[1,2]William M. Baker

1]Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge, CB3 OHA, UK

2]Cavendish Laboratory - Astrophysics Group, University of Cambridge, 19 JJ
Thomson Avenue, Cambridge, CB3 OHE, UK

3]Department of Physics and Astronomy, University College London, Gower
Street, London WC1E 6BT, UK

4]Department of Physics, Florida International University, 11200 SW 8th
Street, Miami, FL, USA

5]INAF— Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125,
Florence, Italy

6]European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748
Garching bei Muenchen, Germany

7]Cahill Center for Astronomy and Astrophysics, California Institute of
Technology, Pasadena, CA 91125, USA

8]Jodrell Bank Centre for Astrophysics, University of Manchester, Oxford Road,
Manchester M13 9PL, UK

# The black hole mass metallicity relation and insights into galaxy quenching

<EMAIL_ADDRESS>Roberto Maiolino  Asa F. L. Bluck  Francesco Belfiore  Mirko
Curti  Francesco D’Eugenio  Joanna M. Piotrowska  Sandro Tacchella  James A.
A. Trussler  [  [  [  [  [  [  [  [

###### Abstract

One of the most important questions in astrophysics is what causes galaxies to
stop forming stars. Previous studies have shown a tight link between
quiescence and black hole mass. Other studies have revealed that quiescence is
also associated with ‘starvation’, the halting of gas inflows, which results
in the remaining gas being used up rapidly by star formation and in rapid
chemical enrichment. In this work we find the final missing link between these
two findings. Using a large sample of galaxies, we uncover the intrinsic
dependencies of the stellar metallicity on galaxy properties. In the case of
the star-forming galaxies, the stellar metallicity is driven by stellar mass.
However, for passive galaxies the stellar metallicity is primarily driven by
the black hole mass, as traced by velocity dispersion. This result finally
reveals the connection between previous studies, where the integrated effect
of black hole feedback prevents gas inflows, starving the galaxy, which is
seen by the rapid increase in the stellar metallicity, leading to the galaxy
becoming passive.

###### keywords:

Galaxies: abundances, Galaxies: evolution, Galaxies: stellar content,
Galaxies: black holes, Galaxies: passive

## Introduction

The chemical enrichment of galaxies is a powerful proxy of their history of
star formation and of the evolutionary processes and mechanisms that have
shaped them. This chemical enrichment is traced by the ‘metallicity’, the
ratio of heavier elements to the Hydrogen and Helium content of either gas or
stars. Metals are produced by stellar nucleosynthesis are then released via
supernovae explosions and stellar winds, hence increasing the metallicity of
the ISM [1]. However, the metallicity also traces other processes, such as
inflows of low-metallcity gas (causing metal dilution) and outflows (often
metal loaded [2, 3]).

Depending on the galaxy type and on the available data, the metallicity in
galaxies can be measured for the gas-phase of the ISM, and for the stellar
population. Both of these metallicities follow well known scaling relations
with other galactic properties. Specifically, the metallicity is known to
increase with the stellar mass of a galaxy, up until $\rm
M_{*}=10^{10.5}M_{\odot}$ where it starts to plateau, in a relationship known
as the mass-metallicity relation (MZR, [4, 5, 1]). At fixed mass the
metallicity is also observed to decrease with increasing star-formation rate
(i.e. anti-correlate), leading to an overall three-dimensional relationship
between the quantities, known as the Fundamental Metallicity Relation (FMR,
[6, 7, 8]).

The advantage of studying stellar metallicities is that they can be estimated
for both star-forming and passive galaxies, whereas, due to being calculated
from emission lines, gas-phase metallicities can only accurately be obtained
for star-forming galaxies (although see also [9]). Star-forming galaxies and
passive galaxies show large differences in stellar metallicity with passive
galaxies having larger values at a given stellar mass [10, 11, 12]. This is
also seen as a continuous sequence in [8], where galaxies with the lower SFR
(i.e. more passive) have higher stellar metallicity. This trend is interpreted
as quenching via starvation, where galaxies quench because their gas supply
(via inflows) is cut off [10, 11]. Starvation implies that the remaining gas
of the galaxy is converted into stars in a closed-box manner, resulting in the
metallicity rising steeply because it is not diluted from accreting (low-
metallicity) gas. However, even if starvation is identified as playing a key
role in galaxy quenching, its causes could not be identified based solely on
the metallicity analysis.

Many different causes for the quenching of star-formation have been proposed
in the past, such as, AGN feedback, morphological quenching, cosmic rays,
magnetic fields, and environmental effects [13]. Of particular interest, are
the recent results of [14, 15, 16] where they find that the strongest link to
quenching in a galaxy is the mass of the supermassive black hole at the centre
(this applies for central galaxies or massive satellites). This has led [15]
to conclude that it is the integrated effect of black hole accretion feedback
that causes quenching. The proposed scenario, also supported by simulations
[14], is that black hole accretion heats the circum-galactic medium (via
radio-jets or outflows) hence preventing accretion of cold gas, and resulting
in delayed quenching via starvation.

A key difficulty when investigating relationships between quantities that
could cause quenching is that many of them are inter-correlated, hence
correlation between quiescence and some galaxy properties does not necessarily
imply causation. In the recent works discussed above [14, 15, 16] this issue
was tackled by using machine learning (random forest) and partial correlation
techniques, to identify direct and primary connections from indirect relations
induced by spurious connections resulting from the fact that most galactic
properties are intercorrelated. The same issue applies also for the
metallicity scaling relations: it is hard to separate intrinsic fundamental
relationships from secondary relations resulting from indirect byproducts.
Within this context [17, 18, 7, 19] utilised multiple techniques to break
these inter-correlations and determine the fundamental dependencies of gas-
phase metallicities and other scaling relations.

In this paper we extend those studies to explore the intrinsic dependence of
stellar metallicity as a function of galactic properties, finding the final
connection between the mechanism of quenching, starvation, and the cause,
integrated black hole feedback.

## Definition of the sample

Figure 1:  The importance of various galactic properties in determining the
stellar metallicity (Z) for star-forming (left panel) and passive (right
panel) galaxies. The galactic parameters evaluated are: the stellar mass
($M_{*}$), the star formation rate (SFR), the black hole mass ($M_{\rm
BH}(\sigma_{c})$, as inferred from the central velocity dispersion), the
gravitational potential (traced by $\phi_{g}=M_{*}/r_{e}$), a control uniform
random variable (R), the overdensity of galaxies (1+$\delta$), and the halo
mass ($M_{H}$). The left hand panel shows that for star-forming galaxies the
primary dependence of stellar metallicity is on stellar mass via the mass-
metallicity relation (MZR). However, for passive galaxies (right hand panel)
the stellar metallicity dependence is almost purely on black hole mass.

To investigate the dependencies of the stellar metallicity on galaxy
parameters, we use galaxies from the Sloan Digital Sky Survey (SDSS) data
release 7 [20], with data on their stellar masses, star-formation rates,
stellar metallicities, halo masses, overdensities and velocity dispersions
(for more details see Methods section 1.1). To ensure reliable stellar
metallicity measurements, we employ a strict median signal to noise cut of 20
per spectral pixel [as in 11]. We also limit the sample to those galaxies with
reliable redshifts in the range $0.02\leq z\leq 0.085$ to both reduce aperture
effects (i.e. the projected size of the SDSS fibre should be larger than 1
kpc) and reduce the effects of cosmological evolution [again, as in 11].

We divide our sample of galaxies into star-forming and passive based on their
distance from the main sequence (using the MS definition of [21]). Star-
forming galaxies are those with $\rm\Delta MS\geq-0.75$, whilst passive
galaxies are those with $\rm\Delta MS\leq-0.75$. This gives us 16505 star-
forming galaxies and 37241 passive galaxies.

Our primary analysis tool is random forest (RF) regression, a supervised
machine learning algorithm that utilises the power of many different decision
trees to accurately and simultaneously assess the contribution of several
parameters in determining a target feature. In this case the target feature is
the stellar metallicity of a galaxy. The parameters that we explore are:
stellar mass ($M_{*}$), star formation rate (SFR), the gravitational potential
as approximately traced by $\Phi_{g}=M_{*}/R_{e}$, where $R_{e}$ is the
circularized half-light radius in the r-band, the mass of the dark matter halo
($M_{H}$) and the strength of the overdensity of galaxies (1+$\delta$). Among
the parameters we also include the central black hole mass as traced by the
central velocity dispersion ($M_{\rm BH}=M_{\rm BH}(\sigma_{c})$). Various
authors have shown that the central velocity dispersion is tightly linked to
the central black hole mass [22, 14], although we note that the central
velocity dispersion might correlate with the dynamical mass or gravitational
potential. However, later on we will discuss (and exclude) that neither
dynamical mass nor gravitational potential play a role in determining the
stellar metallicity. Finally, we also include a uniform random variable as a
control. Summarising, with our analysis we can explore whether the stellar
metallicity depends on internal properties of the individual galaxy (e.g.
stellar mass, black hole mass, SFR) or environmental properties (e.g. halo
mass, overdensity strength). We apply the random forest regression separately
to our star forming and passive samples. For each of these, half the sample is
used to train the random forest while the other half is used to test the
predictions.

### The key parameters according to Random Forest regression

Fig. 1 is a bar-chart showing the relative importance of the various
quantities in determining the stellar metallicities, as inferred from the
random forest analysis, for star-forming galaxies (left panel, blue) and
passive galaxies (right panel, red). For the star-forming galaxies, the
stellar metallicity is primarily driven by stellar mass (i.e. the MZR) with a
secondary contribution from the SFR (i.e. the FMR, as in [8]). However, for
passive galaxies the stellar metallicity is primarily driven by the black hole
mass (as traced by $\sigma_{c}$) with a small secondary importance on $M_{*}$
and SFR. Hence, the relative importance between stellar mass and black hole
mass swap depending on whether the galaxies are star-forming or passive.

One key aspect to note is that we see little to no dependence on
$\Phi_{g}=M_{*}/R_{e}$, which is a proxy of the gravitational potential - this
indicates the importance of black hole mass as traced by $\sigma_{c}$ is not
simply linked to the gravitational potential instead. Additionally, later on
we use two smaller samples, but with spatially resolved spectroscopic
information from the MaNGA survey [23], for which the dynamical masses and
gravitational potential could be estimated accurately. That analysis will show
that neither the dynamical mass nor the gravitational potential of the galaxy
drive the stellar metallicity, while also in that sample, black hole mass is
the primary driver.

Finally, Fig. 1 shows that there is no significant dependence of the stellar
metallicity on environment ($M_{H}$ or $(1+\delta)$), for either the star-
forming or passive galaxies. In the Methods section we also verify that this
result holds also when separating the sample between central and satellite
galaxies.

Figure 2: Black hole mass versus stellar mass colour coded by median stellar
metallicity for star-forming galaxies (left panel) and passive galaxies (right
panel). The arrow angle points in the direction of the greatest increasing
gradient of the stellar metallicity (i.e. it quantifies the colour-shading).
The arrows and colour-shading gradients show that, in the left panel for star-
forming galaxies, increasing stellar metallicity is linked to increasing
stellar mass (i.e. the arrow angle is closer to the horizontal), whilst in the
right panel for passive galaxies, the stellar metallicity is linked more to
increasing black hole mass (the arrow angle is closer to vertical than
horizontal). This means that star-forming galaxies’ stellar metallicity shows
a stronger dependence on stellar mass than black hole mass, whilst for passive
galaxies this is reversed.

### Direction of the relationship - partial correlation coefficients

We next investigate more visually the observed dependence of stellar
metallicity on stellar and black hole mass in star-forming and passive
galaxies, respectively, by reducing the dimensionality of the problem,
specifically using ‘simple’ 3D diagrams [24], which also allow us to assess
the sign of the relation, i.e. whether positive or negative, which is not
possible via the random forest. Figure 2 shows black hole mass (as traced by
$\sigma_{c}$) versus stellar mass, colour-coded by the median stellar
metallicity in each bin for star-forming galaxies (left panel) and passive
galaxies (right panel). Contours show the underlying density distribution of
galaxies, where each outer density contour encloses 90% of the galaxy
population. The arrow angle shows the direction of the steepest metallicity
gradient (i.e. following the colour shading) and is calculated (from the
horizontal) by the ratio of the partial correlation coefficients between the
stellar metallicity, stellar mass and black hole mass. The left panel shows
that both the arrow and the colour-shading follow the stellar mass (x axis)
more than the black hole mass (y axis) meaning that the primary driver in
increasing the stellar metallicity of star-forming galaxies is the stellar
mass. However, the right panel shows the opposite, where the stellar
metallicity gradient traced by arrow and color shading follow more closely the
black hole mass (y axis) than stellar mass (x axis). This confirms the result
of the random forest in a more visual way. More importantly, these diagrams
provide the direction of this correlation, i.e. for passive galaxies the
greater the black hole mass, the greater the stellar metallicity.

We note that in these 3D diagrams we have reduced the dimensionality of the
problem, by considering the dependence of the stellar metallicity only on
stellar and black hole mass. This means that the role of additional secondary
parameters (SFR, $\phi_{g}$, MH, $(1+\delta)$) is somehow taken and embedded
in the dependences on stellar and black hole mass, and this helps explain the
fact that the partial correlation arrow and colour shading in Figure 2 does
not exactly reproduce the (stronger) relative role of M∗ and MBH that one
would expect from the random forest analysis (Fig.1).

Figure 3: Upper: stellar metallicity versus black hole mass binned by tracks
of stellar mass for star-forming galaxies (left, blue) and passive galaxies
(right, red). Lower: stellar metallicity versus stellar mass binned by tracks
of black hole mass, again for star-forming (left, blue) and passive galaxies
(right, red). The left hand plots show that the evolution in stellar
metallicity for star-forming glaxies is primarily driven by stellar mass,
whilst the right hand plots show that for passive galaxies the stellar
metallicity is primarily driven by black hole mass.

### The Black Hole-Metallicity Relation

Fig. 3 provides yet another visualisation of these relative dependencies, by
showing 2D cuts of the 3D surface shown in Fig.2. Specifically, Fig. 3 shows
the stellar metallicity versus black hole mass binned by tracks of stellar
mass for star-forming (upper left) and passive (upper right) galaxies, and
stellar metallicity versus stellar mass binned by tracks of black hole mass
(lower left and right, same format). The upper figures show that, there is a
tight correlation between stellar metallicity and black hole mass (BHZR) for
passive galaxies, where the change in stellar mass between the tracks barely
affects the relationship. On the contrary, for star-forming galaxies the
change in stellar mass results in a significant offset of the tracks. The
lower figures show that this is reversed for stellar metallicity versus
stellar mass whilst binning in tracks of black hole mass. Summarising, for
star forming galaxies at a given BH mass the metallicity depends strongly on
stellar mass, while at a given stellar mass the stellar metallicity depends
little on BH mass. These dependencies are swapped for passive galaxies: at a
given BH mass the stellar metallicity does not depend on the stellar mass,
while at any given stellar mass the stellar metallicity depends strongly on BH
mass.

This result confirms that for star-forming galaxies we recover the standard
MZR. However, for passive galaxies we find strong evidence for a Black Hole-
Metallicity Relation (BHZR), between black hole mass and stellar metallicity.

Figure 4: Stellar metallicity versus black hole mass (as traced by central
velocity dispersion) for passive galaxies. The outer density contour contains
90% of the galaxy population. The black squares show the median values in bins
of black hole mass with the errorbars corresponding to the 16th and 84th
percentiles of the distributions in each bin. The red line is the best-fit to
these bins, showing the black hole metallicity relation (BHZR).

We quantify this Black Hole-Metallicity Relation (BHZR) with the following
linear fit in logarithmic quantities: $\rm
log(Z/Z_{\odot})=0.106^{+0.031}_{-0.031}\,[log(M_{BH}/M_{\odot})\,-\,8]\,+\,0.106^{+0.033}_{-0.033}$
by using LMFIT [25]. Fig. 4 shows the best fit to the BHZR. We fit to the
median bins of the BHZR so as to avoid being heavily biased by the
overpopulated centre of the contours. This means that according to the fit, a
passive galaxy with $\rm log(M_{BH}/M_{\odot})=8$ would be predicted to have
stellar metallicity of $\rm log(Z/Z_{\odot})=0.106$.

## Is it black hole mass or gravitational potential?

Figure 5: Importance of various galactic properties in determining the stellar
metallicity for star-forming (left panels) and passive (right panels) galaxies
for two samples from the MaNGA survey ([26] top panels, [27] bottom panels),
for which spatially resolved spectroscopy provides accurate dynamical masses
and gravitational potentials. Left panels: star forming galaxies; right
panels: passive galaxies. The following galactic parameters are considered:
stellar mass ($M_{*}$), black hole mass as traced by $\sigma_{c}$, star
formation rate (SFR), the dynamical mass, a control uniform random variable
(Random), gravitational potential traced by $\rm\phi_{g}=M_{dyn}/r_{e}$ and
specific gravitational potential traced by $\rm\phi_{gs}=V^{2}+3\sigma^{2}$.
The left hand panels show that for star-forming galaxies there is no clear
primary dependence for the stellar metallicity, due to a combination of a
small sample size and difficulty accurately measuring stellar metallicities of
star-forming galaxies. However, they key result is that for passive galaxies
(right hand panels) the dependence is almost purely on black hole mass, even
with the addition of the dynamical mass, and both definitions of the
gravitational potential $\phi_{g}$.

One important consideration is to properly analyse whether $\sigma_{c}$ is
actually tracing black hole mass or is in fact tracing the dynamical mass or
underlying gravitational potential. We conduct a test of this using $\sim$2000
galaxies from the MaNGA survey with accurate dynamical masses (for more
details see section 1.2). We use two different samples of dynamical
measurements. The first uses the dynamical masses from [26], which we also use
to define the quantity $\rm\phi_{g}=M_{dyn}/r_{e}$ (where $r_{e}$ is the half-
light radius), which is proportional to the gravitational potential. Fig. 5
upper panel shows the random forest regression results for these galaxies. We
also use a similar sized sample of kinematic measurements from [27]. In this
case the proxy for the (specific) gravitational potential is taken as the
quantity $\rm\phi_{g}=V^{2}+3\sigma^{2}$, where V is the rotation velocity and
$\sigma$ is the velocity dispersion. Fig. 5 lower panel shows the random
forest regression results for this sample of galaxies. The results for the two
samples appear to agree, at least qualitatively. Essentially the star-forming
sample shows a null result (although the stellar mass is still formally the
most predictive quantity for the stellar metallicity), likely due to the small
sample size of around 1000 galaxies, and in particular, the difficulty in
measuring stellar metallicities for star-forming galaxies due to their weak
stellar continuum and absorption lines. However, the key insight is from the
passive sample. Clearly the black hole mass, as traced by $\sigma_{c}$, is the
primary driver of the stellar metallicity, even with the addition of dynamical
mass and both definitions of $\phi_{g}$. This shows that the dependence seen
on black hole mass (as traced by $\sigma_{c}$) is not a by product of a
dependence on the gravitational potential or with the dynamical mass, rather
it is actually direct dependence on black hole mass.

Summarising, the test performed in this section (although based on lower
statistics than the SDSS DR7 sample), disfavours the scenario whereby the
stellar metallicity dependence on black hole mass (via central velocity
dispersion) is simply tracing deeper gravitational potentials in more massive
galaxies. Rather this dependence is actually due to the black hole mass itself
resulting from the integrated effect of AGN feedback.

## Discussion

In summary, we have found that in star-forming galaxies the stellar
metallicity is driven primarily by the stellar mass in the MZR. This is likely
tracing the integral of metal formation (as seen for gas-phase metallicities
in [19]): more massive galaxies have produced a greater mass of stars, hence a
greater amount of metals. We also find a secondary dependence on SFR
consistent with the FMR. However, we have discovered that for passive galaxies
the stellar metallicity is driven primarily by black hole mass (as traced by
$\sigma_{c}$) in a BHZR. The black hole mass provides an indication of the
integrated history of feedback associated with black hole accretion, so this
relation likely stems from the black hole quenching galaxies via halo heating
(either preventative or delayed feedback), which suppresses gas accretion and
results in ’starvation’. Residual star formation during starvation results in
a rapid increase of metallicity as a consequence of closed box evolution
without any diluting inflows.

This result crucially bridges the previous gap between studies of the stellar
metallicities, finding evidence for starvation being a key quenching
mechanism, and the causal relation between black hole mass and quenching.

##  1 Methods

###  1.1 SDSS Data

Table 1: Number of galaxies contained within each SDSS sample once all cuts and matches have been made. Samples are star-forming stellar metallicities (SF Z sample) and non-star-forming stellar metallicities (NSF Z sample). Can see that there are many more passive galaxies than SF, hence the need to separate out these classes to properly explore the drivers.  | # of galaxies | log($M_{*}/M_{\odot}$) | log(SFR$/M_{\odot}$yr) | Overdensity
---|---|---|---|---
SF | 16505 | 9.0 - 11.6 | -1.0 - 1.5 | -1.2 - 2.1
Q | 37241 | 9.6 - 11.7 | -2.1 - 0 | -1.2 - 2.7

We use data about stellar masses and star-formation rates from the Sloan
Digital Sky Survey (SDSS) data release 7 [20] MPA-JHU catalogue [28, 4]. We
then match this catalogue to another used in [11], containing mass and light
weighted stellar metallicities fitted by the spectral fitting code FireFly
[29]. FIREFLY is a spectral fitting code that uses $\chi^{2}$ minimization
using a linear combination of simple stellar populations which vary in stellar
ages and metallicities. The stellar population models used are those of [30]
as well as input spectra from MILES [31] and a Kroupa IMF [32]. We also use
mass-weighted halo masses and central satellite classifications from [33, 34],
and overdensities from [35]. We also match to a catalogue of central velocity
dispersions from the NYU value added catalogue [36]. We sigma clip the
resulting dataset to 3$\sigma$ to remove outliers that could bias the random
forest or partial correlation coefficients. We show the resulting sample for
SDSS in Table 2. This dataset enables us to explore the drivers of the stellar
metallicity for a large number of galaxies (15,000+ in each sample) whilst
also being able to explore many parameters of interest relating to each
individual galaxy and its environment.

###  1.2 MaNGA Data

We also use data from the Mapping nearby Galaxies at Apache Point Observatory
(MaNGA) Survey [23], which enables us to also include accurately measured
dynamical masses from [26] and kinematic measurements from [27]. We use
stellar metallicities, stellar masses and star-formation rates from [37, 38].
This significantly cuts down our sample statistics to around 1000 galaxies
each for the star-forming and passive samples, but enables us to control for
these quantities which were unavailable in the larger SDSS sample.

###  1.3 Random Forest Regression

Table 2: Table showing random forest regression hyper-parameters (from left to right), size of the test sample compared to training sample, the number of estimators, the minimum sample size at the end (leaf) node, the maximum depth of the decision trees, and the number of times the forest was bootstrapped for the percentiles. | $\rm test\\_size$ | $\rm n\\_est$ | $\rm min\\_samp\\_leaf$ | $\rm max\\_depth$ | $\rm n\\_times$ | MSE test | MSE train
---|---|---|---|---|---|---|---
SDSS | 50% | 200 | 15 | 100 | 100 | 0.007 | 0.005
MaNGA | 20% | 200 | 15 | 100 | 100 | 0.008 | 0.005

A random forest is a collection of many different decision trees each of which
attempts to reduce a quantity called mean squared error (MSE, which itself
measures the quality of each split) which enables it to calculate parameter
importances in driving a target variable. We split the larger SDSS sample
50-50 into a training and test sample (we use an 80-20 split for the much
smaller MaNGA samples to ensure a larger training set). The random forest is
applied to the training sample to build a model which is then applied to the
test sample to find the parameter importances. Errors for the parameter
importances are calculated by bootstrap random sampling the random forest
regression 100 times - we then take the 16th and 84th percentiles of the
resulting distribution. We compare the test and training sample’s mean squared
errors (MSE) to ensure that the model is not overfitting. For the random
forest regression for the passive galaxies (shown in the right panel of Fig.
1), we find a MSE of 0.007 for the testing sample and 0.005 for the training
sample. For the star-forming galaxies, it is 0.017 and 0.013 respectively.
This shows that there has been no significant overfitting or underfitting. One
of the key benefits of random forest regression is that it can probe several
intercorrelated parameters simultaneously (allowing us to investigate many
parameters at once). It can also find non-monotonic relationships between
them. Furthermore, random forests have been found previously to be able to
find intrinsic dependencies among many intercorrelated variables [providing
the quantity the target truly depends on is included, 15]. They have also been
shown to be able to successfully reverse engineer simulations in [15, 14]. We
include the random forest hyper-parameters and the passive galaxy sample mean
squared errors of the test and training sample in Table LABEL:tab:rf_params.

###  1.4 Partial correlation coefficients

Partial correlation coefficients allow us to take the correlation between two
quantities A and B whilst controlling for any further correlation with
quantity C. This means that if A and C are truly correlated, and B has a
correlation with C this correlation will not introduce a an unphysical
observed correlation between A and B. This is important in determining whether
relationships are intrinsic or stem from relationships between other
quantities that are not considered. We use partial correlation coefficients to
determine the direction of any found relationship, i.e. as the random forest
only gives us their importances, the PCCs can tell us whether two quantities
correlate or anti-correlate. In addition to this it can provide an additional,
although less powerful, method for checking the random forest results. The
ratios of partial correlation coefficients can be used to define a gradient
arrow on a 2D colour plot [24, 17]

$\rm\text{tan}(\theta)=\frac{\rho_{yz|x}}{\rho_{xz|y}}$ (1)

where $\theta$ is the arrow angle measured anticlockwise from the horizontal
while $\rm\frac{\rho_{yz|x}}{\rho_{xz|y}}$ is the ratio of the partial
correlation coefficients between the y axis and colour-coded (z axis)
quantities (whilst controlling for the x axis quantity), to the x axis and
colour-coded (z axis) quantities (whilst controlling for the y axis quantity).
The gradient arrow then points in the direction of the greatest increase in
the colour-coded quantity - it enables the dependence of that quantity on the
x and y axis quantities to be identified visually, whilst the angle of the
arrow enables it to be determined quantitatively. As an example, If the x and
y axis quantities contributed equally to determining the colour-coded
quantity, we would expect an arrow angle of 45 degrees.

###  1.5 Centrals vs Satellites

Figure 6:  The parameter importances of various galactic properties in
determining the stellar metallicity (Z) for star-forming (left panel) and
passive (right panel) galaxies, split into central (blue or red, wider bars)
or satellite galaxies (orange, thinner bars) . The galactic parameters
evaluated are stellar mass ($M_{*}$), star formation rate (SFR), black hole
mass ($M_{\rm BH}$), the gravitational potential ($\phi_{g}$), a control
uniform random variable (Random), overdensities (1-$\delta$), and halo mass
($M_{H}$). We see from both panels that there is little difference between
centrals or satellites.

We can explore whether these relations we have found are the same for central
and satellite galaxies. This is another test of whether environment plays a
significant role. The differences between the stellar metallicities of central
and satellite galaxies have been highlighted previously in [10]. We can probe
whether this is the case by using our random forest regression methodology.
The first stage is to split the dataset into central and satellite galaxies.
We use a mass-weighted approach for this with central galaxies being
classified as the most massive galaxy in their group [33, 34], with the
remaining galaxies being classified as satellites. We also test using a light-
weighted classification and find it makes no significant difference. Fig. 6
shows the random forest regression parameter importances, in exactly the same
form as Fig. 1, only this time with the star-forming and passive samples split
into central or satellite galaxies based upon whether they are the most
massive galaxy in their group. We can see from Fig. 6 that the results remain
the same regardless of central or satellite classification, i.e. again we find
stellar metallicity is not driven by environmental effects. Interestingly this
is different to what was found in [10], which we attribute to being able
analyse almost all relevant parameters simultaneously with the random forest,
which was not possible in the case of [10].

###  1.6 Reliability of $\sigma$ as a tracer of black hole mass

Another question is exactly how reliable $\sigma_{c}$ (the central stellar
velocity dispersion) is as a tracer of black hole mass. This has previously
been explored in [14], where they found that out of stellar mass, bulge mass
and stellar velocity dispersion it was the stellar velocity dispersion that
most tightly correlated with the black hole mass (as obtained from the
directly measured black hole masses in [39]). Velocity dispersion has also
been found to follow the local relation with black hole mass at higher
redshifts [40] suggesting this is likely a universal relationship.

## Acknowledgements

WB, RM, acknowledge support by the Science and Technology Facilities Council
(STFC), by the ERC Advanced Grant 695671 ‘QUENCH’, and by the UKRI Frontier
Research grant RISEandFALL. RM also acknowledges funding from a research
professorship from the Royal Society.

### Availability of data and materials

The SDSS DR7 data is publicly available. The MaNGA data used is also publicly
available.

### Code availability

AstroPy [41], LMFIT [25] and FireFly [29] are publicly available.

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# Topological Necessary Conditions for Control Dynamics

EFTHIMIOS KAPPOS
School of General Sciences,111Submitted for publication, 2009, under revision
Department of Mathematics,
Faculty of Engineering,
Aristotle University of Thessaloniki, Greece

##  1 Introduction

Suppose a smooth feedback control has been found so that the controlled
dynamics have an asymptotically stable attractor at some point x in state
space. Then local Lyapunov functions exist for the dynamics; these functions
must all have a unique minimum at the point x, but are otherwise arbitrary. On
a compact level set of any such Lyapunov function, the controlled dynamics
point inwards, in other words in the direction of the negative of the gradient
of the Lyapunov function. As maps to the unit sphere, the two vector fields
thus have the same degree. But the degree of the negative gradient vector
field is known —it is exactly $(−1)^{n}\neq 0$. Hence the map of the
controlled dynamics, restricted to a level set, to the sphere must be onto.
This means roughly that all control directions must be available near a point
that is to be stabilized by control. In control theory, this is known as the
Brockett condition, but such simple de- gree results were widely known before
(Krasnosel’skii’s name is mentioned in conjunction with that of Brockett.)
Such degree-theoretic arguments have been used for some time in topology and
were eventually adopted by control theorists to derive a number of related
necessary conditions for design of controlled dynamics using continuous feed-
back.

In this paper, we give an account of this theory that has two distinguishing
features: first, there is really no reason to limit ourselves to local
results: a global theory is straightforward to obtain. Secondly, we point out
that such necessary conditions are in some funda- mental sense of limited
value; this is because they involve maps from manifolds to spheres of the same
dimension. By the Hopf theory, homotopy equivalence classes of such maps are
completely classified by a single integer (traditionally called the degree,
but better interpreted in terms of homology groups.)

Several comments are in order: generalizations of the Brockett necessary
condition have been obtained by Coron and others. These suffer to some extent
by the problem mentioned above, but have helped clarify the fact that
_surjectivity_ is not enough: the vector field cannot ‘twist’ too much either
(examples use the degree $k$ maps $\dot{z}=z^{k}$ as part of a control system
decomposition.) Moreover, extensions to the case of _dynamic feedback_ have
been derived.

Perhaps more importantly, many systems cannot be stabilized using continuous
feedback, but can be easily stabilized with discontinuous feedback. The
discontinuity is rather mild: it is usually limited to a ‘thin’ subset of
state space. Recent work of Sontag, Clarke, Subbotin and others has led to a
theory of discontinuous feedback controls and a methodology for obtaining
nonsmooth Lyapunov functions. Now it is possible to interpret this theory in a
_hamiltonian_ context: The discontinuities correspond to jumps between locally
nonsingularly projected lagrangian levels. This way of examining possibly
discontinuous feedback controls is conceptually easier to understand and is in
step with the philosophy of the book [3] which is to give, as far as is
possible, _geometric_ accounts of _analytical_ points.

We begin by giving an outline of the algebraic topological machinery needed
for a discussion of necessary conditions. Here, we depart from the practice of
delegating mathematical background to an appendix, because we believe that
this theory is quite accessible and elegant.

A collection of _global necessary conditions_ is then given, directly based on
the topological results. Essentially, it is argued that if certain dynamics
are achieved, then index-theoretic conditions can be deduced by counting the
equilibrium points and their stability (Euler-type arguments) and _degree-
theoretic_ results are obtained by the Hopf theorems using the Gauss maps of
the dynamics and the gradient vector field of Lyapunov functions.

Finally, let us point out that this paper by no means exhausts the theory of
necessary conditions in control design. Much more crucial limitations on
achievable control dynamics arise from the theory of feedback invariant
objects, a theory that is developed in the book [3].

##  2 Some Background and Methods from Algebraic Topology

A thumb-nail sketch of a number of concepts and methods from algebraic
topology will now be given. There is no effort to be rigorous, but we do hope
to explain enough about the computational methods so that a non-expert reader
can use them in concrete situations.

_Algebraic topology_ is based on a simple principle: attach algebraic objects
to topological spaces that are invariants of the homotopy type of the space.
Thus, more precisely we assign algebraic objects to homotopy equivalence
classes of spaces and this assignment is ‘functorial’ in the sense that maps
of spaces induce homomorphisms of the algebraic objects. This already gives
useful tests: since homotopy equivalent spaces have isomorphic algebraic
objects, two spaces are definitely not homotopy equivalent if their algebraic
objects are not isomorphic. The bulk of algebraic topology consists of
deriving finer and finer such objects so as to be able to better distinguish
spaces and in making clever use of its basic constructions to aid the analysis
of global aspects of other subjects (such as complex analysis, pdes, geometry
etc.)

#### Singular homology

The easiest algebraic object we can attach to a space is the graded abelian
group $H_{*}(X)$ called the singular homology group of $X$ (with integer
coefficients.) One can get quite far with only a vague understanding of what
the singular homology measures and the reason is that powerful and effective
methods for the computation of $H_{*}(X)$ exist. We outline the Mayer-Vietoris
sequence and explain the concept of a long exact sequence of a pair and its
relation to excision.

A graded abelian group $G=\oplus G_{k}$ is a direct sum of groups $G_{k}$,
$k\in{\mathbb{Z}}_{+}$, such that the group addition is ‘component-wise’, in
other words we add elements belonging to the same graded component together.
The notation

$g=\ldots+g^{0}+g^{1}+\ldots$

for an element of $G$, where $g^{k}\in G_{k}$ is therefore unambiguous.

For a topological space, the $k$th homology group $H_{k}(X;{\mathbb{Z}})$
measures in some sense the _‘holes’_ of $X$ that are like $k$ spheres $S^{k}$
(think of a boundary-less space, like a sphere, that does not actually bound
anything itself in $X$.) The $0$th group $H_{0}(X)$ is equal to ${\mathbb{Z}}$
if $X$ is path connected. (Recall that a $0$-sphere is the boundary of an
interval, i.e. the union of two points.) There is a way of defining _reduced
homology groups_ $\tilde{H}_{k}(X)$ so that $\tilde{H}_{0}(X)=0$ for a
connected space and so that all higher dimensional groups coincide with the
non-reduced ones.

Let us give some examples (we omit the zeroth homology group.) The singular
homology of the circle $S^{1}$ is $H_{1}(S^{1})\simeq{\mathbb{Z}}$ and zero
for $k>1$. Since $\pi_{1}(S^{1})={\mathbb{Z}}$ also, the homology group
contains the same information as the fundamental group of the circle. Note the
difference in interpretation, though: In the former case (for $\pi_{1}$), we
are thinking of maps from the circle to itself, classifed by the number of net
encirclements In the latter, we are thinking of a fixed circle —coinciding in
this case with the whole space $S^{1}$— as the generator of a free abelian
group; in this sense, we can write

$H_{1}(S^{1})={\mathbb{Z}}[S^{1}]\simeq{\mathbb{Z}}.$

For the sphere $S^{m}$ of dimension $m>1$, $H_{m}(S^{m})\simeq{\mathbb{Z}}$ is
the only nonzero homology group in positive dimension. Since we also have that
the $m$th homotopy group of an $m$ sphere is ${\mathbb{Z}}$, we have not yet
obtained anything new, compared with homotopy theory. This is a little
misleading: homotopy is both subtler than homology and far more difficult to
compute: we do not, even today, have a complete list of the homotopy groups of
spheres. Moreover, $\pi_{m+k}(S^{m})$ may very well be nonzero for some $k>0$,
while $H_{m+k}(S^{m})=0$ always.

The ‘coincidence’ is really due to a nontrivial theorem, the Hurewicz
isomorphism that states that homotopy and homology groups are isomorphic at
the first level when one, and hence both, are nontrivial (the abelianization
of the possibly nonabelian fundamental group is to be considered, if this
happens at the first level.)

A quick check that homology theory does indeed give something new is to
compute the homology of the torus $T^{2}$. We have that
$H_{2}(T^{2})\simeq{\mathbb{Z}}$ even though $\pi_{2}(T^{2})=0$! (it may be
profitable to spend a minute or two pondering the difference.)

#### The Mayer-Vietoris sequence

Supose a space $X$ is the union of two opne subsets, $X=A\cup B$, with $A,B$
opne and $A\cap B\neq\emptyset$. Then there is a _long exact sequence_
involving the homology groups of the three spaces

$\ldots\to H_{k}(A\cap B)\to H_{k}(A)\oplus H_{k}(B)\to H_{k}(X)\to
H_{k-1}(A\cap B)\to\ldots$ (1)

This gives a surprisingly powerful tool for the computation of homology. Even
without knowing what the maps at each stage are (for which we refer the reader
to standard accounts such as [2]) the exactness allows the computation in
concrete cases, such as that of the spheres. For this, decompose an $m$-sphere
into two slightly overlapping hemispheres $A$ and $B$ so that their
intersection $A\cap B$ is deformable to a sphere of dimension $m-1$. We can
start an induction with dimension $m=1$ and use the Mayer-Vietoris sequence to
obtain

$\ldots\to H_{k}(S^{m-1})\to H_{k}(A)\oplus H_{k}(B)\to H_{k}(S^{m})\to
H_{k-1}(S^{m-1})\to\ldots$ (2)

yielding, for $k=m$, and since disks have no homology

$\ldots\to 0\to 0\oplus 0\to H_{m}(S^{m})\to{\mathbb{Z}}\to 0\to\ldots.$ (3)

We conclude that $H_{m}(S^{m})\simeq{\mathbb{Z}}$, since any exact sequence of
the form

$0\to C\to D\to 0$

implies that the middle map is one-to-one and onto, i.e. an isomorphism.

#### Long exact sequence of a pair and excision

A second very useful method for the computation of homology comes from
considering pairs $(X,A)$, where $A\subset X$ is a subspace. One gets the long
exact sequence for the pair

$\ldots\to H_{k}(A)\to H_{k}(X)\to H_{k}(X,A)\to H_{k-1}(A)\to\ldots$ (4)

where the groups $H_{k}(X,A)$ are the relative homology groups. Without giving
the exact definition, found in the standard texts, let us mention that in many
important cases, these relative groups are isomorphic to the homology groups
of the quotient space $X/A$ (see chapter 3 for the definition.) The long exact
sequence for a pair is thus extremely useful for the computation of the
homological Conley index.

As an example, let us show that the quotient $D^{n}/S^{n-1}$ of a closed ball
by its bounding sphere has the homology of the $n$-sphere $S^{n}$. The long
exact sequence of the pair $(D^{n},S^{n-1})$ is

$\ldots\to H_{k}(D^{n})\to H_{k}(D^{n},S^{n-1})\to H_{k-1}(S^{n-1})\to
H_{k-1}(D^{n})\to\ldots$ (5)

and so, at $k=n$, we get

$\ldots\to 0\to H_{k}(D^{n},S^{n-1})\to{\mathbb{Z}}\to 0\to\ldots$ (6)

hence $H_{k}(D^{n},S^{n-1})\simeq H_{k}(D^{n}/S^{n-1})\simeq{\mathbb{Z}}$.
Similarly, one finds that, for $k\neq n$ (and nonzero),
$H_{k}(D^{n}/S^{n-1})=0$.

#### Maps and homomorphisms

Given a continuous map $f:X\to Y$, there is an induced map in homology, which
we shall denote by $H_{*}(f)$ or $f_{*}$

$H_{*}(f):H_{*}(X)\to H_{*}(Y)$

and one checks that homology is a covariant functor from the category Top
$=(\text{ Top},C^{0})$ of topological spaces and continuous maps to the
category Ab$=(\text{Ab},\text{Hom})$ of abelian groups and homomorphisms
between them. Since $H_{*}(X)$ is graded, the above homomorphism is understood
to mean that it consists of homomorphisms at each level of homology:

$H_{k}(f):H_{k}(X)\to H_{k}(Y)$

for all $k$.

In fact, it would be more precise to say that the functor goes from the
category hTop of homotopy equivalence classes of spaces and homotopic maps to
the category Ab, since

######  Proposition 1.

If the maps $f$ and $g$ are homotopic, then the maps in homology coincide:
$f_{*}=g_{*}$.

and

######  Proposition 2.

If two spaces $X$ and $Y$ are homotopy equivalent and the map $f$ has a
homotopy inverse, then $f_{*}$ is an isomorphism and the homology groups of
$X$ and $Y$, $H_{*}(X)$ and $H_{*}(Y)$ are isomorphic.

######  Corollary 1.

If $f$ is a homeomorphism of spaces, then $f_{*}$ is an isomorphism.

When the space $X$ is a finite-dimensional manifold, the homology groups
$H_{k}(X;{\mathbb{Z}})$ are finitely generated; thus, in this case, the basic
structure theorem for finitely generated abelian groups is applicable.

######  Theorem 1 (Structure Theorem).

Any finitely generated abelian group $G$ decomposes uniquely as the direct sum

$G=F\oplus\tau$

where the abelian group $F$ is free and the group $\tau$ is a torsion
subgroup.

In fact, one can describe the torsion group $\tau$ is more detail (see, for
example, [4].)

The dimension of the free part of the homology group $H_{k}$ is called the
$k$th-Betti number, $b_{k}=\dim H_{k}(X;{\mathbb{Z}})$. The Euler
characteristic $\chi(X)$ is the alternating sum of the Betti numbers

$\chi(X)=\sum_{k}{(-1)}^{k}b_{k}.$

##  3 Collections of topological necessary conditions

The definition of certain Gauss maps is helpful in the statement of our
results. We shall assume that $M^{n}={\mathbb{R}}^{n}$ or is an open subset of
it.

######  Definition 3.1.

  1. 1.

Suppose the vector field $X$ is nowhere zero in $M^{n}$. Then the Gauss map
$G_{X}:M^{n}\to S^{n-1}$ is defined by

$x\mapsto\frac{X(x)}{|X(x)|}.$

  2. 2.

Suppose that $N^{n-1}\subset M^{n}$ is a submanifold such that the restriction
of the vector field $X$ to $N$ is nowhere zero. Then the Gauss map
$G_{X|N}:N^{n-1}\to S^{n-1}$ is obtained by restricting the Gauss map $G_{X}$
to $N$. Note that this is a map between two manifolds of the same dimension,
one of which is a sphere.

  3. 3.

If the submanifold $N^{n-1}\subset M^{n}$ is orientable, we define the Gauss
map $G_{N}:N^{n-1}\to S^{n-1}$ by mapping $x\in N$ to the unit normal vector
to $N$ at $x$ (where an ‘outward’ direction is fixed by choosing an oriented
basis on $N$ and completing it to a basis of ${\mathbb{R}}^{n}$ consistent
with an orientation of ${\mathbb{R}}^{n}$.) Note again that the Gauss map is a
map from an $(n-1)$ dimensional space to the $(n-1)$-sphere.

###  3.1 Index-Theoretic Necessary Conditions

The _topological index_ of equilibrium points leads to a number of necessary
conditions for achieving dynamics with equilibrium points of given stability.
These are _global_ results and are rather classical; our only novelty is in
trying to use as modern an algebraic topological framework as we can to
express them.

If $e\in M^{n}$ is an isolated equilibrium point of the vector field $X$, take
a ball neighborhood $U$ of $e$ (an open set homeomorphic to a ball) such that
$e$ is the _only_ equilibrium of $X$ in $U$ and its boundary $N=\partial U$ is
a closed submanifold homeomorphic to a sphere. Then the Gauss map $G_{X|N}$
gives a map from the sphere $S^{n-1}$ to itself

$S^{n-1}\stackrel{{\scriptstyle h}}{{\rightarrow}}N\stackrel{{\scriptstyle
G_{X|N}}}{{\rightarrow}}S^{n-1}$

where $h^{-1}$ is the homeomorphism from $N$ to the sphere.

At the level of homology, we thus get a homomorphism $\psi=G_{X|N}\circ h$
from $H_{n-1}(S^{n-1})$ to itself. Since this group is isomorphic to
${\mathbb{Z}}$, we get a homomorphism from ${\mathbb{Z}}to{\mathbb{Z}}$. Since
${\mathbb{Z}}$ is a principal ideal domain, such maps are specified by the
image of the generator, say $\alpha\in H_{n-1}(S^{n-1})$. If, say,
$\psi(\alpha)=k\alpha$, then $k$ is the topological index of the equilibrium
$e$.222Confusingly, we are about to give a theorem where the term ‘degree’ is
used instead of ‘toplogical index’; the two terms are equivalent. We shall try
use the qualifier ‘topological’ to avoid confusion with other uses of the term
index. It does not depend on the precise $U$ chosen.

The classical theorem of Hopf describes maps from the sphere to itself.

######  Theorem 2 (Hopf’s Classification Theorem).

Homotopy equivalence classes of maps from $S^{n-1}$ to itself are in a one-to-
one correspondence with the integers. For each integer $k$, the class of maps
corresponding to it is called the class of maps of degree $k$.

For a hyperbolic equilibrium point of _stability index_ $k$, the _topological
index_ (or degree) is equal to ${(-1)}^{n-k}$. Degree $k$ maps are easily
obtained from the degenerate equilibria at the origin of the system in complex
form: $\dot{z}=z^{k}$, for $k\neq 0$.

The Hopf classification of maps from the sphere to itself has a crucial
generalization to maps of an arbitrary compact manifold of dimension $n-1$ to
a sphere of dimension $n-1$ (see Whitehead, [7], p.244)

######  Theorem 3 (Hopf-Whitney).

The homotopy equivalence classes of maps of an $(n-1)$-dimensional compact
manifold $N^{n-1}$ to the sphere $S^{n-1}$ are in one-to-one correspondence
with the elements of the cohomology group $H^{n-1}(N^{n-1};{\mathbb{Z}})$.

######  Corollary 2.

If $N$ is orientable, then the homotopy equivalence classes of maps from
$N^{n-1}$ to $S^{n-1}$ are in one-to-one correspondence with the integers;
they are thus again classified by ‘degree.’

This is, of course, because, for any orientable manifold,
$H^{n-1}(N^{n-1};{\mathbb{Z}})\simeq{\mathbb{Z}}$. If $N$ is not orientable,
then this group is ${\mathbb{Z}}_{2}$ and two maps are homotopic iff they have
the same mod-$2$ degree.

The global version of the Hopf index classification result is the following
theorem of Poincaré-Hopf

######  Theorem 4 (Poincaré-Hopf).

  1. 1.

Suppose $W^{n}\subset{\mathbb{R}}^{n}$ is a compact subset with nonempty
interior such that its boundary is an $(n-1)$-dimensional submanifold of
${\mathbb{R}}^{n}$. Suppose $X$ is a vector field on ${\mathbb{R}}^{n}$ that
is nowhere zero on the boundary $\partial W$ and has a finite set of
equilibrium points $E$. Then

$\deg G_{X|\partial W}=\sum_{e_{i}\in E\cap W}\operatorname{ind}e_{i}$ (7)

  2. 2.

Suppose $M^{n}$ is a compact manifold and $X$ is a vector field on $M^{n}$
with a finite number of isolated equilibria. If the boundary of $M^{n}$ is not
empty, we require the vector field to point inwards at all points. The we have

$\sum_{e_{i}\in E}\operatorname{ind}e_{i}={(-1)}^{n}\chi(M^{n})$ (8)

where $\chi(M^{n})$ is the Euler characteristic of the manifold $M^{n}$ and
$E$ is the set of equilibrium points.

In particular, the sum of the topological indices of the equilibria is a
topological invariant of the manifold and thus is independent of the vector
field chosen.

  3. 3.

Suppose $W^{k}$ is any submanifold of ${\mathbb{R}}^{n}$, with $0\leq k\leq
n-1$. Consider a tubular neighborhood $N_{{\epsilon}}(W^{k})$ so that
$\partial N_{{\epsilon}}(W^{k})$ is an $(n-1)$-dimensional submanifold of
${\mathbb{R}}^{n}$. If $X$ is any vector field on ${\mathbb{R}}^{n}$ such
that, on $W^{k}$, $X$ has a finite number of nondegenerate equilibria, then

$\sum_{e_{i}\in E\cap W}=\deg G_{X|N_{{\epsilon}}(W^{k})}$ (9)

We have collected different versions of this important theorem to help the
reader find the most convenient form for extracting topological information in
applications. Milnor [5] proves versions (2) and (3) and contains a nice
discussion.

######  Remark.

The index already contains considerable topological information for the
purposes of extracting necessary conditions. For the case of an asymptotically
attracting equilibrium, for example, the topological index is equal to
${(-1)}^{n}$, which means that the generator of $H_{n-1}(S^{n-1})$ is mapped
to itself or its negative, depending on the parity of $n$. As a result, the
Gauss map is an isomorphism in homology and we conclude that it must then be
surjective and injective. The surjectivity is essentially the Krasnosel’skii-
Brockett condition and the injectivity was derived by Coron. The form we have
given is, however, considerably more general.

######  Remark.

It must be emphasized that the index is _‘blind’_ to all other dynamical
features except equilibria. Looking at the same point from the other side of
the equalities in Theorem 4, the topological type of the Gauss map in the
large (on the boundary of an enclosing set) affects the configuration of
equilibria inside—and fixes the sum of their indices.

A few examples as simple illustrations of the statements of the theorem:

######  Example 1.

In ${\mathbb{R}}^{n}$, a ball with a vector field pointing inwards at the
boundary must contain equilibria whose index sum is ${(-1)}^{n}$. If these are
all hyperbolic, then the options are

  * •

A single attracting equilibrium.

  * •

Two attractors and a one-saddle.

  * •

If $n$ is even, a single repeller is not ruled out; notice that the two cases
can be distinguished using the Conley index, since the exit set differs for
the two cases.

  * •

Any other configuration of equilibria with the same net index sum.

######  Example 2.

In ${\mathbb{R}}^{3}$, an embedded torus $T^{2}$ gives possible Gauss maps of
arbitrary degree, since its top homology is equal to ${\mathbb{Z}}$. If,
however, we know that there are no enclosed equilibria, as for example in the
case where the torus isolates a limit cycle, then the degree must be zero, by
part (1) of Theorem 4, independently of the stability type of the limit cycle.

This means that the Gauss map is homotopic to the constant map and hence _does
not have to be onto_ (it is not onto in general, for a small enough torus
around the limit cycle). Thus, no necessary condition is derivable in this
case, whether the limit cycle is stable or not.

######  Example 3.

On the torus $T^{2}$ we have, by part (2) of the Theorem, that any vector
field must have total index sum equal to zero, since the Euler characteristic
of the torus is zero. Thus, vector fields that everywhere nonzero are
permissible topologically, as are vector fields with one attractor and one
saddle, one repeller and a saddle, one attractor, one repeller and two saddles
etc.

###  3.2 Necessary conditions using the topological index

It should be clear from the examples how to derive necessary conditions for
achieving global dynamics from the index theorems.

Suppose given a Morse specification of gradient type,
${\mathcal{M}}=(E,h_{0})$, with Morse-lyapunov functions
${\mathcal{F}}({\mathcal{M}})$. In the state space manifold, any choice of an
oriented hypersurface that avoids $|E|$ has a Gauss map degree fixed by the
sum of the indices of the enclosed ‘equilibria’. If this is non-zero, this
implies that there must exist control sections such that the controlled
dynamics give a Gauss map with the desired property. In particular, if the
index sum is equal to plus or minus one, then the Gauss map is onto. Let us
remark that the conditions obtained can iether be used _locally_ to check, for
example, local stabilizability by requiring the map to have degree
${(-1)}^{n}$ for an arbitrarily small neighborhood of the equilibrium, or
_globally_ , since the only relevant information is the position and stability
of the desired equilibria and hence the index/degree results hold for any
compact hypersurface avoiding $|E|$.

A more elegant algebraic topological way of checking _simultaneously_ all
necessary conditions is the following (this does not make it easier to check
in concrete cases):

We do first the case of local asymptotic stabilizability.

Suppose a control section $U\in\Gamma(D)$ is found that locally stabilizes the
origin $0$ in some neighborhood $B$; it will be helpful to consider the set,
for $\epsilon>0$,

$\Sigma_{B}=\\{(x,v)\in D{|}_{B}\;;\;X(x)+v=0\\}$

and the sequence

$\displaystyle
B\setminus\\{0\\}\stackrel{{\scriptstyle\operatorname{graph}U}}{{\longrightarrow}}B\times{\mathbb{R}}^{m}\setminus\Sigma_{B}\stackrel{{\scriptstyle\iota}}{{\hookrightarrow}}T{\mathbb{R}}^{n}{|}_{B}\setminus\\{0\\}$
$\displaystyle\stackrel{{\scriptstyle
G}}{{\longrightarrow}}S{\mathbb{R}}^{n}{|}_{B}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}S^{n-1}$

where $\operatorname{graph}U(x)=(x,U(x))$, $\iota$ is the inclusion map, $G$
is the Gauss map and $\pi$ is the obvious projection in the trivial local
sphere bundle.

Since $0$ is an isolated equilibrium of $X+U$, $X+U\neq 0$ in
$B\setminus\\{0\\}$ and the above is well-defined.

###  3.3 A reinterpretation of Coron’s condition

With the tools we have at our disposal, it is now easy to give a more
geometric interpretation of the necessary condition for local feedback
stabilization given in [1]: We start by noticing that, if $B$ is a ball
neighborhood of the equilibrium $0$, $B\setminus\\{0\\}$ is homotopically
equivalent to $S^{n-1}$ (it actually retracts to the sphere). Thus the
composed map defined by the above sequence, call it $\phi$,

$\phi:B\setminus\\{0\\}\to S^{n-1}$

has a well-defined degree, since $0$ is asymptotically stable for $X+U$ and
this degree is equal to ${(-1)}^{n}$. This means that, at the level of, for
example, homology (or homotopy), the generator, call it $\alpha$, of
$H_{n-1}(S^{n-1})\simeq{\mathbb{Z}}$ is in the image of $\phi$. In other
words, if $0$ is LAS, then there is some local section such that the degree of
the above map is defined and the image of the corresponding homomorphism at
the level of homology is the whole of $H_{n-1}(S^{n-1})$. This is essentially
Coron’s result: Consider the commutative diagram

$\begin{array}[]{ccc}B\setminus\\{0\\}&\rightarrow&S^{n-1}\\\
\downarrow&\nearrow&\\\ D_{\setminus}\Sigma_{V}&&\end{array}$ (10)

where the vertical map is inclusion and the map from
$D_{B}\setminus\Sigma_{B}$ to $S^{n-1}$ will be denoted also by $X+U$ and is
given by the composition $(x,v)\mapsto X(x)+v\mapsto G(X(x)+v)$. We have that

$\phi_{*}(H_{n-1}(B\setminus\\{0\\}))=H_{n-1}(S^{n-1}).$

######  Theorem 5 (Coron, 1990).

If the system $(X,D)$ is locally asymptotically stabilizable, then

${(X+U)}_{*}(H_{n-1}(D_{B}\setminus\Sigma_{B}))=H_{n-1}(S^{n-1}).$

###  3.4 Generalizations

The simple reasoning that led to Coron’s result can be generalized to
equilibrium points that are not attractors, but have a well-defined stability
index.

######  Theorem 6.

Let $0$ be an equilibrium of the state dynamics $X$ of the control pair
$(X,D)$. If there is a continuous local feedback that yields dynamics $X+U$
with $0$ an equilibrium of index $k\,,\,0\leq k\leq n$, then

${(X+U})_{*}(H_{n-1}(D_{B}\setminus\Sigma_{B}))=H_{n-1}(S^{n-1}).$

Finally, necessary conditions applicable to an arbitrary compact, connected
IIS ${\mathcal{S}}$, isolated by the set $B\subset M^{n}$ can be given. More
explicitly, we assume that there is a local feedback $U:B\to D$ such that
$X+U$ has an IIS ${\mathcal{S}}$, whose dynamical structure is known (for
example, ${\mathcal{S}}$ as a set consists of a number of equilibria and limit
cycles and their connecting orbits.) Notice that $X+U\neq 0$ in
$B\setminus{\mathcal{S}}$.

######  Lemma 1.

Endow $M^{n}$ with a Riemannian metric. There is a function $h$ defined on
$B\setminus{\mathcal{S}}$ such that its gradient vector field $\nabla h$ is
topologically equivalent to $X+U$ and such that the Gauss maps of $\nabla h$
and $X_{U}$ induce the same homomorphisms on homology, both $G_{\nabla h}$ and
$G_{X_{U}}$ mapping

$H_{*}(V\setminus{\mathcal{S}})\rightarrow
H_{*}(SM^{n}{|}_{V\setminus{\mathcal{S}}}).$

Now, as we did for the case of local stabilization, we have the map $\phi$
defined by the composite map below

$D_{B\setminus{\mathcal{S}}}\setminus\Sigma\stackrel{{\scriptstyle
X+v}}{{\rightarrow}}TM^{n}\setminus\Sigma\stackrel{{\scriptstyle
G}}{{\rightarrow}}SM^{n}$ (11)

which induces the map $\phi_{*}$ in homology

$H_{*}(D_{B\setminus{\mathcal{S}}})\rightarrow
H_{*}(SM^{n}{|}_{B\setminus{\mathcal{S}}}).$

We now have the result

######  Theorem 7.

If the control pair $(X,D)$ can achieve dynamics with IIS ${\mathcal{S}}$
isolated by the set $B$, then the images of the maps $\phi_{*}$ and
$G_{-\nabla h}$ in $H_{*}(SM^{n}{|}_{B\setminus{\mathcal{S}}})$ coincide.

##  4 Homotopy equivalence and homotopic results

An elementary, but fundamental result forms the key to an alternative approach
to the derivation of necessary conditions. It concerns the Gauss maps of a
gradient vector field of a Lyapunov function for the dynamics $X$ and the
Gauss map of the dynamics on level sets of the Lyapunov function.

######  Theorem 8.

Let ${\mathcal{V}}^{n-1}$ be a compact regular level set of some Lyapunov
function $V$ for the dynamics $X$ on $M^{n}\subset{\mathbb{R}}^{n}$. Then, the
Gauss maps $G_{X|{\mathcal{V}}}$ and $G_{-\nabla V|{\mathcal{V}}}$ os the
vector field $X$ and of the gradient vector field of $V$ with respect to any
riemannian metric are homotopy equivalent.

###### Proof.

Decompose the tangent space $TM^{n}{|}_{{\mathcal{V}}}$ into the tangent space
of ${\mathcal{V}}$ and the span of the gradient vector field $\nabla V$. If
$X_{n}$ is the projection of $X$ to the span of $\nabla V$,we have that
$X_{n}$ is nowhere zero on ${\mathcal{V}}$.

Consider the isotopy of vector field

$Y_{t}(x)=(1-t)X_{n}(x)+tX(x),\;0\leq t\leq 1.$

We have that $Y_{0}=X_{n}$ and $Y_{1}=X$.

Now notice that this gives an isotopy for the corresponding Gauss maps as
well: this is because $Y_{t}(x)\neq 0$ on ${\mathcal{V}}$ and for all $t$. To
see this, write $Y_{t}$ as

$Y_{t}=X_{n}+t(X-X_{n})$

and notice that the vector field $X-X_{n}$ is orthogonal to $X_{n}$, which is
everywhere nonzero.

Define the Gauss maps parametrized by $t$

$G_{t}:{\mathcal{V}}^{n-1}\to S^{n-1},\;x\mapsto\frac{Y_{t}(x)}{|Y_{t}(x)|}.$

Since $Y_{t}(x)$ is everywhere nonzero, this is well defined and gives an
isotopy between

$G_{0}=\frac{X_{n}}{|X_{n}|}=\frac{-\nabla V}{|-\nabla V|}=G_{-\nabla V}$

and

$G_{1}=\frac{X}{|X|}=G_{X}.$

∎

For reference purposes, let us denote the set of homotopy equivalence classes
of maps between two spaces $\Omega$ and $\Omega^{\prime}$ by

$[\Omega,\Omega^{\prime}]$

according to the standard notation. Given a map $f:\Omega\to\Omega^{\prime}$,
we write $[f]$ for its equivalence classs. We thus have, in this notation,
that

$[G_{X}]=[G_{-\nabla V}],\text{in}\;[{\mathcal{V}}^{n-1},S^{n-1}].$

#### Relations to the index

Since the spaces involved are of the same dimension and the target space is a
sphere, we have, by the Hopf theory, that these homotopy equivalence classes
are classified by degree.

#### Limit Cycles

In the case of a limit cycle $\gamma$, we saw that the Gauss map always has
degree zero. Additional necessary conditions are obtained by examining the
Gauss map in more detail.

######  Theorem 9.

Suppose $\gamma$ is a limit cycle for the dynamics $X$ on ${\mathbb{R}}^{n}$.
then

  1. 1.

For any ${\epsilon}>0$, there is a neighborhood $N_{\delta}(\gamma)$ such that

$G_{X}(N_{\delta}(\gamma))\subset N_{{\epsilon}}(G_{X}(\gamma)).$

  2. 2.

The image $G_{X}(\gamma)$ is not contained in any hemisphere: in other words,
for any hyperplane ${\mathcal{P}}\subset{\mathbb{R}}^{n}$,
$G_{X}(\gamma)\cap{\mathcal{P}}\neq\emptyset$. Moreover, for generic
${\mathcal{P}}$, $|G_{X}(\gamma)\cap{\mathcal{P}}|$ is even (here the bars
denote cadinality of a finite set.)

###### Proof.

The first part is proved by continuity and the long flow box (see [6].)

The second part is by contradiction: suppose there exists a hyperplane
${\mathcal{P}}_{a}=\\{v\in{\mathbb{R}}^{n}\;;\;a(v)=0\\}$, for some
$a\in{({\mathbb{R}}^{n})}^{*}$ and is such that
$G_{X}(\gamma)\cap{\mathcal{P}}_{a}=\emptyset$. Since any hyperplane separates
$S^{n-1}$ into two parts, we must have that $a(G_{X}(x))$ is of uniform sign,
say negative, for all $x\in\gamma$.

Choose a basis $b_{1},\ldots,b_{n}$ of ${\mathbb{R}}^{n}$ such that $a$ is the
dual basis vector of $b_{1}$, i.e. $a(b_{1})=1$ and $a(b_{i})=0$ for all
$i\neq 1$. Write $x_{1},\ldots,x_{n}$ for the coordinates in this basis.

######  Claim.

The function $V(x)=\frac{1}{2}x_{1}^{2}$ is a Lyapunov function for $X$ in
some open neighhborhood of $\gamma$.

This is shown by computing $\frac{dV}{dt}{|}_{\gamma}$. We have

$\frac{dV}{dt}=(x_{1},0,\ldots,0)\cdot\dot{\gamma}$

and, since $G_{X}=\frac{X}{|X|}$, this is just $a(X)<0$.

The claim now establishes a contradiction that proves the theorem, since
$G_{X}(\gamma)$ is a closed curve. The last part also follows from this fact
and an elementary transversality argument. ∎

Theorem 9 says roughly that, even though the image of the Gauss map of a limt
cycle is ‘thin,’ still it must curve sufficiently in the target sphere so as
to intersect all possible hyperplanes.

As for the Lyapunov level sets near a limit cycle, we have

######  Theorem 10.

Suppose $\gamma$ is a stable limit cycle for some controlled dynamics. then,
on each level set of a Lyapunov function near $\gamma$, each direction (i.e.
element of the unit sphere) appears at least twice, in other words, for each
$v\in S^{n-1}$,

$|G_{-\nabla V}^{-1}(v)|\geq 2.$

(The proof is a basic topolgical facts about tori and is omitted.) Thus, even
though the Gauss map of the gradient vector field of Lyapunov functions is of
degree zero on any level (as it should be by the Poincaré-Hopf theorem 4), it
covers the unit sphere at least twice.

We see, therefore, that members of the _same_ homotopy equivalence class of
maps can have widely different Gauss images. The trick, as far as control is
concerned, is to find a representative arising from a control section (see
[3].)

##  5 Summary

We have presented ways of deriving collections of necessary conditions for
achieving dynamics of a given type and we also pointed out the limitations of
such topological conditions (due to the simplicity of the Hopf theory of maps
to a sphere.) The basic aim of any analysis is, of course, to arrive at
_constructive_ methodologies. In the treatment of this subject in [3], we find
that conditions that are both necessary and sufficient can be found for
achieving dynamics in a certain class. In this light, the fundamental source
of necessary conditions is the class of control-transverse sections and the
resulting feedback-invariant dynamics.

## References

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