. The region surrounded by the black rectangular boxes is
selected to zoom in the details of the observed light curves and the
theoretical light curves, which are reproduced by the detected frequency
solutions and represented by light green lines.

All the above normalized LC data were prewhitened to extract the frequencies,
amplitudes, and phases of the pulsation modes. This process was cutoff until
an amplitude smaller than 1.3 mmg, and we obtained a total of 23 pulsation
modes (see in Table 2), which is much higher than the typical noise level in
Kepler data of this star ($\sim$ 0.03 mmag). This choice ensured that the
uncertainties of the amplitudes and phases were typically smaller than the
variations in some pulsation modes of interest in 4 years, which was
determined by the subsequent short-time Fourier transformation results. In
each step in the prewhitening process, the Lomb-Scargle algorithm (VanderPlas,
2018) was used to help to find the initial value of frequency with largest
amplitude, and then a non-linear least square fitting was performed to get the
final values of the frequency, amplitude, and phase. The frequencies within
$1.0\leq f\leq 24.4\ \,\mathrm{c}\ \mathrm{d}^{-1}$ were considered in this
work, which was determined by the LC Nyquist frequency (Bowman et al., 2016).

Table 2: Frequency Solution of the First 23 Frequencies with the Largest Amplitudes. $\delta$ is defined as the difference between the observed frequency of F0-F22 mode and the marked one decomposed by $f_{0}$, $f_{1}$, and $f_{2}$. $\,\mathrm{d}A/\,\mathrm{d}t$ denotes the linear variation rate of the amplitudes. ID | Frequency | $\sigma_{f}$ | Amplitude | $\sigma_{A}$ | S/N | Mark | $\delta$ | $(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$ | $(\,\mathrm{d}A/\,\mathrm{d}t)$
---|---|---|---|---|---|---|---|---|---
| $(\,\mathrm{c}\ \mathrm{d}^{-1})$ | $(\,\mathrm{c}\ \mathrm{d}^{-1})$ | $(\mathrm{mmag})$ | $(\mathrm{mmag})$ |  |  | $(\,\mathrm{c}\ \mathrm{d}^{-1})$ | $(\mathrm{yr}^{-1})$ | $(\mathrm{mmag\ days}^{-1})$
F0 | 4.909842 | 0.000001 | 79.9 | 0.2 | 334.9 | $f_{0}$ | $0$ | $(3.0\pm 1.2)\times 10^{-7}$ | $(12\pm 12)\times 10^{-5}$
F1 | 6.431890 | 0.000001 | 77.8 | 0.2 | 377.2 | $f_{1}$ | $0$ | $(3.2\pm 1.0)\times 10^{-7}$ | $(-45\pm 12)\times 10^{-5}$
F2 | 11.341730 | 0.000001 | 28.38 | 0.09 | 320.3 | $f_{0}+f_{1}$ | $-1.9\times 10^{-6}$ | $(2.4\pm 1.6)\times 10^{-7}$ | $(-7\pm 12)\times 10^{-5}$
F3 | 1.522039 | 0.000002 | 16.53 | 0.07 | 221.1 | $-f_{0}+f_{1}$ | $-8.9\times 10^{-6}$ | $(4.6\pm 1.8)\times 10^{-6}$ | $(-1\pm 12)\times 10^{-5}$
F4 | 9.819693 | 0.000001 | 16.11 | 0.06 | 272.0 | $2f_{0}$ | $8.6\times 10^{-6}$ | $(4.0\pm 3.2)\times 10^{-7}$ | $(-3\pm 12)\times 10^{-5}$
F5 | 8.035410 | 0.000002 | 12.54 | 0.06 | 226.5 | $f_{2}$ | $0$ | $(2.4\pm 0.6)\times 10^{-6}$ | $(206\pm 12)\times 10^{-5}$
F6 | 12.863778 | 0.000002 | 11.71 | 0.05 | 246.2 | $2f_{1}$ | $-1.9\times 10^{-6}$ | $(3.8\pm 3.0)\times 10^{-7}$ | $(-16\pm 12)\times 10^{-5}$
F7 | 16.251580 | 0.000002 | 8.51 | 0.04 | 200.9 | $2f_{0}+f_{1}$ | $5.7\times 10^{-6}$ | $(3.4\pm 3.2)\times 10^{-7}$ | $(-11\pm 12)\times 10^{-5}$
F8 | 7.953927 | 0.000002 | 6.94 | 0.04 | 162.2 | $-f_{0}+2f_{1}$ | $-1.1\times 10^{-5}$ | $(6.8\pm 8.4)\times 10^{-7}$ | $(10\pm 12)\times 10^{-5}$
F9 | 3.387805 | 0.000004 | 5.26 | 0.06 | 84.5 | $2f_{0}-f_{1}$ | $1.1\times 10^{-5}$ | $-(0.6\pm 3.0)\times 10^{-6}$ | $(11\pm 12)\times 10^{-5}$
F10 | 17.773617 | 0.000003 | 4.23 | 0.04 | 114.8 | $f_{0}+2f_{1}$ | $-5.7\times 10^{-6}$ | $(8.0\pm 6.2)\times 10^{-7}$ | $(-20\pm 12)\times 10^{-5}$
F11 | 12.945261 | 0.000004 | 3.37 | 0.03 | 103.0 | $f_{0}+f_{2}$ | $9.5\times 10^{-6}$ | $(7.8\pm 10.6)\times 10^{-7}$ | $(58\pm 15)\times 10^{-5}$
F12 | 22.683468 | 0.000004 | 3.24 | 0.03 | 98.3 | $2f_{0}+2f_{1}$ | $3.8\times 10^{-6}$ | $(4.6\pm 6.2)\times 10^{-7}$ | $(-2\pm 12)\times 10^{-5}$
F13 | 14.467309 | 0.000004 | 3.19 | 0.03 | 92.7 | $f_{1}+f_{2}$ | $8.6\times 10^{-6}$ | $(16.0\pm 10.0)\times 10^{-7}$ | $(53\pm 12)\times 10^{-5}$
F14 | 1.603523 | 0.000007 | 2.98 | 0.05 | 55.2 | $-f_{1}+f_{2}$ | $3.1\times 10^{-6}$ | $(3.8\pm 11.4)\times 10^{-6}$ | $(35\pm 12)\times 10^{-5}$
F15 | 3.125582 | 0.000008 | 2.58 | 0.06 | 45.4 | $-f_{0}+f_{2}$ | $1.4\times 10^{-5}$ | $(7.6\pm 5.4)\times 10^{-6}$ | $(54\pm 12)\times 10^{-5}$
F16 | 3.30630 | 0.00001 | 2.18 | 0.06 | 39.2 | $f_{0}+f_{1}-f_{2}$ | $-2.2\times 10^{-5}$ | $-(12.0\pm 6.0)\times 10^{-6}$ | $(55\pm 15)\times 10^{-5}$
F17 | 14.729543 | 0.000006 | 2.10 | 0.03 | 65.4 | $3f_{0}$ | $1.6\times 10^{-5}$ | $-(5.0\pm 14.8)\times 10^{-7}$ | $(8\pm 12)\times 10^{-5}$
F18 | 9.557459 | 0.000006 | 1.99 | 0.03 | 60.7 | $-f_{0}+f_{1}+f_{2}$ | $9.5\times 10^{-7}$ | $(7.0\pm 23.0)\times 10^{-7}$ | $(17\pm 12)\times 10^{-5}$
F19 | 19.295654 | 0.000007 | 1.93 | 0.03 | 57.9 | $3f_{1}$ | $-1.5\times 10^{-5}$ | $(3.6\pm 12.0)\times 10^{-7}$ | $(2\pm 12)\times 10^{-5}$
F20 | 24.205505 | 0.000006 | 1.87 | 0.03 | 59.5 | $f_{0}+3f_{1}$ | $-5.7\times 10^{-6}$ | $(4.4\pm 9.8)\times 10^{-7}$ | $(-4\pm 12)\times 10^{-5}$
F21 | 21.161431 | 0.000009 | 1.37 | 0.03 | 43.7 | $3f_{0}+f_{1}$ | $1.3\times 10^{-5}$ | $(0.4\pm 15.4)\times 10^{-7}$ | $(2\pm 12)\times 10^{-5}$
F22 | 3.04408 | 0.00001 | 1.38 | 0.05 | 25.8 | $-2f_{0}+2f_{1}$ | $-1.6\times 10^{-5}$ | $-(2.4\pm 1.2)\times 10^{-5}$ | $(12\pm 12)\times 10^{-5}$

In order to show the detailed structures in the 23 pulsation modes in
frequency domain, a zoom-in of them is shown in Figure 6, 7, 8, and 9.

Figure 6: Zoom-in of the 23 pulsation modes in the frequency domain, Part I.
The vertical dash lines mark the position of the 23 frequencies and that are
$\Delta\omega=0.0815$ away from them on both sides. Gray indicates there is no
obvious peaks in that position; red indicates the pulsation mode in that
position shows a positive value of $(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$; blue
indicates the pulsation mode in that position shows a negative value of
$(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$; green indicates the pulsation mode in
that position shows an uncertain sign of $(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$.
The blue circle on the top indicates the amplitude of the pulsation mode is
uncertain; the upward-pointing red triangle indicates the amplitude of the
pulsation mode is increasing; the downward-pointing blue triangle indicates
the amplitude of the pulsation mode is decreasing.

Figure 7: Zoom-in of the 23 pulsation modes in the frequency domain, Part II.

Figure 8: Zoom-in of the 23 pulsation modes in the frequency domain, Part III.
Figure 9: Zoom-in of the 23 pulsation modes in the frequency domain, Part IV.

The short-time Fourier transformation (Bowman et al., 2016; Zong et al., 2016,
2018) was then performed to the normalized LC data to get the variation in the
amplitudes and phases. In this process, a time window of 120 days was moving
from the start to the end time of the LC data, with a step of 20 days. In each
step, the prewhitening process was performed to extract the amplitudes and
phases of the specific 23 pulsation modes, while the frequencies were fixed as
the values obtained in the complete LC data in Table 2.

At last, the amplitude and phase (subtracted by its average value) for each of
the 23 pulsation modes in each of the moving window were collected (with the
times that were defined as the midpoints of the window). The variations in the
amplitudes and phases of the 23 pulsation modes are presented in Figure 10,
11, 12, 13, and 14.

Figure 10: Variation in the amplitudes and phases of the 23 pulsation modes,
Part I. The best-fit results to Equations B3 and B2 are presented by the black
solid lines. The $2\sigma$ (deep red and blue) and $3\sigma$ (light red and
blue) bounds are also shown in the panels. In the lower panel of each
subfigure, the $\sigma_{\mathrm{eff}}$ is defined as
$(Q_{\mathrm{obs}}-Q_{\mathrm{cal}})/{\sigma}$, where $Q_{\mathrm{obs}}$ and
$Q_{\mathrm{cal}}$ are the values that come from the observation and model
calculation, respectively; $\sigma$ is the uncertainty of the observed points.

Figure 11: Variation in the amplitudes and phases of the 23 pulsation modes,
Part II.

Figure 12: Variation in the amplitudes and phases of the 23 pulsation modes,
Part III.

Figure 13: Variation in the amplitudes and phases of the 23 pulsation modes,
Part IV.

Figure 14: Variation in the amplitudes and phases of the 23 pulsation modes,
Part V.

In this work, the prewhitening process was performed by the Fourier
decomposition, which can be presented by the formula

$m=m_{0}+\sum A_{i}\sin\left[2\pi(f_{i}t+\phi_{i})\right]~{},$ (A1)

where $m_{0}$ is the shifted value, $A_{i}$ is the amplitude, $f_{i}$ is the
frequency, and $\phi_{i}$ is the corresponding phase.

The uncertainties all through the work were estimated by the framework
introduced in Zong et al. (2016), in which the uncertainties of amplitudes
($\sigma_{A}$) are defined as the median value of the amplitudes within a
Lomb-Scargle spectral window of 2 $\,\mathrm{c}\ \mathrm{d}^{-1}$ equally
divided by the frequency peak, while the uncertainties of the frequencies
($\sigma_{f}$) and phases ($\sigma_{\phi}$) are estimated following the
formalism proposed by Montgomery & O’Donoghue (1999); Aerts (2021).

##  Appendix B Variations in the Amplitudes and Phases

Considering the quasi-periodic signals in most of the amplitudes and phases,
which are not the major goals in this work, we fit the variation in the
amplitudes by the following formula,

$a_{A}+b_{A}\cdot t+c_{A}\cdot
t^{2}+10^{d_{A}}\cdot\sin[2\pi(\frac{1}{P_{A}}\cdot t+\phi_{A})].$ (B1)

The variation in the phases are fit by a similar formula,

$a_{\phi}+b_{\phi}\cdot t+c_{\phi}\cdot
t^{2}+10^{d_{\phi}}\cdot\sin[2\pi(\frac{1}{P_{\phi}}\cdot t+\phi_{\phi})].$
(B2)

In such case, the main quasi-periodic signals (from 200 to 800 days) can be
fully handled, and the linear and quadratic variations in the amplitudes and
phases can be obtained.

The Markov Chain Monte Carlo (MCMC) algorithm (Sharma, 2017) was used to
determine the coefficients and their uncertainties in the above expressions,
and the fitted coefficients are listed in Table 3 and 4.

The variations in amplitudes and phases (subtracted by the average values) are
presented in Figure 10, 11, 12, 13, and 14. In Figure 10, 11, 12, 13, and 14,
we find that almost all these amplitudes and frequencies are modulated by
quasi-periodic signals, which have the periods from about 300 to 760 days. It
might come from the orbital perturbation and other annual systematic
modulation from the Kepler space telescope, or the light-time perturbation
from a companion in a binary system. Because we focus on the linear variation
of the pulsation modes in this work, the detailed discussion about the quasi-
periodic signals will be proposed in our upcoming works.

Based on the Fourier-phase diagram method (Xue et al., 2018), the linear
period variation rates can be directly derived via the values of $c_{\phi}$
(which are equal to the half value of the linear frequency variation rates in
$\mathrm{c\ d}^{-2}$).

Because there exist overlaps in the time domain between the data points in
Figure 10, 11, 12, 13 and 14 (a time window of 120 days and a step of 20 days)
and these data points are not independent, the uncertainties of the parameters
in Table 3 and 4 are underestimated. Consequently, we simulated the above
process using the data points without overlaps (selecting the next point 120
days away from the last one), and find that the uncertainties of the quadratic
term coefficient ($c_{A}$ and $c_{\phi}$) are about 2-3 times larger than the
original ones. For convenience, a uniform factor $\sqrt{6}\approx 2.5$ is used
to rescale the uncertainties of $(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$ and
$\,\mathrm{d}A/\,\mathrm{d}t$ in Table 2. $(1/P)(\,\mathrm{d}P/\,\mathrm{d}t)$
and $\,\mathrm{d}A/\,\mathrm{d}t$ are also used as the criteria to mark the
amplitudes and frequencies variations in Figure 6, 7, 8, and 9.

One should note that, as the criterion of the increase or decrease in the
amplitudes, the $\,\mathrm{d}A/\,\mathrm{d}t$ in Table 2 is different from the
$b_{A}$ in Table 3. The $\,\mathrm{d}A/\,\mathrm{d}t$ (which is equivalent to
the $b_{A0}$) is obtained by the following formula:

$a_{A0}+b_{A0}\cdot t+10^{d_{A0}}\cdot\sin[2\pi(\frac{1}{P_{A0}}\cdot
t+\phi_{A0})].$ (B3)

As a comparison of the results based on short-time Fourier transformation, the
frequencies and amplitudes of the 23 pulsation modes in each year (2009-2013)
are summarized in Table 5.

Table 3: Fitting results of the amplitudes of the 23 pulsation modes. ID | $a_{A}$ | $b_{A}$ | $c_{A}$ | $d_{A}$ | $P_{A}$ | $\phi_{A}$ | $\chi^{2}/\mathrm{d.o.f.}$
---|---|---|---|---|---|---|---
F0 | $79.45\pm 0.10$ | $(1.32\pm 0.27)\times 10^{-3}$ | $(-7.40\pm 1.63)\times 10^{-7}$ | $-0.19\pm 0.02$ | $354\pm 3$ | $-0.08\pm 0.02$ | 183.89/61
F1 | $78.05\pm 0.10$ | $(2.30\pm 2.70)\times 10^{-4}$ | $(-4.18\pm 1.62)\times 10^{-7}$ | $-0.20\pm 0.02$ | $355\pm 3$ | $-0.07\pm 0.02$ | 110.41/61
F2 | $28.49\pm 0.10$ | $(-2.04\pm 2.70)\times 10^{-4}$ | $(0.85\pm 1.61)\times 10^{-7}$ | $-0.67\pm 0.05$ | $363\pm 8$ | $-0.03\pm 0.06$ | 15.50/61
F3 | $16.19\pm 0.11$ | $(1.22\pm 0.28)\times 10^{-3}$ | $(-7.51\pm 1.67)\times 10^{-7}$ | $-0.80\pm 0.08$ | $345\pm 11$ | $-0.18\pm 0.08$ | 41.55/61
F4 | $15.99\pm 0.10$ | $(4.32\pm 2.66)\times 10^{-4}$ | $(-2.80\pm 1.60)\times 10^{-7}$ | $-0.88\pm 0.09$ | $356\pm 12$ | $-0.06\pm 0.09$ | 10.09/61
F5 | $11.87\pm 0.09$ | $(-1.55\pm 0.26)\times 10^{-3}$ | $(2.22\pm 0.16)\times 10^{-6}$ | $-3.53\pm 1.44$ | $484\pm 175$ | $-0.11\pm 0.23$ | 72.28/61
F6 | $11.74\pm 0.10$ | $(1.60\pm 2.70)\times 10^{-4}$ | $(-1.96\pm 1.62)\times 10^{-7}$ | $-0.78\pm 0.07$ | $358\pm 11$ | $-0.06\pm 0.08$ | 28.88/61
F7 | $8.35\pm 0.10$ | $(6.17\pm 2.64)\times 10^{-4}$ | $(-4.51\pm 1.60)\times 10^{-7}$ | $-1.19\pm 0.53$ | $724\pm 54$ | $0.33\pm 0.09$ | 18.59/61
F8 | $6.92\pm 0.10$ | $(-1.81\pm 2.60)\times 10^{-4}$ | $(1.71\pm 1.58)\times 10^{-7}$ | $-3.09\pm 1.56$ | $503\pm 155$ | $-0.12\pm 0.21$ | 20.90/61
F9 | $4.90\pm 0.10$ | $(9.44\pm 2.62)\times 10^{-4}$ | $(-5.11\pm 1.57)\times 10^{-7}$ | $-0.84\pm 0.09$ | $375\pm 11$ | $0.18\pm 0.07$ | 34.36/61
F10 | $4.21\pm 0.10$ | $(3.28\pm 2.80)\times 10^{-4}$ | $(-3.28\pm 1.71)\times 10^{-7}$ | $-1.95\pm 1.43$ | $598\pm 134$ | $0.05\pm 0.20$ | 28.32/61
F11 | $3.03\pm 0.10$ | $(0.16\pm 2.63)\times 10^{-4}$ | $(3.49\pm 1.58)\times 10^{-7}$ | $-2.44\pm 1.61$ | $435\pm 143$ | $-0.12\pm 0.19$ | 19.65/61
F12 | $3.47\pm 0.10$ | $(-7.21\pm 2.62)\times 10^{-4}$ | $(4.37\pm 1.58)\times 10^{-7}$ | $-3.23\pm 1.52$ | $524\pm 165$ | $-0.07\pm 0.23$ | 20.93/61
F13 | $3.01\pm 0.10$ | $(-3.45\pm 2.63)\times 10^{-4}$ | $(5.36\pm 1.59)\times 10^{-7}$ | $-1.13\pm 0.20$ | $399\pm 31$ | $0.08\pm 0.12$ | 11.77/61
F14 | $3.02\pm 0.10$ | $(-7.28\pm 2.81)\times 10^{-4}$ | $(6.62\pm 1.68)\times 10^{-7}$ | $-1.16\pm 0.24$ | $466\pm 58$ | $-0.36\pm 0.14$ | 22.95/61
F15 | $2.12\pm 0.10$ | $(5.17\pm 2.71)\times 10^{-4}$ | $(0.15\pm 1.63)\times 10^{-7}$ | $-0.75\pm 0.07$ | $320\pm 6$ | $-0.42\pm 0.05$ | 31.77/61
F16 | $2.13\pm 0.10$ | $(-6.88\pm 2.69)\times 10^{-4}$ | $(7.52\pm 1.61)\times 10^{-7}$ | $-1.08\pm 0.23$ | $341\pm 19$ | $-0.35\pm 0.09$ | 43.41/61
F17 | $2.00\pm 0.10$ | $(1.68\pm 2.59)\times 10^{-4}$ | $(-0.51\pm 1.56)\times 10^{-7}$ | $-3.28\pm 1.53$ | $501\pm 176$ | $-0.11\pm 0.24$ | 17.21/61
F18 | $2.00\pm 0.10$ | $(-3.07\pm 2.66)\times 10^{-4}$ | $(2.95\pm 1.60)\times 10^{-7}$ | $-3.46\pm 1.45$ | $513\pm 170$ | $-0.10\pm 0.23$ | 16.29/61
F19 | $2.06\pm 0.10$ | $(-4.00\pm 2.63)\times 10^{-4}$ | $(2.58\pm 1.59)\times 10^{-7}$ | $-3.49\pm 1.43$ | $503\pm 172$ | $-0.12\pm 0.23$ | 23.28/61
F20 | $1.92\pm 0.10$ | $(-0.97\pm 2.65)\times 10^{-4}$ | $(0.38\pm 1.60)\times 10^{-7}$ | $-3.51\pm 1.47$ | $514\pm 171$ | $-0.10\pm 0.24$ | 14.06/61
F21 | $1.38\pm 0.10$ | $(-0.44\pm 2.60)\times 10^{-4}$ | $(0.41\pm 1.57)\times 10^{-7}$ | $-3.59\pm 1.40$ | $505\pm 173$ | $-0.10\pm 0.23$ | 14.07/61
F22 | $1.57\pm 0.10$ | $(-7.52\pm 2.74)\times 10^{-4}$ | $(5.30\pm 1.66)\times 10^{-7}$ | $-2.70\pm 1.57$ | $467\pm 169$ | $-0.09\pm 0.23$ | 38.77/61
Table 4: Fitting results of the phases (subtracted by its average) of the 23 pulsation modes. ID | $a_{\phi}$ | $b_{\phi}$ | $c_{\phi}$ | $d_{\phi}$ | $P_{\phi}$ | $\phi_{\phi}$ | $\chi^{2}/\mathrm{d.o.f.}$
---|---|---|---|---|---|---|---
F0 | $(-1.07\pm 0.20)\times 10^{-3}$ | $(3.33\pm 0.54)\times 10^{-6}$ | $(-2.00\pm 0.32)\times 10^{-9}$ | $-3.46\pm 0.07$ | $359\pm 10$ | $-0.05\pm 0.07$ | 43.53/61
F1 | $(-1.72\pm 0.20)\times 10^{-3}$ | $(4.97\pm 0.54)\times 10^{-6}$ | $(-2.83\pm 0.33)\times 10^{-9}$ | $-5.00\pm 1.04$ | $495\pm 151$ | $-0.08\pm 0.19$ | 33.30/61
F2 | $(-2.31\pm 0.54)\times 10^{-3}$ | $(6.48\pm 1.47)\times 10^{-6}$ | $(-3.72\pm 0.89)\times 10^{-9}$ | $-3.24\pm 0.14$ | $377\pm 17$ | $0.49\pm 0.10$ | 29.75/61
F3 | $(-5.83\pm 0.91)\times 10^{-3}$ | $(1.64\pm 0.25)\times 10^{-5}$ | $(-9.34\pm 1.49)\times 10^{-9}$ | $-4.88\pm 1.20$ | $506\pm 172$ | $-0.06\pm 0.21$ | 37.37/61
F4 | $(-3.40\pm 1.01)\times 10^{-3}$ | $(9.21\pm 2.78)\times 10^{-6}$ | $(-5.28\pm 1.68)\times 10^{-9}$ | $-4.72\pm 1.24$ | $498\pm 170$ | $-0.08\pm 0.20$ | 23.19/61
F5 | $(-13.50\pm 1.36)\times 10^{-3}$ | $(4.24\pm 0.36)\times 10^{-5}$ | $(-2.60\pm 0.21)\times 10^{-8}$ | $-2.94\pm 0.18$ | $324\pm 10$ | $-0.32\pm 0.08$ | 69.78/61
F6 | $(-2.33\pm 1.30)\times 10^{-3}$ | $(9.34\pm 3.55)\times 10^{-6}$ | $(-6.56\pm 2.14)\times 10^{-9}$ | $-4.97\pm 1.17$ | $493\pm 174$ | $-0.05\pm 0.20$ | 24.14/61
F7 | $(-3.82\pm 1.78)\times 10^{-3}$ | $(1.18\pm 0.48)\times 10^{-5}$ | $(-7.25\pm 2.92)\times 10^{-9}$ | $-4.91\pm 1.22$ | $500\pm 171$ | $-0.06\pm 0.20$ | 21.68/61
F8 | $(-4.78\pm 2.21)\times 10^{-3}$ | $(1.33\pm 0.61)\times 10^{-5}$ | $(-7.30\pm 3.66)\times 10^{-9}$ | $-4.85\pm 1.23$ | $518\pm 174$ | $-0.06\pm 0.20$ | 13.50/61
F9 | $(0.33\pm 3.59)\times 10^{-3}$ | $(-4.40\pm 9.36)\times 10^{-6}$ | $(2.87\pm 5.45)\times 10^{-9}$ | $-2.32\pm 0.10$ | $309\pm 12$ | $-0.41\pm 0.15$ | 39.28/61
F10 | $(-9.83\pm 3.64)\times 10^{-3}$ | $(3.15\pm 0.99)\times 10^{-5}$ | $(-1.93\pm 0.60)\times 10^{-8}$ | $-4.01\pm 1.46$ | $564\pm 160$ | $-0.13\pm 0.20$ | 19.46/61
F11 | $(-7.09\pm 4.88)\times 10^{-3}$ | $(2.28\pm 1.29)\times 10^{-5}$ | $(-1.36\pm 0.76)\times 10^{-8}$ | $-4.55\pm 1.38$ | $501\pm 166$ | $-0.04\pm 0.21$ | 15.55/61
F12 | $(-7.02\pm 4.72)\times 10^{-3}$ | $(2.23\pm 1.29)\times 10^{-5}$ | $(-1.43\pm 0.78)\times 10^{-8}$ | $-4.58\pm 1.35$ | $516\pm 170$ | $-0.04\pm 0.20$ | 13.13/61
F13 | $(-12.38\pm 5.05)\times 10^{-3}$ | $(4.57\pm 1.32)\times 10^{-5}$ | $(-3.07\pm 0.77)\times 10^{-8}$ | $-4.75\pm 1.29$ | $503\pm 175$ | $-0.05\pm 0.20$ | 16.64/61
F14 | $(-0.50\pm 6.09)\times 10^{-3}$ | $(0.88\pm 1.69)\times 10^{-5}$ | $(-0.81\pm 1.01)\times 10^{-8}$ | $-2.49\pm 0.38$ | $588\pm 80$ | $0.42\pm 0.14$ | 30.29/61
F15 | $(-22.33\pm 5.95)\times 10^{-3}$ | $(5.84\pm 1.58)\times 10^{-5}$ | $(-3.20\pm 0.94)\times 10^{-8}$ | $-3.72\pm 1.51$ | $581\pm 178$ | $-0.09\pm 0.18$ | 35.60/61
F16 | $(10.85\pm 7.11)\times 10^{-3}$ | $(-6.06\pm 1.88)\times 10^{-5}$ | $(4.56\pm 1.12)\times 10^{-8}$ | $-2.13\pm 0.32$ | $761\pm 38$ | $-0.12\pm 0.06$ | 22.16/61
F17 | $(6.64\pm 7.44)\times 10^{-3}$ | $(-1.83\pm 2.00)\times 10^{-5}$ | $(1.01\pm 1.20)\times 10^{-8}$ | $-4.82\pm 1.27$ | $497\pm 173$ | $-0.05\pm 0.20$ | 5.48/61
F18 | $(-4.89\pm 7.59)\times 10^{-3}$ | $(1.52\pm 2.06)\times 10^{-5}$ | $(-0.90\pm 1.23)\times 10^{-8}$ | $-4.62\pm 1.36$ | $499\pm 172$ | $-0.03\pm 0.20$ | 14.21/61
F19 | $(-2.97\pm 7.67)\times 10^{-3}$ | $(1.32\pm 2.11)\times 10^{-5}$ | $(-0.96\pm 1.29)\times 10^{-8}$ | $-4.64\pm 1.33$ | $490\pm 174$ | $-0.05\pm 0.20$ | 13.31/61
F20 | $(-8.05\pm 8.22)\times 10^{-3}$ | $(2.48\pm 2.23)\times 10^{-5}$ | $(-1.46\pm 1.34)\times 10^{-8}$ | $-4.67\pm 1.36$ | $511\pm 173$ | $-0.06\pm 0.20$ | 9.56/61
F21 | $(-1.36\pm 11.10)\times 10^{-3}$ | $(0.17\pm 3.03)\times 10^{-5}$ | $(-0.12\pm 1.83)\times 10^{-8}$ | $-4.42\pm 1.46$ | $517\pm 171$ | $-0.07\pm 0.20$ | 12.13/61
F22 | $(49.17\pm 12.24)\times 10^{-3}$ | $(-15.47\pm 3.37)\times 10^{-5}$ | $(9.56\pm 2.01)\times 10^{-8}$ | $-1.89\pm 0.15$ | $525\pm 31$ | $0.19\pm 0.10$ | 26.45/61
Table 5: Frequencies and amplitudes of the 23 pulsation modes in the year from
2009 to 2013. Frequency is in $\,\mathrm{c}\ \mathrm{d}^{-1}$; Amplitude is in
mmag.

| 2009 (Q0-Q3) | 2010 (Q4-Q7) | 2011 (Q8-Q10) | 2012 (Q11-Q14) | 2013 (Q15-Q17)
---|---|---|---|---|---
ID | Frequency | Amplitude | Frequency | Amplitude | Frequency | Amplitude | Frequency | Amplitude | Frequency | Amplitude
F0 | $4.90984\pm 0.00002$ | $79.4\pm 0.6$ | $4.909843\pm 0.000007$ | $80.0\pm 0.4$ | $4.90984\pm 0.00001$ | $80.0\pm 0.5$ | $4.909844\pm 0.000008$ | $79.9\pm 0.4$ | $4.90985\pm 0.00002$ | $80.0\pm 0.6$
F1 | $6.43189\pm 0.00001$ | $77.8\pm 0.4$ | $6.431890\pm 0.000005$ | $78.1\pm 0.2$ | $6.431887\pm 0.000008$ | $77.8\pm 0.3$ | $6.431884\pm 0.000005$ | $77.7\pm 0.3$ | $6.43188\pm 0.00001$ | $77.6\pm 0.4$
F2 | $11.34175\pm 0.00002$ | $28.4\pm 0.2$ | $11.341738\pm 0.000008$ | $28.4\pm 0.2$ | $11.34173\pm 0.00001$ | $28.4\pm 0.2$ | $11.341727\pm 0.000009$ | $28.4\pm 0.2$ | $11.34173\pm 0.00002$ | $28.4\pm 0.3$
F3 | $1.52203\pm 0.00003$ | $16.2\pm 0.2$ | $1.52204\pm 0.00001$ | $16.6\pm 0.1$ | $1.52204\pm 0.00002$ | $16.6\pm 0.2$ | $1.52203\pm 0.00001$ | $16.6\pm 0.1$ | $1.52205\pm 0.00004$ | $16.5\pm 0.3$
F4 | $9.81969\pm 0.00002$ | $16.0\pm 0.1$ | $9.819694\pm 0.000009$ | $16.15\pm 0.09$ | $9.81968\pm 0.00002$ | $16.1\pm 0.1$ | $9.81969\pm 0.00001$ | $16.1\pm 0.1$ | $9.81969\pm 0.00003$ | $16.0\pm 0.2$
F5 | $8.03546\pm 0.00003$ | $11.5\pm 0.1$ | $8.03545\pm 0.00001$ | $11.68\pm 0.07$ | $8.03540\pm 0.00002$ | $12.2\pm 0.1$ | $8.03540\pm 0.00001$ | $13.24\pm 0.09$ | $8.03547\pm 0.00003$ | $14.5\pm 0.2$
F6 | $12.86381\pm 0.00002$ | $11.8\pm 0.1$ | $12.86379\pm 0.00001$ | $11.77\pm 0.08$ | $12.86377\pm 0.00002$ | $11.8\pm 0.1$ | $12.86378\pm 0.00001$ | $11.61\pm 0.08$ | $12.86383\pm 0.00003$ | $11.5\pm 0.2$
F7 | $16.25157\pm 0.00003$ | $8.45\pm 0.09$ | $16.25158\pm 0.00001$ | $8.57\pm 0.08$ | $16.25158\pm 0.00002$ | $8.57\pm 0.09$ | $16.25158\pm 0.00001$ | $8.43\pm 0.08$ | $16.25166\pm 0.00004$ | $8.6\pm 0.1$
F8 | $7.95391\pm 0.00004$ | $7.0\pm 0.1$ | $7.95394\pm 0.00001$ | $6.84\pm 0.07$ | $7.95394\pm 0.00003$ | $6.9\pm 0.1$ | $7.95392\pm 0.00002$ | $7.01\pm 0.07$ | $7.95396\pm 0.00005$ | $7.0\pm 0.2$
F9 | $3.38771\pm 0.00008$ | $5.0\pm 0.2$ | $3.38779\pm 0.00003$ | $5.23\pm 0.09$ | $3.38777\pm 0.00006$ | $5.4\pm 0.1$ | $3.38780\pm 0.00003$ | $5.3\pm 0.1$ | $3.3878\pm 0.0001$ | $5.4\pm 0.2$
F10 | $17.77368\pm 0.00004$ | $4.24\pm 0.07$ | $17.77358\pm 0.00003$ | $4.29\pm 0.07$ | $17.77363\pm 0.00003$ | $4.29\pm 0.07$ | $17.77360\pm 0.00003$ | $4.12\pm 0.07$ | $17.77367\pm 0.00008$ | $4.2\pm 0.1$
F11 | $12.94528\pm 0.00006$ | $3.06\pm 0.08$ | $12.94523\pm 0.00003$ | $3.15\pm 0.07$ | $12.94531\pm 0.00005$ | $3.30\pm 0.08$ | $12.94523\pm 0.00003$ | $3.56\pm 0.06$ | $12.94522\pm 0.00009$ | $3.9\pm 0.1$
F12 | $22.68354\pm 0.00005$ | $3.32\pm 0.06$ | $22.68347\pm 0.00003$ | $3.24\pm 0.07$ | $22.68341\pm 0.00005$ | $3.15\pm 0.07$ | $22.68347\pm 0.00003$ | $3.22\pm 0.06$ | $22.68343\pm 0.00009$ | $3.3\pm 0.1$
F13 | $14.46749\pm 0.00006$ | $2.88\pm 0.07$ | $14.46735\pm 0.00003$ | $3.01\pm 0.07$ | $14.46727\pm 0.00005$ | $3.14\pm 0.07$ | $14.46730\pm 0.00003$ | $3.33\pm 0.06$ | $14.46745\pm 0.00008$ | $3.7\pm 0.1$
F14 | $1.6038\pm 0.0001$ | $2.9\pm 0.2$ | $1.60347\pm 0.00005$ | $2.76\pm 0.09$ | $1.6036\pm 0.0001$ | $2.9\pm 0.1$ | $1.60349\pm 0.00005$ | $3.1\pm 0.1$ | $1.6035\pm 0.0002$ | $3.2\pm 0.2$
F15 | $3.1255\pm 0.0002$ | $2.4\pm 0.2$ | $3.12562\pm 0.00006$ | $2.28\pm 0.09$ | $3.1256\pm 0.0001$ | $2.7\pm 0.1$ | $3.12550\pm 0.00005$ | $2.7\pm 0.1$ | $3.1256\pm 0.0002$ | $2.9\pm 0.2$
F16 | $3.3060\pm 0.0002$ | $2.2\pm 0.2$ | $3.30620\pm 0.00007$ | $1.90\pm 0.09$ | $3.3063\pm 0.0001$ | $2.0\pm 0.1$ | $3.30632\pm 0.00006$ | $2.4\pm 0.1$ | $3.3063\pm 0.0002$ | $2.6\pm 0.2$
F17 | $14.72953\pm 0.00008$ | $1.98\pm 0.06$ | $14.72952\pm 0.00005$ | $2.11\pm 0.07$ | $14.72961\pm 0.00007$ | $2.05\pm 0.07$ | $14.72956\pm 0.00004$ | $2.16\pm 0.06$ | $14.7295\pm 0.0001$ | $2.2\pm 0.1$
F18 | $9.5574\pm 0.0001$ | $1.94\pm 0.08$ | $9.55751\pm 0.00004$ | $1.91\pm 0.05$ | $9.55733\pm 0.00009$ | $1.91\pm 0.08$ | $9.55746\pm 0.00004$ | $2.09\pm 0.06$ | $9.5576\pm 0.0002$ | $2.1\pm 0.1$
F19 | $19.29575\pm 0.00008$ | $1.97\pm 0.06$ | $19.29574\pm 0.00006$ | $1.91\pm 0.07$ | $19.29569\pm 0.00008$ | $1.90\pm 0.07$ | $19.29567\pm 0.00005$ | $1.99\pm 0.06$ | $19.2957\pm 0.0002$ | $1.9\pm 0.1$
F20 | $24.20562\pm 0.00007$ | $1.84\pm 0.06$ | $24.20550\pm 0.00005$ | $1.89\pm 0.07$ | $24.20554\pm 0.00008$ | $1.76\pm 0.07$ | $24.20549\pm 0.00005$ | $1.87\pm 0.06$ | $24.2055\pm 0.0001$ | $2.0\pm 0.1$
F21 | $21.1616\pm 0.0001$ | $1.37\pm 0.06$ | $21.16138\pm 0.00008$ | $1.32\pm 0.07$ | $21.1613\pm 0.0001$ | $1.40\pm 0.07$ | $21.16142\pm 0.00006$ | $1.36\pm 0.06$ | $21.1616\pm 0.0002$ | $1.4\pm 0.1$
F22 | $3.0438\pm 0.0003$ | $1.3\pm 0.2$ | $3.0441\pm 0.0001$ | $1.31\pm 0.09$ | $3.0443\pm 0.0002$ | $1.3\pm 0.1$ | $3.0440\pm 0.0001$ | $1.43\pm 0.09$ | $3.0436\pm 0.0003$ | $1.6\pm 0.2$

##  Appendix C interaction diagram

The interaction diagram is a useful tool to show the global interaction
structures between the pulsation modes based on their amplitude, frequency,
and phase variations, and clusters the modes with similar behaviors together
when they interact with all the other modes.

The interaction strength of a characteristic quantity
(amplitude/frequency/phase) between two pulsation modes is measured by the
correlation coefficient of their variation over time. In this work, we use the
Spearman correlation coefficient (which is a measure of how well the
relationship between two variables can be described by a monotonic function)
rather than the Pearson correlation coefficient because of (i) the dataset is
nonnormal; (ii) it is more robust for the outliers; (iii) it is more suitable
for the measurement of the nonlinear relations, which focus on the
relationships of the trends rather than the magnitudes of the variations.

The correlation coefficients of the $\text{F}i\ (0\leq i\leq
n-1,i\in\mathbb{Z})$ mode with all the $n$ pulsation modes (including itself)
form a vector
($\mathbf{k}^{i}=(\rho^{i}_{0},\rho^{i}_{1},\rho^{i}_{2},...,\rho^{i}_{n-1}),\
(0\leq i\leq n-1,i\in\mathbb{Z})$) in the interaction space, which is a
$n$-dimensional Euclid space spanned by $n$ unit vectors
$\mathbf{e}_{i}=(0,...,1^{(i\text{-th})},...,0),\ (0\leq i\leq
n-1,i\in\mathbb{Z})$.131313These $\mathbf{e}_{i}$ present the $n$ imaginary
pulsation modes independent with each other. Based on the above structures,
one can use the Euclidean distance between the $\mathbf{k}^{i}$s to measure
the similarities of each pulsation modes in the interaction space. Moreover,
the agglomerative hierarchical clustering (AHC) process (Murtagh & Contreras,
2012) can be performed to cluster the pulsation modes who have similar
behaviors in the interaction space together (i.e., group a set of modes in
such a way that modes in the same group (called a cluster) are more similar
(in some sense) to each other than to those in other groups (clusters)).

The color of a small square represents the correlation coefficient of the
characteristic quantity between the two labeled pulsation modes whose column
and row intersect at the square. The dendrograms on the upper and left
represent the AHC process in the interaction space, which reorders the
pulsation modes in order to cluster those who have similar behaviors together.

Figure 15: Interaction diagram of phases of the 23 pulsation modes.

All the relationships and structures shown in the interaction diagram of
amplitudes indicate that there exist interactions or energy transfer between
the independent pulsation modes and their harmonics/combinations. Moreover,
the interaction diagram of amplitudes help us find the particularity of
$f_{2}$ related modes, discover their rotationally locked modes, and identify
$f_{2}$ as a nonradial $l=1,m=1$ mode, which indicates its potential in future
research. On the other hand, comparing with the complicated interactions shown
in Figure 1, the Interactions Diagram of the phases (see Figure 15) shows that
almost all the pulsation modes have the same variation trend except for the
F16, F22, and F17 modes, which are caused by the decreasing trend of their
periods. Excluding the period variation rate of F17 with large uncertainty,
that of F16 can be interpreted by its $-f_{2}$ component and that of F22 could
be interpreted by its interaction with F15.

##  Appendix D Theoretical Model Calculation

In order to determine the stellar mass and evolutionary stage based on the
single-star evolutionary models, the open-source 1D stellar evolution code
Modules for Experiments in Stellar Astrophysics (MESA, r15140, Paxton et al.
(2011, 2013, 2015, 2018, 2019)) was used to construct the structural and
evolutionary models. At each step along with the evolutionary tracks, the
pulsation frequencies of the specific structure were calculated by the stellar
oscillation code GYRE (Townsend & Teitler, 2013; Townsend et al., 2018;
Goldstein & Townsend, 2020).

The initial parameters used to construct pre-main sequence evolutionary models
of KIC 6382916 were configured as follows. Different metallicity [Fe/H] with
values of $0.040$ (Mathur et al., 2017), $0.111$ (Xiang et al., 2019), and
$0.266$ (Luo et al., 2019) dex were considered as the initial metallicity of
the evolutionary models. The following formulas were used to calculate the
initial heavy element abundance $Z$ and initial hydrogen abundance $X$:

$\mathrm{[Fe/H]}=\log\frac{Z}{X}-\log\frac{Z_{\odot}}{X_{\odot}},$ (D1)
$Y=0.24+3Z,$ (D2) $X+Y+Z=1,$ (D3)

where $X_{\odot}=0.7381$ and $Z_{\odot}=0.0134$ (Asplund et al., 2009).
Eq.(D2) was provided by Mowlavi et al. (1998). Based on the given values of
$\mathrm{[Fe/H]}$, we got ($X=0.704$, $Z=0.014$), ($X=0.695$, $Z=0.016$), and
($X=0.672$, $Z=0.022$), as the initial inputs of the evolutionary models. At
the same time, the values of the option Zbase (which provides the reference
metallicity necessary to calculate element variations) were set to be the
initial metallicity of the star (0.014, 0.016, and 0.022, respectively), and
initial_zfracs = ’AGSS09_zfracs’ was selected. For the opacity,
kap_file_prefix = ’a09’, kap_CO_prefix = ’a09_co’ and kap_lowT_prefix =
’lowT_fa05_a09p’ were selected.

The initial mass of the models was set in the interval from 1.5
$\mathrm{M_{\odot}}$ to 2.5 $\mathrm{M_{\odot}}$ with a step of $0.01\
\mathrm{M_{\odot}}$, covering the typical mass range of HADS. The rotation of
the star was not considered in the model calculation because of the relative
slow rotation of the star ($\Delta\omega=0.0815$). The mixing-length parameter
was set to be a value of $\alpha_{\rm{MLT}}=1.89$ (Niu et al., 2017). The
exponential scheme proposed in Herwig (2000) was adopted to account for the
overshooting, and the overshooting parameter was set to the fixed value
$f_{\mathrm{ov}}=0.015$ (Magic et al., 2010).

All the evolutionary tracks were calculated from the pre-MS to the red giant
branch.

Based on the pulsation frequencies calculated in every step along with the
evolutionary tracks, we got the best-fit seismic models (which have the
smallest $\chi^{2}$ with respect to the observed values of $f_{0}$ and
$f_{1}$) with the different $Z$ values (see in Table 1 for detailed
information). The H-R diagram, $\log\,T_{\mathrm{eff}}$ vs $\log g$ diagram,
and Petersen diagram are shown in Figure 2, Figure 16 and Figure 17,
respectively.

Figure 16: $\log\,T_{\mathrm{eff}}$ vs. $\log g$ diagram with all the
evolutionary tracks and the best-fit seismic models of KIC 6382916 with
different $Z$ values. The colored solid evolutionary tracks show from the
zero-age MS to the post-MS evolutionary stage, with initial masses from 1.50
$M_{\odot}$ to 2.50 $M_{\odot}$. The regions surrounded by the black
rectangular boxes are selected to zoom in the best-fit seismic models, which
are represented by the red stars. The colored dash lines present the
evolutionary tracks with initial mass steps of 0.01 $M_{\odot}$.

Figure 17: Petersen diagram with all the evolutionary tracks and the best-fit
seismic models of KIC 6382916 with different $Z$ values. The colored solid
evolutionary tracks show the stages from the zero-age MS to the post-MS
evolutionary stage, with initial masses from 1.50 $M_{\odot}$ to 2.50
$M_{\odot}$. The regions surrounded by the black rectangular boxes are
selected to zoom in the best-fit seismic models, which are represented by the
red stars. The colored dashed lines present the evolutionary tracks with
initial mass steps of 0.01 $M_{\odot}$.

##  Appendix E Variation of $\Delta\omega$

Based on the results in Table 4, five pairs (F3-F14, F5-F8, F6-F11, F9-F16,
and F15-F22) can be used to calculate the variation rate of the rotation
frequency ($\,\mathrm{d}\Delta\omega/\,\mathrm{d}t$), which can be obtained by
the differences in the frequency variation rates ($2\cdot c_{\phi}$) in each
of the pairs.

At the same time, all the above pairs follow the frequency relation

$\Delta\omega=f_{0}-2f_{1}+f_{2},$ (E1)

which can be differentiated on both sides with respect to time:

$\frac{\,\mathrm{d}\Delta\omega}{\,\mathrm{d}t}\equiv\dot{\Delta\omega}=\dot{f}_{0}-2\dot{f}_{1}+\dot{f}_{2}.$
(E2)

$\dot{f}_{0}$, $\dot{f}_{1}$, and $\dot{f}_{2}$ can also be found in Table 4.

About the uncertainty estimation, all the related frequency variation rates
were taken as independent variables, and a uniform factor $\sqrt{6}\approx
2.5$ was used to rescale the uncertainties as that in Appendix B. The
numerical results were collected in Table 6.

For the best-fit seismic models of different values of $Z$, the variation
rates of the rotation frequency were calculated based on an assumption of the
conservation of angular momentum in this evolutionary phase and a uniform
rotation of the star, whose numerical results were also collected in Table 6.

Table 6: Variation rate of $\Delta\omega$. Origins | $\,\mathrm{d}\Delta\omega/\,\mathrm{d}t\ (\mathrm{c\ d^{-2}})$
---|---
F3-F14 | $-(3\pm 50)\times 10^{-9}$
F5-F8 | $-(37\pm 20)\times 10^{-9}$
F6-F11 | $-(14\pm 40)\times 10^{-9}$
F9-F16 | $-(85\pm 63)\times 10^{-9}$
F15-F22 | $-(255\pm 110)\times 10^{-9}$
F0-F1-F5 | $-(45\pm 10)\times 10^{-9}$
Model (Z=0.014) | $-9\times 10^{-10}$
Model (Z=0.016) | $-1.1\times 10^{-9}$
Model (Z=0.022) | $-1.4\times 10^{-9}$