# Quadratic non-stochastic operators: Examples of splitted chaos

U.A. Rozikov, S.S. Xudayarov  U.Rozikova,b,c a V.I.Romanovskiy Institute of
Mathematics of Uzbek Academy of Sciences; b AKFA University, 1st Deadlock 10,
Kukcha Darvoza, 100095, Tashkent, Uzbekistan; c Faculty of Mathematics,
National University of Uzbekistan<EMAIL_ADDRESS>S. Xudayarov
Bukhara State University, The department of Mathematics, 11, M.Iqbol, Bukhara
city. Bukhara, Uzbekistan<EMAIL_ADDRESS>
###### Abstract.

There is one-to-one correspondence between quadratic operators (mapping
$\mathbb{R}^{m}$ to itself) and cubic matrices. It is known that any quadratic
operator corresponding to a stochastic (in a fixed sense) cubic matrix
preserves the standard simplex. In this paper we find conditions on the (non-
stochastic) cubic matrix ensuring that corresponding quadratic operator
preserves simplex. Moreover, we construct several quadratic non-stochastic
operators which generate chaotic dynamical systems on the simplex. These
chaotic behaviors are _splitted_ meaning that the simplex is partitioned into
uncountably many invariant (with respect to quadratic operator) subsets and
the restriction of the dynamical system on each invariant set is chaos in the
sense of Devaney.

###### Key words and phrases:

quadratic stochastic dynamics; cubic matrix; time; Kolmogorov-Chapman equation

###### 2010 Mathematics Subject Classification:

17D92; 17D99; 60J27

##  1\. Introduction

Non-linear dynamical systems arise in many problems of biology, physics and
other sciences. In particular, such dynamical systems describe the behavior of
populations of different species with population
models111https://en.wikipedia.org/wiki/Chaos-theory [20].

It is known that there are some populations with regular behavior and other
ones with chaotic behavior [1]. The chaos means sensitivity of behavior of the
population to the tiniest changes in initial conditions (the initial state of
the population) and unpredictable behavior. Mathematically studying a chaotic
behavior is useful to our understanding of chaos as a phenomenon. In this
paper we consider several quadratic (non-linear) mappings arising in
population dynamics, which may generate a chaotic behavior. Let us give some
basic notations:

Chaos. For discrete-time dynamical systems a mathematical definition of chaos
is as follows [4]. Let $f$ be a function defined on some state space $X$.
Denote $f^{n}(x)$, meaning $f$ is applied to $x\in X$ iteratively $n$ times.

Furthermore, let $A$ be a subset of $X$. Then $f(A)=\\{f(x):x\in A\\}$. If
$f(A)\subset A$, then $A$ is an invariant set under function $f$.

A continuous map $f:X\to X$ is said to be topologically transitive if, for
every pair of non-empty open sets $A,B\subset X$, there exists an integer $n$
such that $f^{n}(A)\cap B\neq\varnothing.$

Devaney’s definition (see222https://plato.stanford.edu/entries/chaos/ [4], [7]
for more details) of chaos is stated as follows:

A continuous map $f$ is chaotic if $f$ has an invariant set $A\subset X$ such
that

1) $f$ satisfies weak sensitive dependence on its initial conditions on $A$,

2) The set of points initiating periodic orbits are dense in $A$,

3) $f$ is topologically transitive on $A$.

In [2] it was observed that sensitive dependence on initial conditions follows
as a mathematical consequence of the other two properties.

Even simple processes can lead to chaos. This is reason why so hard to predict
the weather and the stock market. One beautiful example is the game of
billiards [25]. Chaotic models are used in certain populations [10] and in the
population growth [9]. Chaos can also be found in ecological systems, such as
hydrology [29]. Some biological application is found in cardiotocography.
Models of warning signs of fetal hypoxia can be obtained through chaotic
modeling [30].

Time evolution operators333https://en.wikipedia.org/wiki/Time-evolution. Time
evolution is the change of state by the passage of time. In general, time is
not required to be a continuous parameter, but may be discrete or even finite.

Consider a system with state space $X$ for which evolution is deterministic
and reversible. For concreteness let us suppose time set $\mathbb{T}$ is the
set $\mathbb{R}$ or $\mathbb{N}_{0}=\\{0\\}\cup\mathbb{N}$.

Then time evolution is given by a family of state mappings

$F_{{t,s}}:X\rightarrow X,\quad\forall t,s\in{\mathbb{T}},$

where $F_{t,s}(x)$ is the state of the system at time $t$, whose state at time
$s$ is $x$. The following identity holds

${F}_{{u,t}}({F}_{{t,s}}(x))={F}_{{u,s}}(x).$ (1.1)

The mappings $F_{t,s}(x)$ are called evolution operators.

A state space with a distinguished evolution operators is called a dynamical
system.

Examples:

_1\. Markov process of square matrices._ One of well studied time evolution is
Markov process, which is defined by linear mappings as follows. A family of
stochastic matrices $\\{F_{s,t}=\mathcal{U}^{[s,t]}:s,t\geq 0\\}$ is called a
_Markov process_ if it satisfies the Kolmogorov-Chapman equation (i.e.
equation (1.1)):

$\mathcal{U}^{[s,t]}=\mathcal{U}^{[s,\tau]}\mathcal{U}^{[\tau,t]},\qquad\text{for
all}\ \ 0\leq s<\tau<t.$ (1.2)

Let $E=\\{1,2,\dots,m\\}$. A _distribution_ (or _state_) of the set $E$ is a
probability measure $x=(x_{1},\dots,x_{m})$, i.e. an element of the simplex:

$S^{m-1}=\left\\{x\in\mathbb{R}^{m}:x_{i}\geq 0,\
\sum_{i=1}^{m}x_{i}=1\right\\}.$

Let $x^{(0)}=(x_{1}^{(0)},\dots,x_{m}^{(0)})\in S^{m-1}$ be an initial
distribution on $E$. Denote by $x^{(t)}=(x_{1}^{(t)},\dots,x_{m}^{(t)})\in
S^{m-1}$ the distribution of the system at the moment $t$. For arbitrary
moments of time $s$ and $t$ with $s<t$ the matrix
$\mathcal{U}^{[s,t]}=\left(U^{[s,t]}_{ij}\right)$ gives the transition
probabilities from the distribution $x^{(s)}$ to the distribution $x^{(t)}$.
Moreover $x^{(t)}$ depends linearly from $x^{(s)}$:

$x^{(t)}_{k}=\sum_{i=1}^{m}U_{ik}^{[s,t]}x^{(s)}_{i},\qquad k=1,\dots,m.$

_2\. Quadratic stochastic process._ Following [3] denote by $\mathcal{S}$ the
set of all possible kinds of stochasticity and denote by $\mathbb{M}$ the set
of all possible multiplication rules of cubic matrices.

Let $\mathcal{M}^{[s,t]}=\left(P_{ijk}^{[s,t]}\right)_{i,j,k=1}^{\
\underset{m}{}}$ be a cubic matrix with two parameters.

A family $\\{F_{s,t}=\mathcal{M}^{[s,t]}:\ s,t\in\mathbb{T}\\}$ is called (see
[3], [12], [20] for details) a Markov process of cubic matrices (or a
quadratic stochastic process) of type $(\sigma|\mu)$ if for each time $s$ and
$t$ the cubic matrix $\mathcal{M}^{[s,t]}$ is stochastic in sense
$\sigma\in\mathcal{S}$ and satisfies the Kolmogorov-Chapman equation (for
cubic matrices):

$\mathcal{M}^{[s,t]}=\mathcal{M}^{[s,\tau]}*_{\mu}\mathcal{M}^{[\tau,t]},\qquad\text{for
all}\ \ 0\leq s<\tau<t.$ (1.3)

with respect to the multiplication $\mu\in\mathbb{M}$.

Quadratic stochastic processes arise naturally in the study of biological and
physical systems with interactions. Assume that the matrix
$\left(P_{ijk}^{[s,t]}\right)$ is 3-stochastic (i.e., $P_{ijk}^{[s,t]}\geq 0$
and $\sum_{k}P_{ijk}^{[s,t]}=1$), then the probability distribution $x^{(t)}$
(for the quadratic process) can be found by the formula of the total
probability as

$x^{(t)}_{k}=\sum_{i,j=1}^{m}P_{ijk}^{[s,t]}x^{(s)}_{i}x^{(s)}_{j},$ (1.4)

where $k=1,\dots,m,\ \ 0\leq s<t$.

For case when $P_{ijk}^{[s,t]}$ does not depend on $s,t$ the theory of
corresponding quadratic stochastic operator is well developed ([5], [6], [8],
[11], [13] \- [27] and references therein).

_3\. Discrete-time quadratic dynamical systems._ In this paper we consider
discrete time, i.e. $\mathbb{T}=\mathbb{N}_{0}=\\{0\\}\cup\mathbb{N}$ and in
the equality (1.4) we assume the coefficients $P_{ijk}^{[s,t]}$ do not depend
on $s$ and $t$, but the cubic matrix $\mathbb{P}=\left(P_{ijk}\right)$ is not
assumed to be stochastic.

In general, a quadratic operator $V$, $V:x\in\mathbb{R}^{m}\to
x^{\prime}=V(x)\in\mathbb{R}^{m}$ corresponding to a cubic matrix $\mathbb{P}$
is defined by:

$V:x^{\prime}_{k}=\sum\limits_{i,j=1}^{m}P_{ij,k}x_{i}x_{j},\ \ k=1,\dots,m.$
(1.5)

Without loss of generality we assume $P_{ij,k}=P_{ji,k}$. Indeed, if this
equality is not satisfied then we can introduce

$\overline{P}_{ij,k}=\frac{1}{2}(P_{ij,k}+P_{ji,k}).$

The aim of this paper is to find conditions on the cubic matrix $\mathbb{P}$
ensuring that corresponding operator preserves simplex $S^{m-1}$. Moreover, we
want to construct several quadratic (non-stochastic) operators which generate
chaotic dynamical systems on the simplex.

##  2\. Quadratic non-stochastic operators

The following theorem gives conditions for coefficients of $V$ to preserve the
simplex.

######  Theorem 1.

For a quadratic operator $V$ (given by (1.5)), to preserve a simplex $S^{m-1}$
it is sufficient that

  * i)

$\sum\limits_{k=1}^{m}P_{ij,k}=1,\,\,\ i,j=1,\dots,m;$

  * ii)

$0\leq P_{ii,k}\leq 1,\,\,\ i,k=1,\dots,m;$

  * iii)

$-{1\over m-1}\sqrt{P_{ii,k}P_{jj,k}}\leq P_{ij,k}.$

and necessary that the conditions i), ii) and

  * iii’)

$-\sqrt{P_{ii,k}P_{jj,k}}\leq P_{ij,k}\leq 1+\sqrt{(1-P_{ii,k})(1-P_{jj,k})}$

are satisfied.

###### Proof.

Sufficiency. Let $x\in S^{m-1}$. We show that
$x^{\prime}=V(x)=(x_{1}^{\prime},\dots,x_{m}^{\prime})\in S^{m-1}$. Using
condition i) we get

$\sum_{k=1}^{m}x^{\prime}_{k}=\sum_{k=1}^{m}\sum\limits_{i,j=1}^{m}P_{ij,k}x_{i}x_{j}=\sum\limits_{i,j=1}^{m}\left(\sum_{k=1}^{m}P_{ij,k}\right)x_{i}x_{j}$
$=\sum\limits_{i,j=1}^{m}x_{i}x_{j}=\sum\limits_{i=1}^{m}x_{i}\sum\limits_{j=1}^{m}x_{j}=1,\
\ \forall k=1,\dots,m.$

The condition ii) is needed to have $V(e_{i})\in S^{m-1}$, for vertices
$e_{i}=(0,...,0,1,0,...0)$ (here 1 is the $i$th coordinate). Because in this
case $V(e_{i})=(P_{ii,1},\dots,P_{ii,m}).$ Now using iii) we show that
$x_{k}^{\prime}\geq 0$:

$x^{\prime}_{k}=\sum\limits_{i,j=1}^{m}P_{ij,k}x_{i}x_{j}=\sum_{i<j}\left({P_{ii,k}\over
m-1}x_{i}^{2}+2P_{ij,k}x_{i}x_{j}+{P_{jj,k}\over m-1}x_{j}^{2}\right)$
$\geq\sum_{i<j}\left({P_{ii,k}\over m-1}x_{i}^{2}-{2\over
m-1}\sqrt{P_{ii,k}P_{jj,k}}x_{i}x_{j}+{P_{jj,k}\over m-1}x_{j}^{2}\right)$
$=\sum_{i<j}\left(\sqrt{P_{ii,k}\over m-1}x_{i}-\sqrt{P_{jj,k}\over
m-1}x_{j}\right)^{2}\geq 0.$

Therefore the quadratic operator $V$ preserves the simplex.

######  Remark 1.

  * 1.

In [26], for $m=2$, it is proven that the conditions i), ii) and iii’) are
sufficient and necessary to preserve the simplex.

  * 2.

In case $m\geq 3$ the conditions i), ii) and iii’) are not sufficient to
preserve the simplex. Indeed, consider the example (satisfying i), ii) and
iii’)):

$\begin{array}[]{lll}P_{ii,1}=1,\ \ P_{ii,k}=0,\ \ \forall i=1,\dots,m;\ \
\forall k=2,\dots,m;\\\\[5.69054pt] P_{ij,1}=-\sqrt{P_{ii,1}P_{jj,1}}=-1,\ \
\forall i\neq j;\\\\[5.69054pt] P_{ij,k}\in[0,2],\forall i\neq j,\,k\geq 2\ \
\mbox{with}\ \ \sum\limits_{k=2}^{m}P_{ij,k}=2.\end{array}$

Then for the first coordinate of the corresponding quadratic operator $V$ we
have

$x_{1}^{\prime}=\sum_{i=1}^{m}x_{i}^{2}-2\sum_{1\leq i<j\leq
m}x_{i}x_{j}=\left(x_{1}-\sum_{i=2}^{m}x_{i}\right)^{2}-4\sum_{2\leq i<j\leq
m}x_{i}x_{j}$ $=(2x_{1}-1)^{2}-4\sum_{2\leq i<j\leq m}x_{i}x_{j}.$

Take $x\in S^{m-1}$ such that $x_{1}=1/2$ and $x_{j}>0$, $j\geq 2$ with
$\sum_{j=2}^{m}x_{j}=1/2$. Then $x_{1}^{\prime}<0$, i.e.
$x^{\prime}=V(x)\notin S^{m-1}$.

Necessity. Now following [27] we prove necessity of conditions i)-iii’).

Assume $V$ preserves the simplex. As we mentioned above condition ii) is
needed for $V(e_{i})=(P_{ii,1},\dots,P_{ii,m})\in S^{m-1}$, which also
requires $\sum_{k=1}^{m}P_{ii,k}=1$ (a particular case of the condition i)).
To show that i) is necessary, let us take $x=\alpha e_{i}+\beta e_{j}$, where
$\alpha,\beta\geq 0$, $\alpha+\beta=1$. Then for $V(x)$ we have

$x_{k}^{\prime}=P_{ii,k}\alpha^{2}+2P_{ij,k}\alpha\beta+P_{jj,k}\beta^{2},\ \
k=1,\dots,m.$

Since $V$ preserves simplex, we have

$0\leq P_{ii,k}\alpha^{2}+2P_{ij,k}\alpha\beta+P_{jj,k}\beta^{2}\leq 1,\ \
k=1,\dots,m.$ (2.1)

and since $\beta=1-\alpha$ we have

$\sum_{k=1}^{m}x_{k}^{\prime}=\sum_{k=1}^{m}\left(P_{ii,k}\alpha^{2}+2P_{ij,k}\alpha(1-\alpha)+P_{jj,k}(1-\alpha)^{2}\right)=1$

which for $\alpha=1/2$ by $\sum_{k=1}^{m}P_{ii,k}=1$ gives i).

To obtain condition iii’) in (2.1) we denote

$a=P_{ii,k},\ \ b=P_{ij,k},\ \ c=P_{jj,k}.$ (2.2)
$f(\alpha)=(a-2b+c)\alpha^{2}+2(b-c)\alpha+c.$

Then inequalities (2.1) equivalent to find conditions on parameters $a,b,c$
such that $f(\alpha)\in[0,1]$ for each $\alpha\in[0,1]$. We have
$f(0)=c\in[0,1]$ and $f(1)=a\in[0,1]$.

Case: $a-2b+c=0$. That is $b=(a+c)/2\geq 0$. Then $f(\alpha)=2(b-c)\alpha+c$.
The graph of this linear function connects points $(0,c)$ and $(1,a)$ and
contained in $[0,1]^{2}$.

Case: $a-2b+c\neq 0$. In this case the function $f(\alpha)$ has its extremum
point

$\alpha_{0}={c-b\over a-2b+c}.$

Therefore, $f(\alpha)\in[0,1]$ if and only if one of the following conditions
holds

  * (1)

$\alpha_{0}\notin(0,1)$

  * (2)

$\alpha_{0}\in(0,1)$ and $f(\alpha_{0})\in[0,1]$.

By elementary analysis it is easy to see that solution to the inequalities
associated with conditions (1) and (2) is

$-\sqrt{ac}\leq b\leq 1+\sqrt{(1-a)(1-c)}.$

This by using notations (2.2) shows that condition iii’) is necessary too. ∎

######  Definition 1.

A quadratic operator (1.5), preserving a simplex, is called non-stochastic if
at least one of its coefficients $P_{ij,k}$, $i\neq j$ is negative.

######  Definition 2.

Let $V:S^{m-1}\rightarrow S^{m-1}$ be a quadratic non-stochastic operator
(QnSO). It is called a Volterra QnSO (VQnVO) if

iv) $P_{ij,k}=0,$ for all $k\notin\\{i,j\\}$.

######  Theorem 2.

If for a quadratic operator the conditions i)-iii) of Theorem 1 are satisfied
then it is not a VQnSO.

###### Proof.

Assume i)-iii) are satisfied then we get

$P_{kk,k}=1,\ \ \ P_{ki,k}+P_{ik,k}=1.$ $P_{ii,k}=0,\,\,\ i\neq k.$

Moreover, by iii) we have

$-{1\over m-1}\sqrt{P_{ii,k}}\sqrt{P_{kk,k}}\leq P_{ik,k}\quad\Rightarrow
0\leq P_{ik,k},\,\ i\neq k.$

Similarly we get $0\leq P_{ki,k}$. Thus all coefficients of Volterra quadratic
operator are non-negative. Hence it is not VQnSO. ∎

##  3\. One-dimensional QnSO.

Here we give a review of results related to the one-dimensional QnSO. Consider
arbitrary QnSO on $S^{1}$:

$\begin{array}[]{ll}x^{\prime}=ax^{2}+2bxy+cy^{2}\\\
y^{\prime}=(1-a)x^{2}+2(1-b)xy+(1-c)y^{2},\end{array}$ (3.1)

where

$a,c\in[0,1],\ \ b\in[-\sqrt{ac},\ \ 1+\sqrt{(1-a)(1-c)}].$ (3.2)

Using $x+y=1$, the operator (3.1) can be reduced to the function

$f(x)=(a-2b+c)x^{2}+2(b-c)x+c.$

Under condition (3.2) we have $f:[0,1]\to[0,1]$. The dynamical system
generated by $f$ can be fully studied. In [16, Section 2.2], the case $b\geq
0$ of this (stochastic) operator was studied. In [27] a class of QnSOs
$V:S^{1}\to S^{1}$ were studied.

Here to avoid several cases, we consider the case $a=c=1$, then $b\in[-1,1]$.
The operator is QnSO iff $b\in[-1,0)$. Therefore, using $x+y=1$ from the
second equality of (3.1) we get

$y^{\prime}=2(1-b)y(1-y),\ \ b\in[-1,0).$

Denote $\mu=2(1-b)$. From $b\in[-1,0)$ it follows that $\mu\in(2,4]$.

The function $g(y)\equiv g_{\mu}(y)=\mu y(1-y)$ is well-known as a logistic
map. For an initial point $x_{0}\in[0,1]$ consider the trajectory (dynamical
system):

$x_{n+1}=g(x_{n}),\ \ n=0,1,2,\dots$ (3.3)

For $\mu\in(2,4]$ this dynamical system has the following
properties444https://en.wikipedia.org/wiki/Logistic-map:

######  Remark 2.

  * •

The function $g(y)$ has two fixed points $0$ and $1-{1\over\mu}$.

  * •

If $\mu$ between 2 and 3, the trajectory will eventually approach the fixed
point $1-{1\over\mu}$, but first will fluctuate around that value for some
time.

  * •

If $\mu$ between 3 and $1+\sqrt{6}\approx 3.44949$, from almost all initial
point $x_{0}$ the trajectory will approach 2-periodic orbit. These two values
are dependent on $\mu$.

  * •

With $\mu$ between 3.44949 and 3.54409 (approximately), from almost all
initial point $x_{0}$ the trajectory will approach 4-periodic orbit (permanent
oscillations among four values).

  * •

With $\mu$ increasing beyond 3.54409, from almost all initial points the
trajectory will approach oscillations among 8 values, then 16, 32, etc.

  * •

At $\mu$ approximately equal to $3.56995$ from almost all initial points, we
no longer see oscillations of finite period. Slight variations in the initial
point yield dramatically different results over time, a prime characteristic
of chaos.

  * •

A rough description of chaos is that chaotic systems exhibit a great
sensitivity to initial points. The logistic map for most values of
$\mu>3.56995$ exhibits chaotic behavior.

######  Remark 3.

We do not know any quadratic stochastic operator with chaotic behavior of
trajectories. In the (above considered) case: $a=c=1$, $b\in[-1,0)$, the
operator (3.1) has the form

$\begin{array}[]{ll}x^{\prime}=x^{2}+2bxy+y^{2}\\\
y^{\prime}=2(1-b)xy.\end{array}$ (3.4)

Since $P_{12,1}=b<0$, this operator is non-stochastic. For arbitrary initial
point $(x_{0},1-x_{0})\in S^{1}$, its trajectory has the form
$(x_{n},1-x_{n})$, where $x_{n}$ is defined by (3.3). Therefore, from above-
mentioned properties of the logistic map, it follows that when $-1\leq
b<-0.784975$ the operator (3.4) generates a chaotic dynamical system on the
one-dimensional simplex.

##  4\. Examples of two-dimensional QnSO

1\. Consider the following example of QnSO on the two-dimensional simplex
$S^{2}$:

$\begin{array}[]{lll}x^{\prime}=x^{2}+y^{2}+z^{2}-axy-axz+2yz\\\\[5.69054pt]
y^{\prime}=(2+a)xy\\\\[5.69054pt] z^{\prime}=(2+a)xz,\end{array}$ (4.1)

where $a\in[0,2]$. Note that $P_{12,1}=P_{13,1}=-a/2$.

1.1. Fixed points. The fixed points are solutions to the system

$\begin{array}[]{lll}x=x^{2}+y^{2}+z^{2}-axy-axz+2yz\\\\[5.69054pt]
y=(2+a)xy,\\\\[5.69054pt] z=(2+a)xz.\end{array}$ (4.2)

In case $y=z=0$ we get the fixed point $p_{0}=(1,0,0)$. For $y+z\neq 0$ we get
$x={1\over 2+a}$. Consequently, the following is a family of fixed points

$p_{y}=\left({1\over 2+a},y,1-{1\over 2+a}-y\right),\ \ {\rm where}\ \
y\in\left[0,1-{1\over 2+a}\right].$

1.2. On invariant sets. Recall that a set $M$ is called invariant with respect
to an operator $V$ if $V(M)\subset M$.

It is easy to see that the following sets are invariant with respect to (4.1):

$M_{0}=\\{(x,y,z)\in S^{2}:y=0\\},\ \ M_{1}=\\{(x,y,z)\in S^{2}:z=0\\},$
$M_{\omega}=\\{(x,y,z)\in S^{2}:y=\omega z\\},\ \ \omega\in[0,+\infty).$

Denoting $t=y+z$ we reduce the operator (4.1) to the following

$\begin{array}[]{ll}x^{\prime}=x^{2}+t^{2}-axt\\\\[5.69054pt]
t^{\prime}=(2+a)xt.\end{array}$ (4.3)

Since $x=1-t$. The operator (4.3) coincides with (3.4). Therefore, under
condition $1.56995<a\leq 2$ this operator and (4.1) generate chaotic dynamical
systems. Moreover, using $y+z=t$ one can study trajectories of (4.1) by
related trajectories of (4.3). For example, if a trajectory $(x_{n},t_{n})$ of
(4.3) has a limit, say $(\alpha,\beta)$, then the corresponding trajectory
$(x_{n},y_{n},z_{n})$ has property that

$\lim_{n\to\infty}x_{n}=\alpha,\ \ \lim_{n\to\infty}(y_{n}+z_{n})=\beta.$

The fixed point $t_{*}=1-{1\over 2+a}$ of the function $(2+a)t(1-t)$ gives an
invariant set with $y+z=t_{*}$, i.e.,

$X=\\{(x,y,z)\in S^{2}:x={1\over 2+a}\\}.$

Note that the above mentioned fixed point $p_{y}$ is the following

$p_{y}=X\cap M_{\omega},\ \ {\rm where}\ \ \omega={(2+a)y\over 1+a-(2+a)y}.$

Moreover,

$S^{2}=M_{0}\cup M_{1}\cup(\bigcup_{\omega\in(0,+\infty)}M_{\omega}).$

Therefore, it suffices to study restrictions of the operator (4.1) on each
these invariant sets.

Restriction of the operator (4.1) on $M_{0}$ (using $x=1-z$) can be written as
the function $z^{\prime}=(2+a)z(1-z)$. Similarly, on $M_{1}$ one has the same
function $y^{\prime}=(2+a)y(1-y).$

For each $\omega\in(0,+\infty)$, on the set $M_{\omega}$ the restriction of
(4.1) can be written as

$z^{\prime}=(2+a)(1-(\omega+1)z)z.$ (4.4)

Multiply both side of (4.4) to $1+\omega$ and denote $\zeta=(1+\omega)z$ then
we get $\zeta^{\prime}=(2+a)\zeta(1-\zeta).$

Therefore, the trajectories of the operator on the invariants $M_{0}$, $M_{1}$
and $M_{\omega}$ are given by the same logistic function. For $\mu=2+a$ using
facts of Remark 2 one can give dynamics of these functions (i.e., the dynamics
of operator (4.1) on the above mentioned invariants). In particular, under
condition $1.56995<a\leq 2$ each one-dimensional dynamical system is chaotic.

######  Remark 4.

As we have seen, the operator (4.1) is chaotic for $1.56995<a\leq 2$, but it
is not chaos on the simplex in the sense of Devaney. Because, it is not
topologically transitive. It is _splitted_ chaos meaning that the simplex is
partitioned into uncountably many invariant subsets and the restriction of the
operator (4.1) on each invariant set is chaos in the sense of Devaney.

2\. Consider the following example:

$\begin{array}[]{lll}P_{ii,1}=1,\ \ P_{ii,k}=0,\ \ \forall i=1,2,3;\ \ \forall
k=2,3;\\\\[5.69054pt] P_{ij,1}=-{1\over 2}\sqrt{P_{ii,1}P_{jj,1}}=-{1\over
2},\ \ \forall i\neq j;\\\\[5.69054pt] P_{ij,k}\in[0,{3\over 2}],\forall i\neq
j,\,k=2,3\ \ \mbox{with}\ \ P_{ij,2}+P_{ij,3}={3\over 2}.\end{array}$

Then taking some parameters equal to zero we get the following quadratic
operator $V$:

$\begin{array}[]{lll}x^{\prime}=x^{2}+y^{2}+z^{2}-xy-xz-yz\\\\[5.69054pt]
y^{\prime}=3xy+ayz\\\\[5.69054pt] z^{\prime}=3xz+(3-a)yz,\end{array}$ (4.5)

where $a\in(0,3)$.

######  Remark 5.

In the operator (4.5) one can also consider the cases $a=0$ and $a=3$. These
cases are more simple than the case $a\in(0,3)$. Because, for example, if
$a=0$ then $z^{\prime}=3z(x+y)=3z(1-z)$, i.e., the variable $z$ has dynamics
independent from other variables. Therefore, below we consider the case $a\neq
0$, $a\neq 3$.

2.1. Fixed points. It is easy to see that the fixed points of the operator
(4.5) are

$\begin{split}s_{1}=(1,0,0),s_{2}=\left({1\over 3},0,{2\over
3}\right),s_{3}=\left({1\over 3},{2\over 3},0\right),\\\\[8.53581pt]
s_{4}=\left({a^{2}-3a+3\over a^{2}-3a+9},{2a\over a^{2}-3a+9},{2(3-a)\over
a^{2}-3a+9}\right).\end{split}$ (4.6)

######  Definition 3.

[4]. A fixed point $x^{*}$ of the operator $V$ is called hyperbolic if its
Jacobian $J$ at $x^{*}$ has no eigenvalues on the unit circle.

######  Definition 4.

[4]. A hyperbolic fixed point $x^{*}$ is called:

  * i)

attracting if all the eigenvalues of the Jacobian $J(x^{*})$ are less than 1
in absolute value;

  * ii)

repelling if all the eigenvalues of the Jacobian $J(x^{*})$ are greater than 1
in absolute value;

  * iii)

a saddle otherwise.

To study the type of each fixed point rewrite operator (4.5) (using $x=1-y-z$)
as

$W:\begin{array}[]{ll}y^{\prime}=3xy+ayz=y(3-3y+(a-3)z)\\\\[5.69054pt]
z^{\prime}=3xz+(3-a)yz=z(3-ay-3z).\end{array}$ (4.7)

Note that $W$ maps the set $T=\\{(y,z)\in[0,1]^{2}:y+z\leq 1\\}$ to itself.

The Jacobian of $W$ at point $(y,z)$ is

$J_{W}(y,z)=\left(\begin{array}[]{cc}3-6y+(a-3)z&(a-3)y\\\\[5.69054pt]
-az&3-ay-6z\end{array}\right).$

For eigenvalues of the Jacobian at fixed points we have

  * Case $s_{1}$:

$\lambda_{1}=\lambda_{2}=3$

  * Case $s_{2}$:

$\lambda_{1}=1+{2a\over 3}\in[1,3],\ \ \lambda_{2}=-1$

  * Case $s_{3}$:

$\lambda_{1}=-1,\ \ \lambda_{2}=3-{2a\over 3}\in[1,3]$

  * Case $s_{4}$:

In this case $\lambda_{1}$ has a bulky form. But using Maple one can plot its
graph (see Fig. 1). Therefore, $0<\lambda_{1}<1$ for any $a\in(0,3)$.
Moreover, one can see that $\lambda_{2}=-1$.

Figure 1. The graph of the eigenvalue $\lambda_{1}(s_{4})$ as function of
parameter $a\in[0,3]$.

Thus we have proved the following

######  Proposition 1.

Fixed point $s_{1}$ is repeller. Points $s_{2}$ and $s_{3}$ are non hyperbolic
(but semi-repeller555meaning that the second eigenvalue is greater than 1 in
absolute value.). The fixed point $s_{4}$ is non-hyperbolic (but semi-
attracting666meaning that the second eigenvalue is less than 1 in absolute
value.).

2.2. Invariant sets. Let $a\in(0,3)$. Introduce the following sets:

$M_{1}=\\{(y,z)\in T:y=0\\},\ \ M_{2}=\\{(y,z)\in T:z=0\\},$
$M_{3}=\\{(y,z)\in T:z={3-a\over a}y\\}.$ $M_{4}=\\{(y,z)\in T:z<{3-a\over
a}y\\},\ \ M_{5}=\\{(y,z)\in T:z>{3-a\over a}y\\}.$

######  Lemma 1.

The sets $M_{i}$, $i=1,\dots,5$ are invariant with respect to the operator
$W$, i.e. (4.7).

###### Proof.

It is easy to see that $W(M_{i})\subset M_{i}$, $i=1,2$. For the case
$i=3,4,5$ assume $(y,z)\in M_{i}$, we shall show that
$(y^{\prime},z^{\prime})=W(y,z)\in M_{i}$. From the first and second
equalities of (4.7) we find

$yz={1\over a}(y^{\prime}-3xy),\ \ \ yz={1\over 3-a}(z^{\prime}-3xz).$

Consequently

${1\over a}(y^{\prime}-3xy)={1\over 3-a}(z^{\prime}-3xz)\ \ \Leftrightarrow\ \
(3-a)y^{\prime}-az^{\prime}=3x((3-a)y-az).$ (4.8)

From the last equality it follows that

$(3-a)y-az=0\ \ \Leftrightarrow\ \ (3-a)y^{\prime}-az^{\prime}=0.$

Thus $M_{3}$ is an invariant. Moreover, from (4.8) it follows that if $x=0$
then $(3-a)y^{\prime}-az^{\prime}=0$, this means that any point $(y,z)$ with
$y+z=1$ (i.e. $x=0$) after first iteration goes inside of the invariant set
$M_{3}$. Therefore, consider $x>0$, then from (4.8) we get

$(3-a)y-az>0\ \ \Leftrightarrow\ \ (3-a)y^{\prime}-az^{\prime}>0.$
$(3-a)y-az<0\ \ \Leftrightarrow\ \ (3-a)y^{\prime}-az^{\prime}<0.$

Thus $M_{4}$ and $M_{5}$ are invariant sets. ∎

Note that

$T=\bigcup_{i=1}^{5}M_{i}.$ (4.9)

2.3. Trajectories. In this subsection for any initial point
$(y^{(0)},z^{(0)})\in T$ we investigate behavior of the trajectories
$(y^{(n)},z^{(n)})=W^{n}(y^{(0)},z^{(0)}),\ \ n\geq 1.$

By (4.9) it suffices to study the trajectories on each invariant set.

Case $M_{1}$: Reducing (4.7) on $M_{1}$ we get one-dimensional dynamical
system generated by $z^{\prime}=3z(1-z)$ which is a logistic map with
parameter $\mu=3$. For this function it is known (see Remark 2 and [28, page
10]) that it has repeller fixed point $z=0$ and attracting fixed point
$z=2/3$. Consequently, for trajectory of the operator (4.7) on the invariant
set $M_{1}$ we have

$\lim_{n\to\infty}(y^{(n)},z^{(n)})=\lim_{n\to\infty}W^{n}(y^{(0)},z^{(0)})=\left\\{\begin{array}[]{ll}(0,0),\
\ \mbox{if}\ \ (y^{(0)},z^{(0)})=(0,0)\\\\[5.69054pt] (0,2/3),\ \ \mbox{if}\ \
(y^{(0)},z^{(0)})=(0,z^{(0)}),z^{(0)}>0.\end{array}\right.$

Case $M_{2}$: is similar to the case $M_{1}$.

Case $M_{3}$: Restricting $W$ on $M_{3}$ we get
$y^{\prime}=3y(1-{9-3a+a^{2}\over 3a}y)$. Denoting $t={9-3a+a^{2}\over 3a}y$
and $t^{\prime}={9-3a+a^{2}\over 3a}y^{\prime}$ the last mapping can be
written as $t^{\prime}=3t(1-t)$. Therefore, this case also conjugate to the
above cases and the following holds

$\lim_{n\to\infty}(y^{(n)},z^{(n)})=\left\\{\begin{array}[]{ll}(0,0),\ \
\mbox{if}\ \ (y^{(0)},z^{(0)})=(0,0)\\\\[5.69054pt] \left({2a\over
a^{2}-3a+9},{2(3-a)\over a^{2}-3a+9}\right),\ \ \mbox{if}\ \
(y^{(0)},z^{(0)})=(y^{(0)},{3-a\over a}y^{(0)}),y^{(0)}>0.\end{array}\right.$

Case $M_{4}$: In this case for $y>0$ we have ${z\over y}<{3-a\over a}$ and
${z^{\prime}\over y^{\prime}}<{3-a\over a}$. Iterating the inequalities we get

$P_{n}:={z^{(n)}\over y^{(n)}}<{3-a\over a},\ n\geq 1.$ (4.10)

By (4.7) we have

$P_{n+1}=P_{n}\cdot{3-ay^{(n)}-3z^{(n)}\over 3-3y^{(n)}+(a-3)z^{(n)}}>P_{n},$

because:

${3-ay^{(n)}-3z^{(n)}\over 3-3y^{(n)}+(a-3)z^{(n)}}>1\ \ \Leftrightarrow\ \
-ay^{(n)}-3z^{(n)}>-3y^{(n)}+(a-3)z^{(n)}\ \ \Leftrightarrow\ \
P_{n}<{3-a\over a}.$

Hence $P_{n}$ is strictly increasing and with the upper bound ${3-a\over a}$.
Consequently, its limit exists and equal to ${3-a\over a}$ as the supremum of
$P_{n}$ in $M_{3}\cup M_{4}$.

Case $M_{5}$: This case is similar to the case $M_{4}$, now $P_{n}>{3-a\over
a}$ and $P_{n}$ is strictly decreasing with limit ${3-a\over a}$ too.

Thus for any initial point from $M_{4}\cup M_{5}$ the set of limit points of
its trajectory is subset of $M_{3}$.

Now we give limit points of the operator (4.5). To do this we introduce:

$\hat{M}_{i}=\\{(x,y,z)\in S^{2}:(y,z)\in M_{i}\\},\ \ i=1,2,3,4,5.$

Then

$S^{2}=\bigcup_{i=1}^{5}\hat{M}_{i}.$

Summarizing above-mentioned results about trajectories of $W$, we obtain the
following

######  Theorem 3.

If $(x^{(0)},y^{(0)},z^{(0)})\in\hat{M}_{i}$ for some $i=1,2,3,4,5$ then for
the operator (4.5) the following holds

$\lim_{n\to\infty}V^{n}(x^{(0)},y^{(0)},z^{(0)})=\left\\{\begin{array}[]{lllll}s_{1}\
\ \mbox{if}\ \ x^{(0)}=1\\\\[5.69054pt] s_{2}\ \ \mbox{if}\ \
i=1,z^{(0)}>0\\\\[5.69054pt] s_{3}\ \ \mbox{if}\ \
i=2,y^{(0)}>0\\\\[5.69054pt] s_{4}\ \ \mbox{if}\ \
i=3,y^{(0)}>0\\\\[5.69054pt] \in\hat{M}_{3}\ \ \mbox{if}\ \
i=4,5,\end{array}\right.$

where $s_{i}$, $i=1,2,3,4$ are defined in (4.6).

Based on numerical analysis we make the following

Conjecture. If $(x^{(0)},y^{(0)},z^{(0)})\in\hat{M}_{4}\cup\hat{M}_{5}$ then
for the operator (4.5) the following holds

$\lim_{n\to\infty}V^{n}(x^{(0)},y^{(0)},z^{(0)})=s_{4}.$

Thus the operator (4.5) does not generate a chaotic dynamical system.

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When, Where, and How to Open Data
A Personal Perspective

Benjamin Nachman

Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA
Berkeley Institute for Data Science, University of California, Berkeley, CA
94720, USA

> ABSTRACT
>
>
>
>
> This is a personal perspective on data sharing in the context of public data
> releases suitable for generic analysis. These open data can be a powerful
> tool for expanding the science of high energy physics, but care must be
> taken in when, where, and how they are utilized. I argue that data
> preservation even within collaborations needs additional support in order to
> maximize our science potential. Additionally, it should also be easier for
> non-collaboration members to engage with collaborations. Finally, I advocate
> that we recognize a new type of high energy physicist: the ‘data physicist’,
> who would be optimally suited to analyze open data as well as develop and
> deploy new advanced data science tools so that we can use our precious data
> to their fullest potential.
>
>
> This document has been coordinated with a white paper on open data
> commissioned by the American Physical Society’s (APS) Division of Particles
> and Field (DPS) Community Planning Exercise (‘Snowmass’) Theory Frontier [1]
> and relevant also for the Computational Frontier.

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

##  1 Introduction and Overview

There is increasing interest across many disciplines in “open science”. This
can refer to sharing analysis code, data, experimental machinery, or protocols
with people other than the scientists who did the original work. It can be
motivated by a desire to archive the data for later re-analysis, to drive a
faster pace of discovery by involving more scientists, or to ensure
reproducibility and transparency in science. How to produce policies from
these arguments is an ongoing challenge – one in which modern high energy
physics (HEP) experiments can be instructive***See Ref. [1] for a brief
historical context., as they offer extreme examples of dataset and
collaboration size.

Typical HEP experimental papers have hundreds to thousands of authors. We have
this tradition because data “collection and curation” includes detector design
and operation, simulation development, pattern recognition, and calibration,
and is a significant part of our collective work and is highly integrated with
data analysis. While the growing divide between “theorists” and
“experimentalists” is worrisome for the future of HEP, it currently has
important implications for funding and career advancement. The HEP model is
different from nearby fields with a tradition of open data. For example, the
researchers who build and operate detectors in astronomy are often not the
people who analyze the data.

It is also worth noting that our field has a long history of data sharing,
where anyone in the collaboration (a considerable fraction of all HEP
physicists) has access to the raw data. We also have advanced procedures for
collaborating with external scientists and sharing data for targeted usage via
tools such as HepData [2]. While this setup has been very successful, it is
being challenged by the recent public release of data suitable for generic
data analysis by the CMS collaboration [3] (to be followed by the other LHC
experiments [4]). Henceforth, “open data” will refer to data suitable for
generic data analysis and not a derived data product like digitized figures or
tables. I will also mostly be referring to open data in the context of running
experiments, although I will explore the connection with legacy data analysis
as well.

The motivation for open data seems to be two-fold: ensuring data preservation
and increasing the chances for discovery. Data preservation is of paramount
importance and is critically understaffed for ongoing and finished
experiments. Releasing and stress-testing data during the lifetime of an
experiment is certainly one way to achieve data preservation. However, this is
not the only way. Data preservation can also be accomplished within the
collaboration by ensuring that all older data can be analyzed. This can be
stress-tested automatically with continuous integration workflows or manually
by performing physics analysis on older data, as was done in the case of the
recent $W$ boson mass measurement from the ATLAS collaboration using early Run
1 data [5]. It is not a given that older data are readily accessible
internally; in fact, I would be surprised if any new graduate student on ATLAS
would be able to readily interrogate the data from Run 1 as data formats and
software tools have changed significantly. This is a serious problem that
should be addressed independent of data releases. Some studies with the CMS
open data [6, 7, 1] have discussed the challenges with using these data. These
stories sounded painfully familiar of the typical graduate student struggle.
While most students do not get to write a paper about this struggle (it
usually does not even make their thesis), their challenge is no less important
and indicative of a serious challenge for data access and preservation
internal to experimental collaborations. This does not have to be a necessary
part of the graduate student experience.

In terms of discovery, having more “eyes on the data” is not obviously a
problem for our field - the collaborations at the Large Hadron Collider (LHC)
have many thousands of scientists and there are well-tested mechanisms for
external scientists (including theorists) to join for generic (by becoming an
author) or limited-scope projects. Proposals for new ideas can be readily
carried out with simulations (even “open simulations” if the situation calls
for it).

While it may not be necessary for data preservation and for broadening HEP
science, open data could advance both of these areas. Along with these
potential benefits, there may also be potential costs. If the data are made
available to those who are not involved in the “collection and curation”, what
is the incentive to take part in these important activities? If the incentive
to collect complex data is removed, the complex data may never be collected.
Furthermore, those not involved in the “collection and curation” will likely
not be able to perform accurate and precise experimental measurements because
they are not familiar with the intricate details of the data. Time delays are
often a mechanism proposed to protect the incentive to work on data collection
and curation. However, any time delay other than the lifetime of an experiment
is a statement about what sort of science is most “interesting”. Multi year
delays would likely protect high profile, statistics-limited analyses.
Systematic uncertainty-limited analyses would not be protected (e.g. precision
Standard Model measurements). See Fig. 1 for the actual time delays between
data collection and paper completion - there is a clear shift between searches
and measurements with a heavy tail to the right for the latter. It has also
not been long enough since the end of Run 2 to know what the true tail will
look like in the future.

Figure 1: A histogram showing the time between the end of data taking and the
posting of a paper to arXiv for physics papers from the ATLAS Collaboration,
scraped from https://twiki.cern.ch/twiki/bin/view/AtlasPublic. ‘Searches’
include publications from the SUSY, Exotics, and HDBS groups while
‘Measurements’ includes publications from the Standard Model, Top Quark, and B
Physics groups. Datasets less than 1 fb-1 are excluded. There are 703 papers
in the above histogram.

It is often stated that open data would level the playing field and make HEP
science more equitable. However, open data may also have the opposite effect.
Large collaborations have membership from a diverse set of institutes who do
not all have the same local resources. Analyzing HEP data requires significant
resources (computational and otherwise). Working within a HEP collaboration
ensures that all members have a chance to analyze these data. In contrast,
open data is only analyzable by wealthy groups (where ‘wealth’ has multiple
meanings). There is also the unfortunate reality that researcher diversity
decreases with seniority (see e.g. Ref. [8]) while the bulk of the collection
and curation work within collaborations is performed by relatively early
career researchers.

Lastly, open data may reduce the integrity of HEP science if analysis scope
and reviewer expectations are not clear and calibrated. The release of CMS
open data has already created at least one incident where a paper from a
collaboration was initially rejected from a journal because the reviewers
believed the analysis had already been done by theorists working on CMS open
data [9]. Open data makes it “easy” to compare predictions to data, but if
those data are not carefully corrected for detector effects or if
uncertainties are not quantified, then the comparison may not be meaningful. A
study comparing uncorrected data with predictions is not the same as a proper
“measurement”. The word “measurement” is in quotes because it carries a
certain gravitas with “experimentalists” that it often does not with
“theorists”. This could be addressed with a more integrated training of
“experimentalists” and “theorists” in the future (see below). It is also worth
noting that many applications of open data could simply use simulation (or
even open simulation) - the data themselves do not add to the scientific
aspect of the research. Once again, this is often the case for papers that
propose new methods. There is also a difference between ‘performance
studies†††It is unfortunate that these studies are often only findable on
collaboration wiki pages or at best, the CERN Document Server. This is
changing with many studies findable on Inspire, but it would increase idea
sharing across the community to also post them to arXiv.’ (e.g. Refs. [10,
11]) and ‘measurements’ (e.g. Refs. [12, 13]).

##  2 Experience with Diverse Open Data Models

This section briefly illustrates my personal experiences with different models
of data sharing.

### ATLAS.

ATLAS does not yet make data available for generic data analysis, although a
number of datasets are available for outreach [14] and a growing number of
open simulations have been published (see e.g. Ref. [15]). The goal of this
section is to describe the experience (from the “experimentalist” perspective)
of “theorists” working directly with ATLAS. This is done through short term
associate (STA) positions that are approved by the collaboration management
[16]. I have worked with STAs on a number of occasions [12, 17]. The threshold
to set up one of these positions is quite low and readily allows for the
“theorist” to interact with internal data/simulation. This was highly
effective. It would be useful to hear the perspective from the “theorists” to
see if their experience was as positive.

### CMS.

I do not have any direct experience analyzing CMS Open Data, but I would
completely believe the stories in Refs. [6, 7, 1] about how difficult it is to
analyze them from the public releases (despite the impressive effort by CMS to
make it as easy as possible). On the other hand, the simplified data formats
made available through the EnergyFlow package (https://energyflow.network)
make it very simple to access these data after some processing. I have used
the corresponding simulations in a number of studies [18, 19, 20], where
having full detector simulations was actually necessary to make a point. The
EnergyFlow package is a great service to the community and certainly lowers
the barrier to entry for Open Data/Open Simulation studies.

### HERA/H1.

Unlike the $e^{+}e^{-}$ experiments at the $Z$ pole (LEP/SLD), the deep
inelastic scattering experimental collaborations at HERA continue to exist.
Their software infrastructure has been modernized [21] and it is still
possible to analyze data from two decades ago. These data are precious and new
insights from jet physics and modern machine learning (ML) will yield new and
exciting physics results - see e.g. Ref. 2. I recently became involved in the
H1 experiment because of my interests in quantum chromodynamics (QCD) and in
preparation for the upcoming Electron Ion Collider (EIC). While the H1 data
are proprietary, the collaboration welcomes new members. There is enough
institutional knowledge to enable precise analyses, but not enough
institutional inertia to discourage new ideas. There are simulated datasets
that are hundreds of times the size of the experimental data and significant
institutional knowledge about how to generate more events, even with different
generators. This seems like an ideal model for long term data analysis in HEP.

Figure 2: Differential cross section measurements from the H1 Collaboration
[22, 23, 24], performed in 2021-2022 using data collected in 2006-2007 and an
unfolding method proposed in 2019 [25].

### LEP/SLC.

In contrast to HERA, the experimental collaborations at LEP and SLD ended a
long time ago and their data are not public. I have been fortunate to have the
opportunity to explore the ALEPH dataset from LEP through personal connections
(or rather connections of connections) with former collaboration members.
These data are pristine and just like for HERA, there are many new analyses to
be performed with modern QCD and ML tools. However, the software
infrastructure has not been modernized and there are no standard tools for
generating new events or estimating uncertainties. With significant effort,
some aspects of these tasks can be performed (see e.g. Ref. [26]), as at least
some collaboration software is still accessible. The situation with the SLD
collaboration at SLC is even less positive. As with ALEPH, I have access to
the data through personal connections, but there is little or no simulation
and the software to produce more simulation disappeared when old computers
were discarded. As there is very little we can do without simulations, this
makes it challenging to use these pristine and unique data.

##  3 Resources for Data Preservation

Everyone agrees that data preservation is critical, however, it is often an
unfunded mandate to make this happen. Resources for data preservation should
be available before, during, and after an experiment has ended. Data from any
point in the data taking should be analyzable throughout the course of the
experiment’s lifetime. I am impressed by the HERA model for data preservation,
which is possible due to a modest investment from DESY. It seems like national
laboratories are well-suited to host (which is more than just store) these
important and large datasets and the accompanying software/documentation.
Funding data preservation after an experiment has ended is complicated because
project funding is no longer available. Funding data preservation centers
(like the CERN Open Data Portal) at national laboratories seems like a viable
path forward. It may also be possible that a modest investment now could even
resurrect the unique datasets taken at LEP and SLC while enough former
collaboration members are around, especially as the community prepares for a
future $e^{+}e^{-}$ collider.

I note that not everything about data preservation is in dire straits. A
number of tools and initiatives are now in place to preserve various data
products and analysis logic. While not “open data” as I have defined it above,
these developments are an important success for the future of HEP. For
example, current and future search results in ATLAS are being preserved
through the RECAST [27] system, which makes it straightforward to determine
the sensitivity to new models not tested at the time of the original analysis.
For Standard Model measurements, Rivet [28] and HepData are broadly accepted
as community standards for preserving analysis logic and final measurement
data products. These have enabled many data re-analyses.

##  4 Data Physicists

It is clear that there will be more and not less open data. I have tried to
argue above that alternatives to open data may be preferable, but I want to
now turn my attention to how I believe HEP should react to the growing
availability of these data. In particular, I think the trend in open data
provides further and strong evidence for the need for a new type of HEP
scientist that is neither an “experimentalist” or a “theorist” - they are a
“Data Physicist”‡‡‡This name was coined by David Shih, who gave an
inspirational talk at the recent Snowmass Summer Study.. These scientists have
the core skills to understand and interrogate data as well as the
computational and theoretical background to relate these data to underlying
physical properties. Unlike a traditional “experimentalist”, a Data Physicist
will likely not have extensive (or any) hands-on instrumentation experience
and unlike a traditional “theorist”, they may not have extensive (or any)
experience with complex higher-order cross section calculations. It is not
enough to make data easily accessible and well-documented - we also need to
train a cohort of scientists who are well-equipped to use these data for
science. In particular, I expect the Data Physicist to have a HEP background,
but extensive training in statistics/data science/machine learning and
scientific computing. Software and computing are becoming commensurate with
instrumentation and it is important and necessary physics research to work on
these topics for maximizing the science from our data.

How can we create this cohort? There needs to be career paths (starting in
graduate school) for Data Physicists. This includes degree programs as well as
long term funding prospects (that are not all short grants as is typical for
computing). Important institutions like HEP-CCE, IRIS-HEP, the NSF AI
Institutes (IAFAI, A3D3, etc.) are great examples of interdisciplinary
research on software and computing and I strongly support continuing and
expanding these initiatives. However, funding through individual PIs, as is
currently the case for other areas of experimental and theoretical HEP, will
be important for a sustained effort.

##  5 Conclusions and Outlook

The LHC still has a long and exciting program ahead - only $\mathcal{O}(1\%)$
of the full dataset has been collected. The future data will open up exciting
avenues for research that are not possible with the existing data. At the same
time, the data from the first and second runs of the LHC are unique. The low
instantaneous luminosity of Run 1 may compensate for the improved detectors of
Runs 2 and beyond for certain precision measurements. I would argue that
before we make it a priority to ensure that external users can analyze these
legacy datasets, we should work hard to make sure that collaboration members
are able to analyze them.

It is fantastic that many people are excited about sharing and exploring data.
The reason I became an “experimentalist” in graduate school is precisely
because I wanted to interact directly with data. On the other hand, I have
found through the course of my career so far that the labels “experimentalist”
and “theorist” can be unnecessarily restrictive. With a growing need for
cross-cutting methodology and the growing availability of open datasets, we
need to train and support Data Physicists to make the most of our precious
data. With open minds and the right skill set, we will be ready to make the
discoveries of tomorrow.

## Executive Summary:

1. Significant short term investment for Runs 1 and 2 and modest long term investment going forward to ensure that all relevant previous data collected by a collaboration are analyzable by the collaboration. This could be stress-tested with data challenges, including targeted measurements that would benefit from earlier data. The focus should be on enabling generic data analysis and is complementary to efforts to preserve historical analysis logic and data products. This could benefit from involvement by non-collaboration researchers (see 2 and 3). 2. Make it easier for non collaboration members to engage with the collaboration on data analysis. This should include ease procedurally as well as financially (see 3). 3. Significant and long term investment (‘base funding’) in Data Physicists at all levels (undergraduate, graduate, postdoc, and tenure-track faculty/staff). This funding should be different than existing experimental/theoretical sources; data physicists are not ‘experimentalists’ who do not touch detectors and are not ‘theorists’ who do data analysis - they have dedicated training and a focused research program. Data Physicists are also not engineers, although they are also critical for enabling the science.

## Acknowledgements

I thank Matt Bellis for initiating this white paper (in concert with Ref.
[1]). Many of the ideas expressed here are the direct result of extensive
conversations with Jesse Thaler and Hannah Joo spanning many years. I have
also benefited from enlightening discussions with many other people including
Stephen Bailey, Kees Benkendorfer, Krish Desai, Matt LeBlanc, Radha
Mastandrea, Jennifer Roloff, and David Shih. The views expressed here are my
own, but I would like to thank the ATLAS Collaboration for years of
discussions that have helped form my perspective and for the opportunity to
contribute to and benefit from our dataset. Furthermore, I would like to thank
the H1 Collaboration (including Stefan Schmitt and Daniel Britzger) for being
so welcoming and Miguel Arratia for connecting me to the collaboration in the
first place. I also thank Tony Johnson and Su Dong for helping me explore the
SLD data. Lastly, I thank my ALEPH data collaborators, in particular, Anthony
Badea, Patrick Komiske, Yen-Jie Lee, Eric Metodiev, and Jesse Thaler.

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