# Stochastic stability criteria for two-dimensional linear autonomous systems
perturbed by white noise

M.M. SHUMAFOV11footnotemark: 1 AND V.B. TLYACHEV22footnotemark: 2
11footnotemark: 1 Adygea State University, 385000, Maykop, Pervomayskaya str.
208, Russia,
<EMAIL_ADDRESS>
22footnotemark: 2 Adygea State University, 385000, Maykop, Pervomayskaya str.
208, Russia,
<EMAIL_ADDRESS>
###### Abstract

We give necessary and/or sufficient conditions for stochastic stability of
second-order linear autonomous systems with parameters, which are perturbed by
a random process of the "white noise” type. The Ito’s and Stratonovich’s forms
of stochastic differential equations representing the system with white noise
are considered. The investigation of stochastic stability of the systems
considered is based on the construction of special Lyapunov functions in the
quadratic forms. The bifurcation value of the white noise intensity at which a
system first becomes unstable is presented in the analytical expression. As an
example a damped harmonic oscillator with randomly perturbed parameters is
considered.

Keywords: autonomous system, stochastic stability, Lyapunov function,
quadratic form, damped harmonic oscillator.

##  1 Introduction

The method of Lyapunov functions or Lyapunov’s direct method proposed by
eminent russian mathematician A.M. Lyapunov in his famous memoir [1] initially
for the study of problems of the stability of deterministic systems and later
developed in the works of numerous authors proved to be extremely general and
powerful. The invaluable advantage of this method is that it does not require
the knowledge of the solutions themselves. The existence of a Lyapunov
function with certain properties is sufficient for stability (or instability)
of solutions of differential equations. Lyapunov’s method proved later to be
applicable also to many other problems in the qualitative theory of
differential equations. Details and bibliography about the stability study of
deterministic systems using the Lyapunov functions method can be found, for
example, in monographs [2-9].

The Lyapunov functions method is also widely used in stability theory of
stochastic differential equations. The papers [10, 11] are one of the first
publications where it was considered the problem of stability of systems with
random variables and solved in terms of Lyapunov functions. They stimulated
considerable further research [12-15], see also bibliography in [16, 17].

The books of H.J. Kushner [16] and R.Z. Khasminskii [17] were the first
monographs devoted to problems of stochastic stability by Lyapunov’s functions
method. They gave a powerful impetus to the development of the Lyapunov’s
method for the theory of stochastic differential equations. In [16, 17] method
of Lyapunov functions developed for deterministic systems has been carried
over to the stochastic case. Later this research direction has been developed
in the works of Gikhman and Skorokhod [18], L. Arnold and Schmalfuss [19], Mao
[20], Levakov [21] (see also bibliography in these books). We also refer to
surveys [22–24].

As in deterministic theory the problem is to construct a suitable Lyapunov
function for a system in a given region of the phase space. For deterministic
systems there are some techniques and methods for construction Lyapunov
functions, the descriptions of which can be found, for example, in [7] and
[25]. It should be also noted later works [26, 27], where a topological
approach is used to construct Lyapunov functions.

As for stochastic systems, as far as we know, there are a few works devoted to
the study of stability of concrete classes of stochastic differential
equations by Lyapunov’s functions method in accordance with strategy assigned
in the monographs [16, 17]. We refer to the papers [28, 29, 30] (see also
examples in monographs [16, 17, 18, 19, 20, 21]) where the stochastic
stability conditions are derived by using Lyapunov functions for some classes
of linear and nonlinear first- and second-order stochastic differential
equations and systems. In the paper [31] global asymptotic stability in
probability and $p$–th moment exponential stability are studied for autonomous
linear stochastic differential equations using a Lyapunov function of a
quadratic form of some degree. In general, this field of research is still
largely unstudied in comparison with deterministic case.

Our paper is motivated by Kushner’s paper [28]. The purpose of the present
work is to obtain necessary and/or sufficient conditions of stability in
probability and exponential stability in the mean square of linear two-
dimensional autonomous stochastic systems. We construct special Lyapunov
functions in quadratic forms as the main tools for stochastic stability
analysis. We should note that despite the fact that the existence, in general,
of Lyapunov functions for linear stochastic systems is proved in the class of
quadratic forms in [17], the construction of such functions in the explicit
form is actual, since on the basis of these functions the effective,
coefficient, stochastic stability criteria are established. As an example, we
consider a damped harmonic oscillator parameters (the coefficient of friction
and eigenfrequency) of which are perturbed by white noise.

Khasminskii [17, p. 213 ] states that ”It is extremely interesting to study
”bifurcation” values of the noise intensity, i.e., values for which the system
first becomes unstable”. Following his suggestion, we are also interested in
studying the large bifurcation values of the noise intensity for which the
zero solution of the systems considered are still stochastically stable.

The results obtained in this paper can be used when performing linear
stability analysis of nonlinear two-dimensional stochastic systems in the
neighborhood of the origin. Also the Lyapunov quadratic functions presented
can be useful to construction Lyapunov functions for nonlinear stochastic
systems.

##  2 Problem statement

Consider a linear autonomous homogeneous system whose parameters are perturbed
by white noise

$\left\\{\begin{array}[]{ll}\begin{aligned}
\dot{x}(t)=(a+e\dot{\xi}(t))x(t)+(b+f\dot{\xi}(t))y(t),\\\
\dot{y}(t)=(c+g\dot{\xi}(t))x(t)+(m+h\dot{\xi}(t))y(t),\\\
\end{aligned}\end{array}\right.$ $None$

where $x(t)$ and $y(t)$ are scalar functions in $t$, $t\in[0,+\infty)$,
$\dot{\xi}(t)$ is a Gaussian "white noise" process of unit intensity, $a,$
$b,$ $c,$ $m,$ $e,$ $f,$ $g,$ $h$ are real constants.

The system (2.1) can be understood in different ways. In general, the exact
mathematical meaning to the systems of differential equations under random
perturbations of their parameters was given in the works of S.N. Bernshtein,
I.I. Gikhman and K. Ito. The simplest and most convenient construction of the
solution of stochastic differential equations was suggested by Ito [32].
However, as noted in the monograph [17, p. 162], when considering physical
models, where "white noise” is an idealization of a real process with small
time correlation, it is often naturally to regard the corresponding equation
as a stochastic equation in the Stratonovich form [33]. We will consider the
system (2.1) in the two forms: Ito’s and Stratonovich’s ones.

The system (2.1) considered as a system of stochastic equations in the sense
of Ito can be rewritten in the following form

$\left\\{\begin{array}[]{ll}dx(t)=(ax(t)+by(t))dt+(ex(t)+fy(t))d\xi(t),\\\
dy(t)=(cx(t)+my(t))dt+(gx(t)+hy(t))d\xi(t),\\\ \end{array}\right.$ $None$

where $x(t)=x(t,\omega)$, $y(t)=y(t,\omega)$ are random processes,
$\xi(t)=\xi(t,\omega)$ is Wiener process, $\omega$ is an elementary event,
$dx(t),$ $dy(t),$ and $d\xi(t)$ are stochastic differentials of processes
$x(t),$ $y(t)$ and $\xi(t)$ in the sense of Ito, respectively. It is assumed
that the complete probability space $(\Omega,\Sigma,P)$ is given, where
$\Omega$ is the space of elementary events, $\Sigma$ is $\sigma$-algebra of
subsets of $\Omega$, $P$ is probability measure on $\Sigma.$

There is a connection between the two forms of Ito and Stratonovich stochastic
equations [17, p. 161]. For system (2.1) this relationship is expressed in the
fact that the corresponding stochastic system in the Stratonovich form is
obtained from (2.2) by replacing the coefficients $a,$ $b,$ $c$ and $m$ by the
following

$\left\\{\begin{array}[]{ll}\begin{aligned} a\to a+\frac{e^{2}+fg}{2},\;\;b\to
b+\frac{f(e+h)}{2};\\\ c\to c+\frac{g(e+h)}{2},\;\;m\to
m+\frac{h^{2}+fg}{2}.\\\ \end{aligned}\end{array}\right.$ $None$

We derive the necessary and / or sufficient conditions for stability in
probability and exponential stability in the mean square of the zero solution
of Ito system (2.2) and of Stratonovich system (2.2), (2.3). Necessary and
sufficient conditions for exponential stability in mean square of the zero
solution of the classical damped harmonic oscillator with white noise
parameters perturbation are given. The analytical expressions of the
bifurcation intensity value of the white noise are presented.

##  3 Criteria for stochastic stability

A solution of the system (2.2) is a pair of random processes $(x(t),y(t))$
satisfying with probability 1 integral equations corresponding to the
stochastic system (2.2). Denote the solution of the system (2.2) with the
initial condition $(x(s)=\xi,\,y(s)=\eta)$ by ($x(t,\omega;s,\xi,\eta),$
$y(t,\omega;s,\xi,\eta)).$

Recall the definitions of the stability in probability and exponential
stability in the mean square of a solution of the system (2.2).

######  Definition 1 (17, p. 152)

The trivial solution $(x(t)=0,y(t)=0)$ of the system (2.2) is called stable in
probability for $t\geq 0$ if for any $s\geq 0,$ $\rho>0$ and $\varepsilon>0$
there exists $\delta=\delta(s,\rho,\varepsilon)>0$ such that

$P\\{\sup\limits_{t\geq s}|x(t,\omega;s,\xi,\eta)|>\varepsilon\\}<\rho$

and

$P\\{\sup\limits_{t\geq s}|y(t,\omega;s,\xi,\eta)|>\varepsilon\\}<\rho,$

whenever $|\xi|<\delta,$ $|\eta|<\delta.$

######  Definition 2 (17, p. 171)

The trivial solution $(x(t)=0,y(t)=0)$ of the system (2.2) is called
exponentially stable in the mean square for $t\geq 0$ if there exist positive
numbers $A$ and $\alpha$ such that

$M(|x(t,\omega;s,\xi,\eta)|^{2}+|y(t,\omega;s,\xi,\eta)|^{2})\leq
A(\xi^{2}+\eta^{2})\exp{[-\alpha(t-s)]}.$

Here $M$ denotes an expectation.

In order to avoid bulky expressions, we will restrict ourselves to the
consideration of two cases (the remaining cases can be treated similarly):

1) one of the coefficients $e,$ $f,$ $g,$ $h$ of the "random" part of system
(2.2) is non-zero and the rest are zero;

2) two coefficients are different from zero and equal to each other, and the
other two ones are equal to zero.

In general, a Lyapunov function for a stochastic system
$dx(t)=f(x)dt+\sigma(x)d\xi(t)$ (
$x\in\mathbb{R}^{n},f:\mathbb{R}^{n}\to\mathbb{R}^{n},$
$\sigma:\mathbb{R}^{n}\to\mathbb{R}^{n\times m},$ $\xi(t)$ is a vector Wiener
process) is a scalar function $V:D\to\mathbb{R}$ defined on a region
$D\subset\mathbb{R}^{n}$ that is continuous, positive-definite, twice
continuously differentiable throughout $D,$ except possibly for the origin
$x=0,$ and satisfies the condition $LV\leq 0$ (or $LV<0$), where $L$ is the
differential generator associated with the stochastic system.

For the system (2.2) the operator $L$ have the form

$\begin{array}[]{ll}\begin{aligned}
\hfil\displaystyle\begin{split}L=(ax+by)\frac{\partial}{\partial
x}+(cx+my)\frac{\partial}{\partial
y}+\frac{1}{2}\Big{[}(ex+fy)^{2}\frac{\partial^{2}}{\partial x^{2}}+\\\
+2(ex+my)(gx+hy)\frac{\partial^{2}}{\partial x\partial
y}+(gx+hy)^{2}\frac{\partial^{2}}{\partial y^{2}}\Big{]}.\\\
\end{split}\end{aligned}\end{array}$

In the deterministic case ($e=f=g=h=0$), the operator $L$ coincides with the
operator $\dot{V}$ of "the derivative of $V$ with respect to the system".

Now we turn to the formulations of the theorems on stochastic stability of
systems (2.2) and (2.2), (2.3). Proofs of the proposed theorems are based on
the construction of appropriate Lyapunov functions in quadratic forms and
applying general stability theorems of Kushner [16, p. 39] and Khasminskii
[17, pp. 152, 185].

In the absence of "fluctuating" terms ($e=f=g=h=0$) the stability conditions
obtained below turn into the necessary and sufficient Routh-Hurwitz conditions
for the deterministic part of system (2.2).

1. One of the coefficients of the "random" part of the system (2.2) is nonzero and the others are zero.

###### Theorem 1

Let, in the system (2.2), $e\neq 0,$ $f=g=h=0.$ Then the zero solution
($x(t)\equiv 0,$ $y(t)\equiv 0$) of the system (2.2) is stable in probability
if the following inequalities hold

1) $a+m<0,$ $am-bc>0,$ $b\neq 0,$

2) $e^{2}<[2(a+m)(bc-am)]/[m^{2}+(am-bc)].$

In addition, any solution ($x(t),y(t)$) of system (2.2) possesses the
property: $(x(t),y(t))\to\\{(x,y):x=0\\}$ as $t\to\infty$ with probability 1.

###### Theorem 2

Let $e\neq 0,$ $f=g=h=0.$ Then the zero solution ($x(t)\equiv 0,$ $y(t)\equiv
0$) of the system (2.2), (2.3) is stable in probability if at least one of the
following conditions holds:

1) $m>0,$ $a+m<0,$ $am-bc>0,$ $b\neq 0,$
$\dfrac{e^{2}}{2}<min\\{-(a+m),\lambda_{0}\\},$ where $\lambda_{0}$ is a
positive root of the quadratic equation $Q(\lambda)=0,$

$Q(\lambda)=2m\lambda^{2}+(2m^{2}+3am-2bc)\lambda-(a+m)(bc-am);$

or

2) $m<0,$ $a+m<0,$ $am-bc>0,$ $b\neq 0,$
$\frac{e^{2}}{2}<min\\{-(a+m),\dfrac{bc-am}{m},\lambda_{0}\\},$ where
$\lambda_{0}$ is the smallest positive root of the equation $Q(\lambda)=0$ in
the case

$B\geq 2\sqrt{AC},$

where

$A=2m,\;B=2m^{2}+3am-2bc,\;C=(a+m)(am-bc);$

if $0<B<2\sqrt{AC},$ then $\lambda_{0}$ is absent from hypothesis 2),

3) $m=0,a<0,bc<0,e^{2}<-a.$

In addition, any solution ($x(t),y(t)$) of the system (2.2), (2.3) possesses
the property: $(x(t),y(t))\to\\{(x,y):x=0\\}$ as $t\to+\infty$ with the
probability 1.

###### Remark 1

The inequalities with respect to parameter $e$ in the conditions 1)–3) of
Theorem 1 give upper bounds of the white noise intensity, up to which the
perturbed system (2.2) remains stable in probability. The right-hand sides of
these inequalities determine the lower bound for the bifurcation value of the
white noise intensity.

The Theorems 1 and 2 are proved using the following Lyapunov function

$V_{e}(x,y)=(am-bc)x^{2}+(mx-by)^{2}.$

Mean square exponential stability conditions of zero solution of the systems
(2.2) and (2.2), (2.3) in the case of $e\neq 0,$ $f=g=h=0$ are given by the
following theorems.

###### Theorem 3

Let $e\neq 0,$ $f=g=h=0.$ Then for the zero solution ($x(t)=0,y(t)=0$) of the
system (2.2) to be exponentially stable in mean square it is necessary and
sufficient that the following conditions should satisfy:

1) $a+m<0,\;am-bc>0$;

$2a$) $e^{2}<\min\left\\{\dfrac{2(a+m)(bc-am)}{m^{2}+(am-
bc)},\;\dfrac{p+\sqrt{p^{2}+c^{2}q}}{c^{2}}\right\\}$
if $c\neq 0,$ where

$p=(a+m)(c^{2}+m^{2}+am-bc)-2c(ac+bm),$ $q=4(am-
bc)\left[(a+m)^{2}+(b-c)^{2}\right];$

$2b$) $e^{2}<-2a$ $(a<0),$ if $c=0.$
In this case any solution $(x(t),y(t))$ of the system (2.2) has the property:
$x(t)\to 0,$ $y(t\to 0$ as $t\to 0$ with probability 1.

###### Remark 2

The inequalities $2a$) and $2b$) of theorem 3 give the bifurcation values of
white noise intensity $e^{2}.$

###### Theorem 4

Let $e\neq 0,$ $c=f=g=h=0.$ Then the zero solution $(x(t)\equiv 0,y(t)\equiv
0)$ of system (2.2), (2.3) is exponentially stable in mean square if and only
if $a<0,$ $m<0,$ $e^{2}<-2a.$
In this case any solution ($x(t),y(t)$) of the system (2.2), (2.3) has the
property: $x(t)\to 0,$ $y(t)\to 0,$ as $t\to+\infty$ with probability 1.

###### Theorem 5

Let $c\neq 0,$ $m=0,$ $e\neq 0,$ $f=g=h=0.$ Then the zero solution
$(x(t)\equiv 0,y(t)\equiv 0)$ of system (2.2), (2.3) is exponentially stable
in mean square if and only if at least one of the following two conditions is
satisfied:

1) $a<0,$ $bc<0,$ $(-b)/c\geq 1,$ $e^{2}<-a$ or

2) $a<0,$ $bc<0,$ 0<$(-b)/c<1,$ $\dfrac{e^{2}}{2}<\dfrac{(-a)(b+c)}{b+3c}.$
In this case any solution $(x(t),y(t))\to(0,0)$ as $t\to+\infty$ with
probability 1.

###### Remark 3

The comparison between the stability conditions given by Theorems 1 and 2, and
Theorems 3 and 4 (or 5, in the hypotheses of theorems 4 and 5) shows that, in
general, the stability conditions for the system (2.2) in Ito’s form are wider
than the stability conditions for this system in Stratonovich’s one.

In the case $f\neq 0,$ $e=g=h=0$ the conditions of stability in probability
and mean square exponential stability of zero solution of the (2.2) in the Ito
form coincide with the corresponding stability conditions of the system (2.2),
(2.3) in the Stratonovich form. The following theorems are valid.

###### Theorem 6

Let $f\neq 0,$ $e=g=h=0.$ Then a sufficient condition for stability in
probability of zero solution ($x(t)=0,y(t)=0$) of the system (2.2) is that the
following inequalities should be satisfied:

1) $a+m<0,$ $am-bc>0,$ $c\neq 0;$

2) $f^{2}<[2(a+m)(bc-am)]/c^{2}.$
In this case any solution $(x(t),y(t))\to\\{(x,y):y=0\\}$ as $t\to\infty$ with
probability 1.

The statement of theorem 6 is proved using the Lyapunov function

$V_{f}(x,y)=(cx-ay)x^{2}+(am-bc)y^{2}.$

###### Theorem 7

Let $f\neq 0,$ $e=g=h=0.$ Then a necessary and sufficient condition for mean
square exponential stability of zero solution ($x(t)=0,y(t)=0$) of the system
(2.2) is that the following inequalities should be fulfilled:

1) $a+m<0,\;am-bc>0$;

2) $f^{2}<\min\left\\{\dfrac{2(a+m)(bc-
am)}{c^{2}},\;\dfrac{-p+\sqrt{p^{2}+c^{2}q}}{c^{2}}\right\\}$
in the case $c\neq 0,$ where

$p=(a+m)(c^{2}+m^{2}+am-bc)-2c(ac+bm),$ $q=4(am-
bc)\left[(a+m)^{2}+(b-c)^{2}\right];$

If $c=0,$ then the condition 2) is lacking.

In addition, any solution $(x(t),y(t))\to(0,0)$ as $t\to\infty$ with
probability 1.

The Theorem 7 is proved using the Lyapunov function

$\begin{array}[]{ll}V_{f}(x,y)=[c^{2}+m^{2}+(am-bc)]x^{2}+2(ac+bm-\\\
-c{f^{2}}/{2})xy+[a^{2}+b^{2}+(am-bc)-(a+m){f^{2}}/{2}]y^{2}.\end{array}$

The cases $g\neq 0;$ $e=f=h=0$ and $h\neq 0;$ $e=f=g=0$ are reduced to the
cases discussed above, by replacing $x\to y,$ $y\to x,$ $a\to m,$ $b\to c,$
$c\to b,$ $m\to a$ and $g\to f$ in the first case and $h\to e$ in the second
case, respectively.

2. Two coefficients of the "random" part of the system (2.2) are different from zero and the rest are zero. To avoid bulky expressions, we limit ourselves to the cases when these two non-zero coefficients from $e,f,g,h$ are equal to each other.

The following statement takes place.

###### Theorem 8

Let $e=f\neq 0,$ $g=h=0.$ Then a sufficient condition for the zero solution
($x(t)=0,y(t)=0$) of the system (2.2) to be stable in probability is that the
following inequalities should be satisfied:

1) $a+m<0,$ $am-bc>0,$ $c\neq 0;$

2) $e^{2}<min\\{[2(a+m)(bc-am)]/[(am-bc)+(c-m)^{2}];$

$(m-a-2c)+\sqrt{(a+m)^{2}+4c(a-m+c-b)}\\}.$
In this case any solution $(x(t),y(t))$ of system (2.2)
$(x(t),y(t))\to\\{(x,y):y=0\\}$ as $t\to+\infty$ with probability 1.

To prove Theorem 8 we use the Lyapunov function in the form

$V(x,y)=c^{2}x^{2}-2c(2a+c^{2})xy+\Big{[}a^{2}+(am-
bc)+\frac{e^{2}}{2}(a+m-2c)\Big{]}y^{2}.$

The following theorem gives a necessary and sufficient stability conditions in
the mean square.

###### Theorem 9

Let $e=f\neq 0,$ $g=h=0.$ Then the zero solution $(x(t)\equiv 0,y(t)\equiv 0)$
of the system (2.2) is exponentially stable in the mean square if and only if
the the following inequalities are satisfied:

1) $a+m<0,$ $am-bc>0;$

2a) $e^{2}<\min\\{[2(a+m)(bc-am)]/[(am-bc)+(c-m)^{2}],$
$(p+\sqrt{p^{2}+4m^{2}q})/(2m^{2})\\}$ if $m\neq 0,$ where

$p=(b-c)(b^{2}+c^{2}+am-bc)-2m(ac+bm),$ $q=(am-bc)[(a+m)^{2}+(b-c)^{2}];$

$2b$) $e^{2}<\min\\{(2ab)/(c-b);$ $((-b)[a^{2}+(b-c)^{2}])/((c-b)^{2})\\}$ if
$m=0$ and $b<0,$ $c>0$
(if $b>0,$ $c<0,$ then the condition $2b$) is missing).

In addition, any solution $(x(t),y(t))$ of system (2.2) has the property
$x(t)\to 0,$ $y(t)\to 0$ as $t\to+\infty$ with probability 1.

###### Theorem 10

Let $e=f\neq 0,$ $g=h=0.$ Assume that $m=0,$ $a<0,$ $b<0,$ $c>0.$ Then for the
zero solution ($x(t)=0,y(t)=0$) of the system (2.2), (2.3) to be exponentially
stable in mean square it is necessary and sufficient that

$\frac{e^{2}}{2}<min\Big{\\{}\frac{ab}{c-2b},\frac{ab+(c-b)^{2}-(c-b)\sqrt{c^{2}+2b(a-c)}}{-b}\Big{\\}}.$

In this case for any solution $(x(t),y(t))$ we have: $x(t)\to 0,$ $y(t)\to 0$
as $t\to+\infty$ with probability 1.

###### Theorem 11

Let $e=f\neq 0,$ $g=h=0.$ Suppose that $b=c=m,$ $a<b<0.$ Then in order that
for the zero solution of the system (2.2), (2.3) to be exponentially stable in
the mean square, it is necessary and sufficient that the inequality should be
satisfied

$\frac{e^{2}}{2}<min\Big{\\{}-\frac{a+b}{3},\frac{-3a-\sqrt{a^{2}+8b^{2}}}{4}\Big{\\}}.$

In this case any solution $(x(t),y(t))\to(0,0)$ as $t\to+\infty$ with
probability 1.

The Theorems 9–11 are proved by using the Lyapunov function in the form

$\begin{array}[]{ccc}V(x,y)=(c^{2}+m^{2}+am-bc)x^{2}+2(me^{2}+ac+bm)+\\\
+\left(a^{2}+b^{2}+am-bc+(b-c)e^{2}\right)y^{2}.\end{array}$

###### Remark 4

The right-hand sides of the inequalities in conditions 2$a$) and 2$b$) of
Theorem 9 and the inequalities in Theorems 10 and 11 give a bifurcation value
of the white noise intensity.

The following theorems provide sufficient conditions for stability in
probability.

###### Theorem 12

Let $e=g\neq 0,$ $f=h=0.$ Then sufficient conditions for stability in
probability of the trivial solution $(x(t)\equiv 0,y(t)\equiv 0)$ of the
system (2.2) are that the following inequalities should hold:

1) $a+m<0,$ $am-bc>0$ $b\neq 0;$

2) $e^{2}<[2(a+m)(bc-am)]/[(am-bc)+(b-m)^{2}].$
In this case any solution $(x(t),y(t))$ of the system (2.2) possesses the
property $(x(t),y(t))\to\\{(x,y):y=0\\}$ as $t\to+\infty$ with probability 1.

###### Theorem 13

Let $e=g\neq 0,$ $f=h=0.$ Then the zero solution of the system (2.2), (2.3) is
stable in probability if

$\begin{array}[]{ll}1)\;m<0,\;b\neq 0,\\\ 2)\;e^{2}/2<min\\{(am-
bc)/(-m);\lambda_{0}\\},\end{array}$

where $\lambda_{0}$ is the positive root of the quadratic equation

$2m\lambda^{2}+[2(am-bc)+m(a+m)+(b-m)^{2}]\lambda-(a+m)(am-bc)=0.$

In addition, any solution $(x(t),y(t))\to(0,0)$ as $t\to+\infty$ with
probability 1.

To prove the Theorems 12, 13 the Lyapunov function

$\begin{array}[]{ccc}V(x,y)=(am-bc)x^{2}+(mx-by)^{2}\end{array}$

is used.

The following theorem considers the case $e=h\neq 0,$ $f=g=0.$

###### Theorem 14

Let $e=h\neq 0,$ $f=g=0.$ Then the trivial solution $(x(t)\equiv 0,y(t)\equiv
0)$ of system (2.2) is stable in probability if:

1) $a+m<0,$ $am-bc>0;$ $c\neq 0;$

2) $e^{2}<\min\\{p_{1},p_{2}\\},$
where $p_{1}$ and $p_{2}$ are the smallest positive roots of the equations:

$\lambda^{2}+2(a+m)\lambda+4(am-bc)=0$

under the condition $(a+m)^{2}\geq 4(am-bc)$ and

$\lambda^{3}+3(a+m)\lambda^{2}+2[(a+m)^{2}+2(am-bc)]\lambda+4(a+m)(am-bc)=0,$

respectively.

(If $(a+m)^{2}<4(am-bc),$ then the first equation has no real roots and
therefore $p_{1}$ is absent.)

In addition, any solution $(x(t),y(t))\to\\{(x,y):y=0\\}$ as $t\to+\infty$
with probability 1.

The Theorem 14 is proved by using the following Lyapunov function

$V(x,y)=2c^{2}x^{2}-2c(2a+e^{2})xy+[2(a^{2}+am-bc)+e^{4}+(3a+m)e^{2}]y^{2}.$

The next theorem considers the case $f=g\neq 0,$ $e=h=0.$

###### Theorem 15

Let $f=g\neq 0,$ $e=h=0.$ Then the sufficient conditions for the trivial
solution $(x(t)\equiv 0,y(t)\equiv 0)$ of the system (2.2) to be stable are
that the following inequalities should be satisfied:

1) $a+m<0,$ $am-bc>0;$

$2a$) $m<0,\;f^{2}<\min\\{p_{1},p_{2},p_{3}\\},$
where $p_{1}=[m^{2}+(am-bc)]/(-m),$ $p_{2}$ and $p_{3}$ are the smallest
positive roots of the equations, respectively:

$\lambda^{3}+(a+m)\lambda^{2}-2(b^{2}+c^{2}+2am)\lambda+4(a+m)(bc-am)=0,$
$None$ $\begin{array}[]{ll}2m\lambda^{3}+[c^{2}+2m^{2}+2(am-
bc)+2m(a+m)]\lambda^{2}+\\\ +2[(a+m)(m^{2}+am-bc)-4bm(b+c)]\lambda-4b^{2}(am-
bc)=0.\end{array}$ $None$

(The equations ($A$) and ($B$) can not possibly have positive roots: in this
case $p_{2}$ and/or $p_{3}$ are missing.)

$2b$) $m>0,$ $f^{2}<\min\\{p_{2},p_{3}\\}.$

In addition, any solution $(x(t),y(t))\to\\{(x,y):x=0\\}$ as $t\to+\infty$
with probability 1.

The Theorem 15 is proved by using the Lyapunov function:

$\begin{array}[]{rr}V(x,y)=2[m^{2}+(am-bc)+mf^{2}]x^{2}-2(2bm-cf^{2})xy+\\\
+[2b^{2}-(a+m)f^{2}-f^{4}]y^{2}.\end{array}$

###### Remark 5

The right-hand sides of inequalities in hypotheses 2 and 2$a$), 2$b$) in
Theorems 14 and 15, respectively, give the lower bounds of the bifurcation
value of the white noise intensity $f^{2}.$

The remaining two cases $f=h\neq 0,$ $e=g=0$ and $g=h\neq 0,$ $e=f=0$ are
reduced to the considered cases $e=g\neq 0,$ $f=h=0$ and $e=f\neq 0,$ $g=h=0$
by replacing $x\to y,$ $y\to x;$ $a\to m,$ $b\to c,$ $c\to b$, $m\to a$ and
$h\to e,$ $f\to g,$ $g:=e$ (in the first case) and $h\to e,$ $g\to f,$ $f:=e$
(in the second case).

##  4 Example: The damped harmonic oscillator

Consider a harmonic oscillator with eigenfrequency $\omega$ subject to the
action of a damping force proportional to velocity with coefficient $k.$ In
many problems it seems natural to assume that $k$ and $\omega$ are merely the
mean value of the damping coefficient and eigenfrequency, while their true
values are stochastic processes with small correlation interval. Such a system
can be described by stochastic equation

$\ddot{x}(t)+(k+\sigma_{2}\dot{\xi}(t))\dot{x}(t)+(\omega^{2}+\sigma_{1}\dot{\xi}(t))x(t)=0,$
$None$

where $x(t)$ is a displacement of a material point from the equilibrium
position at the time $t$; $\dot{\xi}(t)$ is a white noise of unit intensity,
$\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ are the intensities of white noises
acting on the frequency and damping coefficient, respectively.

We interpret the equation (4.1) in two senses: as an Ito and a Stratonovich
stochastic equations.

Denoting $\dot{x}(t)=y(t),$ rewrite the equation (4.1) as a system of
stochastic equations

$\left\\{\begin{array}[]{ll}dx(t)=y(t)dt,\\\
dy(t)=-[\omega^{2}x(t)+ky(t)]dt-[\sigma_{1}x(t)+\sigma_{2}y(t)]d\xi(t),\\\
\end{array}\right.$ $None$

where $d\xi(t)$ is the stochastic differential of the Wiener process $\xi(t)$
in the Ito sense.

The equation (4.1) interpreted in the Stratonovich form of stochastic
equations is equivalent to the system of Ito’s form

$\left\\{\begin{array}[]{lr}dx(t)=y(t)dt,\\\
dy(t)=-\Big{[}\Big{(}\omega^{2}-\dfrac{\sigma_{1}\sigma_{2}}{2}\Big{)}x(t)+(k-\dfrac{\sigma_{2}^{2}}{2})y(t)\Big{]}dt-\\\
\;\;\;\;\;\;\;\;\;\;\;\;-[\sigma_{1}x(t)+\sigma_{2}y(t)]d\xi(t).\\\
\end{array}\right.$ $None$

We formulate the conditions of exponential stability in the mean square for
the systems (4.2) and (4.3).

Proposition 1. The trivial solution ($x(t)\equiv 0,y(t)\equiv 0$) of the
system (4.2) is exponentially stable in the mean square if and only if the
following conditions are satisfied:

$\begin{array}[]{lll}a)\;\sigma_{1}^{2}<2k\omega^{2}\textit{ in the case
}\sigma_{1}\neq 0,\,\sigma_{2}=0,\\\ b)\;\sigma_{2}^{2}<2k\textit{ in the case
}\sigma_{1}=0,\,\sigma_{2}\neq 0\\\
c)\;\sigma^{2}<\dfrac{2k\omega^{2}}{\omega^{2}+1}\textit{ in the case
}\sigma_{1}=\sigma_{2}=\sigma.\\\ \end{array}$

The proposition 1 is proved by using the Lyapunov functions:

$\begin{array}[]{lll}a)\;V(x,y)=\dfrac{k^{2}+\omega^{2}(\omega^{2}+1)+k{\sigma_{1}^{2}/2}}{2k\omega^{2}-\sigma_{1}^{2}}x^{2}+\dfrac{2k+\sigma_{1}^{2}}{2k\omega^{2}-\sigma_{1}^{2}}xy+\\\
+\dfrac{\omega^{2}+1}{2k\omega^{2}-\sigma_{1}^{2}}y^{2},\\\
b)\;V(x,y)=\left(\dfrac{\omega^{2}+1}{2k-\sigma_{2}^{2}}+\dfrac{k}{2\omega^{2}}\right)x^{2}+\dfrac{1}{\omega^{2}}xy+\dfrac{\omega^{2}+1}{\omega^{2}(2k-\sigma_{2}^{2})}y^{2},\\\
c)\;V(x,y)=\dfrac{1}{2[\sigma^{2}(\omega^{2}+1)-2k\omega^{2}]}\Big{\\{}2[(\omega^{2}+1)\sigma^{2}-\omega^{4}-\omega^{2}-\\\
-k^{2}]x^{2}-4kxy-2(\omega^{2}+1)y^{2}\Big{\\}}.\\\ \end{array}$

From Proposition 1 and Kushner’s Theorem [16, p. 39] we obtain the following
statement.

Corollary. For each number $p>0$ and for any solution $(x(t),y(t))$ with the
initial condition $x(0)=x_{0},$ $y(0)=y_{0},$ where
$(x_{0},y_{0})\in\\{(x,y):V(x,y)<p\\},$ the following relations hold: $x(t)\to
0,$ $y(t)\to 0$ as $t\to+\infty$ with probability at least $1-(V(x_{0})/p),$
where $V$ is the Lyapunov function corresponding to the cases $a$), $b$),
$c$), respectively.

###### Remark 6

Since the number $p$ is arbitrary we have $x(t)\to 0,$ $y(t)\to 0$ with
probability 1.

Using the Lyapunov functions similar to those $a$)–$c$) above, we get the
following

Proposition 2. The trivial solution of the system (4.3) is exponentially
stable in the mean square if and only if:

$\begin{array}[]{lll}a)\;\sigma_{1}^{2}<2k\omega^{2}\textit{ in the case
}\sigma_{1}\neq 0,\sigma_{2}=0,\\\ b)\;\sigma_{2}^{2}<k\textit{ in the case
}\sigma_{1}=0,\sigma_{2}\neq 0,\\\
c)\;\sigma^{2}<\omega^{2}+\frac{k+1}{2}-\sqrt{\omega^{4}+(1-k)\omega^{2}+\frac{(k+1)^{2}}{4}}\\\
\textit{ in the case }\sigma_{1}=\sigma_{2}=\sigma\neq 0.\\\ \end{array}$

In addition for any solution $(x(t),y(t))$ of the system (4.3) $x(t)\to 0,$
$y(t)\to 0$ as $t\to+\infty$ with probability 1.

###### Remark 7

The right-hand sides of the inequalities $a$)–$c$) of the propositions 1 and 2
determine the exact values of the bifurcation values of the intensities
$\sigma_{1}^{2},$ $\sigma_{2}^{2}$ and $\sigma^{2}$ of the white noise.

###### Remark 8

Comparing the inequalities $a$)–$c$) from propositions 1 and 2 we see that in
general the stability conditions for equation (4.2) in the Ito form are wider
than the corresponding stability conditions for the equation (4.3) in the
Stratonovich form.

##  5 Conclusion

In the paper sufficient conditions for stability in probability, necessary and
sufficient conditions for exponential stability in the mean square of the zero
solution of two-dimensional linear autonomous stochastic systems in the Ito
and Stratonovich forms are obtained. The boundaries of the bifurcation values
of the white noise intensity, which affects the parameters of the stochastic
system are established. As an example a damped harmonic linear oscillator
equation is considered in the case when both the natural frequency and
friction coefficient are affected by white noise. Necessary and sufficient
conditions for exponential stability of trivial solution in the mean square of
the perturbed harmonic oscillator are given.

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# Cycle switching in Steiner triple systems of order 19

Grahame Erskine and Terry S. Griggs
School of Mathematics and Statistics
The Open University
Walton Hall
Milton Keynes MK7 6AA
UNITED KINGDOM
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Cycle switching is a particular form of transformation applied to isomorphism
classes of a Steiner triple system of a given order $v$ (an
$\operatorname{STS}(v)$), yielding another $\operatorname{STS}(v)$. This
relationship may be represented by an undirected graph. An
$\operatorname{STS}(v)$ admits cycles of lengths $4,6,\ldots,v-7$ and $v-3$.
In the particular case of $v=19$, it is known that the full switching graph,
allowing switching of cycles of any length, is connected. We show that if we
restrict switching to only one of the possible cycle lengths, in all cases the
switching graph is disconnected (even if we ignore those
$\operatorname{STS}(19)$s which have no cycle of the given length). Moreover,
in a number of cases we find intriguing connected components in the switching
graphs which exhibit unexpected symmetries. Our method utilises an algorithm
for determining connected components in a very large implicitly defined graph
which is more efficient than previous approaches, avoiding the necessity of
computing canonical labellings for a large proportion of the systems.

MSC2020 classification: 05B07.
Keywords: Steiner triple system; cycle switching; connected graph.

##  1 Introduction

A _Steiner triple system_ of _order_ $v$, denoted by $\operatorname{STS}(v)$,
is an ordered pair $S=(V,\mathcal{B})$ where $V$ is a set of _points_ of
cardinality $v$, and $\mathcal{B}$ is a collection of _triples_ , also called
_blocks_ , which collectively have the property that every pair of distinct
points of $V$ is contained in precisely one triple. Such systems exist if and
only if $v\equiv 1$ or $3\pmod{6}$; first proved by Kirkman in 1847 [8]. In
this paper, we take the point set $V$ to be $\\{0,1,2,\ldots,v-1\\}$; two such
Steiner triple systems $(V,\mathcal{B})$ and $(V,\mathcal{D})$ are said to be
_isomorphic_ if there exists a bijection $\phi$ on their point set such that
every triple $B\in\mathcal{B}$ maps to a triple $\phi(B)\in\mathcal{D}$. Up to
isomorphism, the numbers of $\operatorname{STS}(v)s$ of orders 7, 9, 13 and 15
are 1, 1, 2 and 80 respectively. With the advent of modern computing
techniques, Kaski and Östergård [7] determined the number of non-isomorphic
$\operatorname{STS}(19)$s; there are 11,084,874,829 which are now stored in
compact notation on the Internet [11]. They have been extensively analysed
with the results appearing in [1].

A _switch_ is an operation on an $\operatorname{STS}(v)$ whereby a set of
blocks is removed from the system and replaced by a new set of blocks of the
same cardinality: provided the removed and replacement block sets cover
precisely the same set of pairs of points, the resulting structure will be a
new $\operatorname{STS}(v)$. To avoid redundancy, we usually insist that the
removed and replacement block sets be disjoint. The two sets of blocks in a
switch operation are known as _tradeable configurations_ ; together they
constitute a _trade_. The smallest possible tradeable configuration in a
Steiner triple system is a set of four blocks containing six points in an
arrangement known as a _Pasch_ configuration or 4-_cycle_. Small tradeable
configurations have been classified in [3].

In general, finding particular tradeable configurations in a given
$\operatorname{STS}(v)$ may be an onerous computational task. Most research
into switching of Steiner triple systems has concentrated on a particular
class of tradeable configurations known as _cycles_ which we describe now,
following the notation of [6].

Let $a,b$ be distinct points in an $\operatorname{STS}(v)$, and let $c$ be the
unique point which shares a block with $a$ and $b$. The _cycle graph_ $C_{ab}$
is defined as follows. Let $X_{ab}$ be the set $V\setminus\\{a,b,c\\}$ and let
$\mathcal{B}_{a}$ be the set of blocks in $\mathcal{B}$ containing $a$, apart
from the block $\\{a,b,c\\}$; and similarly for $\mathcal{B}_{b}$. We define
the graph $C_{ab}$ on the vertex set $X_{ab}$ to have edges
$\\{B\setminus\\{a\\}:B\in\mathcal{B}_{a}\\}$ and
$\\{B\setminus\\{b\\}:B\in\mathcal{B}_{b}\\}$. It is easy to see that each
vertex in $C_{ab}$ has valency 2, and so $C_{ab}$ is a 2-regular graph of
order $v-3$. Further, each cycle in $C_{ab}$ has even length at least 4.

Now let $\\{x_{1},x_{2},\ldots,x_{2k}\\}$ be the vertices of some cycle of
length $2k$ in $C_{ab}$. There are $2k$ blocks of the $\operatorname{STS}$ of
the form:

$\quad\\{a,x_{1},x_{2}\\},\\{a,x_{3},x_{4}\\},\ldots,\\{a,x_{2k-1},x_{2k}\\}\in\mathcal{B}_{a}$;
and

$\quad\\{b,x_{1},x_{2k}\\},\\{b,x_{2},x_{3}\\},\ldots,\\{b,x_{2k-2},x_{2k-1}\\}\in\mathcal{B}_{b}$.

The _cycle switch_ relative to the pair $\\{a,b\\}$ and the set
$\\{x_{1},x_{2},\ldots,x_{2k}\\}$ is formed by exchanging the roles of $a$ and
$b$ in these blocks. Note that the cycle switch is uniquely determined by the
pair $\\{a,b\\}$ and any one of the points $\\{x_{1},x_{2},\ldots,x_{2k}\\}$.
For a given pair $\\{a,b\\}$, the cycle graph $C_{ab}$ may consist of a single
Hamiltonian cycle of length $v-3$, or multiple disjoint cycles of length
totalling $v-3$. In the case where $C_{ab}$ consists of exactly two cycles of
length $2k$ and $2\ell$ where $2k+2\ell=v-3$, we call these two cycles a
_complementary pair_. The significance of Hamiltonian cycles and complementary
pairs is shown by the following lemma, proved in [4].

######  Lemma 1.

  1. (i)

Let $C$ be a Hamiltonian cycle in some $\operatorname{STS}(v)$. Then switching
$C$ results in a new $\operatorname{STS}(v)$ of the same isomorphism class as
the original.

  2. (ii)

Let $C_{1}$ and $C_{2}$ be complementary cycles in some
$\operatorname{STS}(v)$. Then switching $C_{1}$ results in a new
$\operatorname{STS}(v)$ of the same isomorphism class as switching $C_{2}$.

######  Remark.

As noted above, the smallest possible cycle length is 4, commonly referred to
as a Pasch configuration. The blocks of a Pasch configuration have the form
$\\{a,b,c\\}$, $\\{a,d,e\\}$, $\\{b,e,f\\}$ and $\\{c,d,f\\}$. This
configuration may be viewed as the blocks arising from a 4-cycle in the cycle
graph $C_{af}$, but also in the graphs $C_{bd}$ and $C_{ce}$. Thus every
4-cycle in an $\operatorname{STS}(v)$ arises from the cycle graphs of exactly
three pairs of points. This situation does not occur for any other cycle
lengths.

Our principal interest in this paper is in equivalence classes of Steiner
triple systems under cycle switching. We seek to determine, for a given
$\operatorname{STS}(v)$, which isomorphism classes can be reached from the
given system by a finite sequence of cycle switches of a given type. It is
immediate that this defines an equivalence relation on the set of isomorphism
classes of $\operatorname{STS}(v)$s, and so partitions the set. Alternatively,
we may regard the set of $\operatorname{STS}(v)$s as the vertices of a graph,
with a (directed) edge from $(V,\mathcal{B})$ to $(V,\mathcal{B}^{\prime})$
whenever the isomorphism class of the latter system can be reached from the
former by a cycle switch. Then the equivalence classes correspond to connected
components in this graph; we shall use these two views of cycle switching
equivalence interchangeably in what follows.

More formally, the graph defined above is a multidigraph, with one directed
edge emanating from a vertex for every cycle of the appropriate type contained
in the corresponding $\operatorname{STS}(v)$. It is of course possible for a
cycle switch to reach the same isomorphism class as the original system, and
for more than one cycle to reach the same isomorphism class. Thus, loops and
multiple directed edges are possible in the multidigraph. Of course, if there
is a directed edge from $(V,\mathcal{B})$ to $(V,\mathcal{B}^{\prime})$, then
there must also be a directed edge in the opposite direction by considering
the reverse switch. As our principal interest here is in connectivity, we may
suppress the multiple edges and loops, and consider a pair of opposite
directed edges as a single undirected edge, yielding a simple undirected
graph. Allowing switching of any possible cycle length, we will denote the
multidigraph of $\operatorname{STS}(v)$s by
$\overrightarrow{\operatorname{S}}(v)$, and the corresponding undirected graph
by $\operatorname{S}(v)$. The above discussion shows that the connected
components of $\operatorname{S}(v)$ and $\overrightarrow{\operatorname{S}}(v)$
are the same, so we will work with whichever is most convenient.

Typically though, we will wish to restrict the allowable cycle switches to
those of a particular length, or set of lengths. The cycle switching graph of
$\operatorname{STS}(v)$s allowing only switching of cycles of length $\ell$
will be denoted by $\operatorname{S}_{\ell}(v)$, and similarly for the
directed version. If we allow a restricted set of lengths
$C\subset\\{4,6,\ldots,v-7,v-3\\}$ then the switching graph will be denoted by
$\operatorname{S}_{C}(v)$.

In [4], the switching graphs for all $v\leq 15$ and all allowable cycle
lengths were determined. Trivially, the graphs $\operatorname{S}_{4}(7)$ and
$\operatorname{S}_{6}(9)$ both consist of a single vertex (with a suppressed
loop). The graphs $\operatorname{S}_{4}(13)$ and $\operatorname{S}_{6}(13)$
both consist of two vertices representing the cyclic and non-cyclic systems
joined by an edge. A suppressed loop on the vertex representing the non-cyclic
system indicates that this system can also be switched to itself. This is not
the case for the cyclic system. The graph $\operatorname{S}_{10}(13)$ consists
of two isolated vertices (again, each with a suppressed loop).

Referencing the standard numbering of the 80 $\operatorname{STS}(15)$s as
given for example in [9], the graph $\operatorname{S}_{4}(15)$ consists of one
connected component consisting of 79 vertices, and an isolated vertex
corresponding to the unique anti-Pasch system of this order (#80). The graph
$\operatorname{S}_{6}(15)$ consists of two connected components, one with 57
vertices and the other with 7 vertices (systems #11, #12, #19, #20, #21, #22
and #61), as well as 16 isolated vertices corresponding to systems with no
6-cycle. The seven systems above all contain an $\operatorname{STS}(7)$
subsystem. The graph $\operatorname{S}_{8}(15)$ consists of one connected
component containing 77 vertices, and three isolated vertices corresponding to
systems #1, #16 and #80 which have no 8-cycle. The graph
$\operatorname{S}_{12}(15)$ consists of 80 isolated vertices, 78 of which have
a suppressed loop; the other two correspond to systems #1 and #2 which have no
12-cycle.

In [6], it was shown that the switching graph $\operatorname{S}(19)$ of
$\operatorname{STS}(19)$s is connected. In Section 3, we extend this result by
showing that $\operatorname{S}_{\ell}(19)$ is disconnected for all
$\ell\in\\{4,6,8,10,12,16\\}$, and determining all the connected components of
these graphs. To our mind the results are a little surprising. particularly
for 4-cycle switching. It was already known that the graph
$\operatorname{S}_{4}(19)$ is disconnected and that there exist components
consisting of just two vertices [5]. Our expectation was that there would be
one component containing nearly all systems and a number of other very small
components each containing possibly only four or fewer systems. So it was
interesting to find that there also exists a component containing 16 systems
with an automorphism group of order 256, yet all systems have only the
identity automorphism! The graph $\operatorname{S}_{6}(19)$ also contains a
component of order 24. We also show that $\operatorname{S}_{\\{4,6\\}}(19)$ is
connected, so the minimum set of cycle lengths required to ensure the
connectivity of the switching graph has cardinality 2. The analysis done in
this paper also posed computational challenges due to the large number of
isomorphism classes of $\operatorname{STS}(19)$s and our algorithm to deal
with these is described in Section 2

When providing listings of particular systems, we will often employ the
following compact notation which is common in the literature, see for example
[2]. The set $V$ of points is taken to be $\\{0,1,2,\ldots,18\\}$. The points
$10,\ldots,18$ are represented by the letters $a,\ldots,i$ respectively. The
57 blocks of an $\operatorname{STS}(19)$ are represented by a string of
symbols $s_{1}s_{2}\ldots s_{57}$. Using the usual lexicographical order, the
symbol $s_{i}$ is the largest element $z_{i}$ in the $i$th triple
$\\{x_{i},y_{i},z_{i}\\}$, where $x_{i}<y_{i}<z_{i}$. The remaining two
elements implicitly have the property that there is no pair
$x^{\prime}_{i}<y^{\prime}_{i}$ such that
$\\{x^{\prime}_{i},y^{\prime}_{i}\\}$ does not appear in an earlier triple,
and either (i) $x^{\prime}_{i}<x_{i}$ or (ii) $x^{\prime}_{i}=x_{i}$ and
$y^{\prime}_{i}<y_{i}$.

##  2 Algorithm

The principal practical problem to be considered when determining connected
components of switching graphs of $\operatorname{STS}(19)$s is the large
number of systems to be considered. There are 11,084,874,829
$\operatorname{STS}(19)$s [7], and so creating a full graph of this order in
computer memory is entirely impractical. A further efficiency issue arises due
to the fact that the vertices in the graph are _isomorphism classes_ of
$\operatorname{STS}(19)$s. To compute the isomorphism class of a given system,
we use the nauty package [10] to produce a canonical labelling of the point-
block incidence graph. If carried out for every $\operatorname{STS}(19)$, this
procedure would represent by far the most computationally expensive part of
the process. We now describe an algorithm which avoids the necessity of
canonically labelling the majority of the systems.

Let $c$ be an allowable cycle length in an $\operatorname{STS}(19)$. Let $u,v$
be $\operatorname{STS}(19)$s, considered as vertices in the switching graph
$\overrightarrow{\operatorname{S}_{c}}(19)$. We write $u\to v$ if there is a
directed edge from $u$ to $v$ in the graph; that is to say, there is some
$c$-cycle in $u$ which reaches the isomorphism class of $v$ when switched. Let
$N_{c}(u)$ denote the number of $c$-cycles in the vertex $u$. We define a
strict partial order $<$ on the vertices of
$\overrightarrow{\operatorname{S}_{c}}(19)$ as follows:

$u<v\text{ iff }u\to v\text{ and }N_{c}(u)<N_{c}(v).$

In other words, $u<v$ if $v$ is reached by a $c$-cycle switch from $u$, _and_
$v$ contains strictly more $c$-cycles than $u$. Note that by definition, every
connected component in $\overrightarrow{\operatorname{S}_{c}}(19)$, and hence
also in $\operatorname{S}_{c}(19)$, contains a vertex which is a maximal
element in this preorder. Thus to compute the connected components of the
graph, it suffices to consider only these maximal elements.

Step 1. For each $\operatorname{STS}(19)$ $u$ we find all the $c$-cycles in
the system, and for each cycle we find the system $v$ reached by the switch.
If $N_{c}(v)>N_{c}(u)$ then we discard $u$.

Step 2. For each system remaining after Step 1, we extend the logic by
discarding a system $u$ if there is some system $v$ at distance at most $k$
from $u$ in $\overrightarrow{\operatorname{S}_{c}}(v)$ with
$N_{c}(v)>N_{c}(u)$. This is done by recursively exploring the switched
systems starting from $u$, stopping to discard $u$ whenever a system with more
$c$-cycles is encountered. This becomes an expensive operation, but typically
a value of $k=4$ is sufficient to reduce the remaining population of vertices
to be considered to manageable levels.

Step 3. The remaining systems after Step 2 will often comprise a number of
small connected components in the switching graph. To detect this, we select a
vertex $u$ in the set and recursively explore the switching graph until a
limit $N$ is reached, or no new vertices are detected. Typically, a value of
$N$ around 100 is sufficient to find most small components.

Step 4. Systems remaining after Step 3 will fall into one or more larger
connected components. Let $V=\\{v_{1},v_{2},\ldots,v_{m}\\}$ be the set of
remaining vertices at this stage. By construction, every vertex of $V$ is
connected to at least one vertex of $\operatorname{S}_{c}(19)$. Now choose for
each $i=1,\ldots,m$ a positive integer $k_{i}$ and construct the closed
$k_{i}$-neighbourhood $\overline{N}_{k_{i}}(v_{i})$, i.e. the set of vertices
in $\operatorname{S}_{c}(19)$ at distance no more than $k_{i}$ from $v_{i}$.
Then define a graph $W$ with vertex set $V$ and an edge from $v_{i}$ to
$v_{j}$ if
$\overline{N}_{k_{i}}(v_{i})\cap\overline{N}_{k_{j}}(v_{j})\neq\emptyset$.
Then if $W$ is connected, it follows that $\operatorname{S}_{c}(19)$ is
connected.

Step 5. If the graph $W$ from Step 4 is not connected, then each connected
component of $W$ is a subset of some connected component of
$\operatorname{S}_{c}(19)$, but two connected components of $W$ may in fact be
in the same connected component of $\operatorname{S}_{c}(19)$. To deal with
this possibility, we select a representative vertex from each connected
component of $W$ and repeat Step 3 with a much larger value of $N$ to detect
any large connected components. In practice, a value of $N=1,000,000$ proved
sufficient to detect all such components.

##  3 Results

Before describing our detailed results, we begin with a small lemma which
relates the connected components of $\operatorname{S}_{10}(19)$ to those of
$\operatorname{S}_{6}(19)$, and the components of $\operatorname{S}_{12}(19)$
to those of $\operatorname{S}_{4}(19)$.

######  Lemma 2.

Let $v\geq 13$ and let $u$ be a vertex in $\operatorname{S}(v)$ representing a
given $\operatorname{STS}(v)$. Then the connected component in
$\operatorname{S}_{v-7}(v)$ containing $u$ is a subgraph of the connected
component of $\operatorname{S}_{4}(v)$ containing $u$; and the connected
component in $\operatorname{S}_{v-9}(v)$ containing $u$ is a subgraph of the
connected component of $\operatorname{S}_{6}(v)$ containing $u$.

###### Proof.

Every $(v-7)$-cycle is necessarily complementary to a 4-cycle, and so by Lemma
1, any isomorphism class reached by a $(v-7)$-cycle switch is also reached by
some 4-cycle switch. A similar argument holds for $(v-9)$-cycles and 6-cycles.
∎

We now summarise the results for the cycle switching graphs
$\operatorname{S}_{\ell}(19)$, where $\ell\in\\{4,6,8,10,12\\}$. Note that by
Lemma 1(i), the graph $\overrightarrow{\operatorname{S}_{16}}(19)$ consists of
a set of isolated vertices, possibly with a number of loops at each vertex.
Thus the connected components of $\operatorname{S}_{16}(19)$ are
11,084,874,829 isolated vertices.

###  3.1 Switching 4-cycles

We carried out the algorithm described in Section 2, using a maximum distance
of 4 in step 2. After steps 2 and 3, there were 2,591 identified systems
without a 4-cycle. An $\operatorname{STS}$ without 4-cycles is also known as
an _anti-Pasch_ system; the 2,591 anti-Pasch systems agrees with the results
in [1]. Step 3 also identified 139 components containing between two and four
systems each; these are illustrated in Figure 1 as types (b) to (f). Component
type (b) consists of two $\operatorname{STS}(19)$s with a single 4-cycle each,
with the property that switching the 4-cycle in one system reaches the
isomorphism class of the other. Such an arrangement is known as a _twin_
system, a concept first introduced in [5]. The count of 126 pairs of twins
agrees with [1].

Component type |  Number
---|---
|  |
(a) |  |  2,591
(b) |  |  126
(c) |  |  1
(d) |  |  9
(e) |  |  1
(f) |  |  2

Type | Example representative system
---|---
(a) | 2468acegiad7ghbei5f89gdih9hbdfibhaiegfiehgcfeigfidcgfhihh
(b) | 2468acegih7cdagifga9hdifef7eibdbe9ghiahigbfciegihfgdhfihh
(c) | 2468acegiefcdhaig578fcdhi9ibfghhg9deiagdifebhicfgeihhgihf
(d) | 2468acegihfceabig9c7adfih8gefiddihgbefheibhdfgifecihgihgh
(e) | 2468acegi85fhideg9ibdacghgachif8dgbfh9hfdiebihgifeigehhif
(f) | 2468acegiceh7gfbii7adbcfhg8aefhchfigde9ifbifghgdidhigehhi

Figure 1: Small connected components in
$\overrightarrow{\operatorname{S}_{4}}(19)$

In addition to these small components, step 3 identified one more component
containing 16 systems, each having exactly six 4-cycles. These 4-cycles form a
structure consisting of six Pasch configurations collectively containing 18
points and 24 blocks. Thus the Pasch configurations are block disjoint. The
structures fall into two isomorphism classes ($A$ and $B$), each having
automorphism group $C_{2}\times C_{2}\times C_{2}\times D_{8}$ of order 64.
This totally unexpected and somewhat mysterious component is illustrated in
Figure 2, with the labelling of the vertices indicating the isomorphism class
of the structure containing the six Pasch configurations of the
$\operatorname{STS}(19)$. The component is bipartite, but does not respect the
isomorphism partitioning of the Pasch configuration structures. Its
automorphism group has order 256 and acts with two orbits, one on the eight
vertices incident with a double directed edge and the other on the remaining
eight vertices. In contrast to the orders of the automorphism groups of the
component and of the Pasch configuration structures, every
$\operatorname{STS}(19)$ in the component has trivial automorphism group!

$A2$$B2$$A0$$B6$$B7$$A1$$B0$$A6$$A7$$B1$$B5$$B4$$A3$$A5$$A4$$B3$

$A0$ | 2468acegi6d8bhcfid5hiafge7cibghiahgefdfighgecdfehfbgiihhi
---|---
$A1$ | 2468acegid7ahfbei75gbehfi9igcehfcieghdhigfe9bdfghidihgifh
$A2$ | 2468acegi6d8bhcfid5iahgef7cibghhiafgedfighgecdfehfbgiihhi
$A3$ | 2468acegid7ahfbei75fciegh9igcehgbehfidhigfe9bdfghidihgifh
$A4$ | 2468acegicha89dfi9gbieahffhabig7f9eiddighgdcefihfihgehhgi
$A5$ | 2468acegic8f9behii9gefchdd7ghfeehdcgiihcbbigfeghfdiighifh
$A6$ | 2468acegifcdbeahiheagibdf7c9digb8aighfgihhfeibgfdiheghihf
$A7$ | 2468acegi97cdfehidi7hagfebecgih98feghhgifficgehigbdihfdih
$B0$ | 2468acegigbd79ihfi8ehdfcg9cfaehad9ighihfggebighedfiidhfhi
$B1$ | 2468acegiad97fheie5ic9hfgdbigchhfegcighiefafdgbdhdihigfhi
$B2$ | 2468acegie8b7fhdiidchbfegfadgchgcfibhihdeegfihcgbdiigifhh
$B3$ | 2468acegie8c9dfhi59fabghi79gdihbihefgefghidbhgcgiiehifhdf
$B4$ | 2468acegi8cf7hedii7gahdfe9bhdgfahebgiedihifdggbficfgiihhh
$B5$ | 2468acegic8f9behiibgefcdhd7ghfeehdgciihcbbigfeghfdiighifh
$B6$ | 2468acegie57fdchiagbfc9hic9igdhh9deifghefibdigbfhieghihfg
$B7$ | 2468acegiec7bdfhiid8hbegffah9dgg8ahfideihfcigiegbifhgdihh

Figure 2: The connected component of order 16 in
$\overrightarrow{\operatorname{S}_{4}}(19)$

At step 4, there were 83 systems remaining. As a first attempt we chose
$k_{i}=4$ for all $i$. The corresponding graph $W$ turned out not to be
connected; three vertices were isolated. However, choosing larger values of
$k_{i}$ for these awkward vertices, we were able to show that the graph $W$ in
step 4 is connected. Thus all the remaining $\operatorname{STS}(19)$s form a
single connected component, and step 5 was not required.

Summarising these results we have the following theorem.

######  Theorem 3.

The Pasch (4-cycle) switch graph $\operatorname{S}_{4}(19)$ consists of 2,732
connected components as follows.

  * •

2,591 anti-Pasch systems as in Figure 1(a);

  * •

126 pairs of twin systems as in Figure 1(b);

  * •

one component as in Figure 1(c);

  * •

nine components as in Figure 1(d);

  * •

one component as in Figure 1(e);

  * •

two components as in Figure 1(f);

  * •

one component as in Figure 2;

  * •

one component containing the remaining 11,084,871,929 systems.

###  3.2 Switching 6-cycles

As in [1], we find a single 6-cycle free system. On base set
$\\{\infty,A_{i},B_{i},C_{i}:i\in\mathbb{Z}_{6}\\}$ the blocks are the orbits
generated by the starters $\\{\infty,A_{0},A_{3}\\}$,
$\\{\infty,B_{0},B_{3}\\}$, $\\{\infty,C_{0},C_{3}\\}$,
$\\{A_{0},A_{1},B_{0}\\}$, $\\{A_{0},A_{2},C_{1}\\}$,
$\\{B_{0},B_{1},C_{1}\\}$, $\\{B_{0},B_{2},A_{4}\\}$,
$\\{C_{0},C_{1},B_{2}\\}$, $\\{C_{0},C_{2},A_{4}\\}$,
$\\{A_{0},B_{3},C_{0}\\}$, $\\{A_{0},B_{1},C_{3}\\}$ under the permutation
$(\infty)(A_{0}\ A_{1}\ldots A_{5})(B_{0}\ B_{1}\ldots B_{5})(C_{0}\
C_{1}\ldots C_{5})$. The system is therefore 3-rotational.

There is one component of order 24, consisting of systems each containing 36
6-cycles. This component is illustrated in Figure 3; because of the large out-
degree of each vertex, we have suppressed multiple edges and loops, so the
figure represents the undirected component of $\operatorname{S}_{6}(19)$. Note
that as an undirected graph, this component is not regular; the vertex degrees
range from 9 to 19. The automorphism group of the undirected component has
order 4. A representative system, from which the component may be
reconstructed by 6-cycle switching, is the following.

2468acegif58aehdi5ha9gdficeidhgegfbdic9ieghhgiffibehhigfh

Figure 3: The connected component of order 24 in $\operatorname{S}_{6}(19)$

This time, we were unable to show that the graph $W$ in step 4 is connected,
but construction of the graph revealed that the remaining systems formed at
most two connected components. Thus we used step 5 and identified a further
large connected component containing 284,433 vertices. We summarise these
results in the following theorem.

######  Theorem 4.

The four connected components of $\operatorname{S}_{6}(19)$ are as follows.

  * •

one 6-cycle free system;

  * •

one component of order 24 as in Figure 3;

  * •

one component containing 284,433 systems;

  * •

one component containing the remaining 11,084,590,371 systems.

It is noteworthy that representatives of the four connected components in
$\operatorname{S}_{6}(19)$ are connected via 4-cycle switching. Since by [1]
there exists an $\operatorname{STS}(19)$ without a $c$-cycle for all
$c\in\\{4,6,8,10,12,16\\}$, it follows that every graph
$\operatorname{S}_{c}(19)$ is disconnected. We thus have the following
theorem.

######  Theorem 5.

The minimum cardinality of a set $C$ such that $\operatorname{S}_{C}(19)$ is
connected is 2, and an example set is $C=\\{4,6\\}$.

###  3.3 Switching 8-cycles

By [1], there are exactly 381 8-cycle free systems and step 3 identified
these; therefore there are 381 isolated vertices in $\operatorname{S}_{8}(19)$
of this form. There is one further isolated vertex. On base set
$V=\\{X,Y,Z,A_{i},B_{i}:i\in\mathbb{Z}_{8}\\}$, the blocks are $\\{X,Y,Z\\}$
together with the orbits generated by the starters $\\{X,A_{0},A_{1}\\}$,
$\\{Y,B_{0},B_{1}\\}$, $\\{Z,A_{0},A_{4}\\}$, $\\{Z,B_{0},B_{4}\\}$,
$\\{A_{0},A_{2},B_{3}\\}$, $\\{A_{0},A_{3},B_{0}\\}$,
$\\{B_{0},B_{2},A_{4}\\}$, $\\{B_{0},B_{3},A_{1}\\}$ under the permutation
$(Z)(X\ Y)(A_{0}\ A_{1}\ldots A_{7})(B_{0}\ B_{1}\ldots B_{7})$. The cycle
graph on the pair $\\{X,Y\\}$ has two 8-cycles in a complementary pair. By
Lemma 1, these both switch to the same isomorphism class, which in this case
is the original system. There are no other 8-cycles. The cycle graphs on the
pairs $\\{X,Z\\}$ and $\\{Y,Z\\}$ have four 4-cycles; on the pairs
$\\{A_{i},B_{i}\\}$, $i\in\mathbb{Z}_{8}$ a 6-cycle and a 10-cycle; and on the
pairs $\\{A_{i},A_{i+3}\\}$ and $\\{B_{i},B_{i+3}\\}$, $i\in\mathbb{Z}_{8}$ a
4-cycle and a 12-cycle. On all other pairs the cycle graph is a Hamiltonian
cycle.

At step 4, we were able to show that all other vertices in
$\operatorname{S}_{8}(19)$ are in a single connected component. We summarise
these results in the following theorem.

######  Theorem 6.

The 383 connected components of $\operatorname{S}_{8}(19)$ are as follows.

  * •

381 8-cycle free systems;

  * •

one further system with two complementary 8-cycles,switching to the original
system;

  * •

one component containing the remaining 11,084,874,447 systems.

###  3.4 Switching 10-cycles

We find 66 10-cycle free systems, in agreement with [1]. By Lemma 1, every
connected component of $\operatorname{S}_{10}(19)$ is a subset of some
connected component of $\operatorname{S}_{6}(19)$. However, there may be
6-cycles in a given $\operatorname{STS}(19)$ which are not complementary to a
10-cycle, and so the components of $\operatorname{S}_{6}(19)$ may be
disconnected in $\operatorname{S}_{10}(19)$.

The unique 6-cycle free system is of course also 10-cycle free. There is also
a connected component of order 24 consisting of the same systems as the
component of the same order in the 6-cycle switching graph. Interestingly, and
as we have already observed, although the graph of this component in the
6-cycle switching graph is not regular (Figure 3), it is regular of degree 36
in the 10-cycle switching graph!

The connected component of cardinality 284,433 in $\operatorname{S}_{6}(19)$
fractures into 76 components, 65 of which are isolated vertices corresponding
to 10-cycle free systems. There are 10 components of “moderate” size which are
summarised in Table 1. In the table, by “degree” we mean the out-degree of a
vertex in $\overrightarrow{\operatorname{S}_{10}}(19)$; in other words, the
number of 10-cycles in each system in the component. It is an unexpected
feature of these components that, with the single exception of the largest
component of order 1,078, all systems have the same number of 10-cycles. The
remaining component has order 282,123.

The largest connected component in $\operatorname{S}_{6}(19)$ fractures into
just two components. One of these has order 56 and is regular of degree 18. A
representative system is the following.

2468acegib6cfgdih5cbhegdicgdhfibi9hgf9feihahifgedcifeghhi

The other component contains the remaining 11,084,590,315 systems.

# | Order | Degree | Representative system
---|---|---|---
1 | 84 | 18 | 2468acegi687hcigf758eghdi8cfghi7difhgideghbeifhfhdeigbfih
2 | 152 | 30 | 2468acegi6a9cedih95ahcgfiaebdhi9gdfihifeghfhiegdbighfghif
3 | 152 | 24 | 2468acegi678ebdih857hcfig7cedhi8fgeihbfhgidhigfgihefidgfh
4 | 152 | 18 | 2468acegi6bcg9fhic5bedhgibdhfigchgidffedhiafgihbicieefhgh
5 | 294 | 30 | 2468acegii6geacdhahf89idghecgdfafbegiidbeafhigghifhhehfii
6 | 1078 | 26 (432) | 2468acegia9bfhegie7gcdhif89fidhedgbhidihfiaghgcefhidegfhi
|  | 28 (432) |
|  | 32 (214) |
7 | 108 | 20 | 2468acegif79cigehg9dfbhiehbicdeeabdihfagiheigedggfhffhihi
8 | 108 | 28 | 2468acegifecgahidcdgbi9hf8iebdhh8bgcifebicdfhhgigeffgihhi
9 | 98 | 27 | 2468acegid7h9bgif7afhibdgig9fehbihefggedcfadecbhigfihihhi
10 | 19 | 27 | 2468acegi578ibfgh867edfhi7hcgif8gihefbfdihdeighcgehifhdig
Table 1: Components in $\overrightarrow{\operatorname{S}_{10}}(19)$

We summarise the results for 10-cycle switching below.

######  Theorem 7.

The 80 connected components of $\operatorname{S}_{10}(19)$ are as follows.

  * •

66 10-cycle free systems;

  * •

one component containing 24 systems, being the same systems as those of the
component of cardinality 24 in $\operatorname{S}_{6}(19)$;

  * •

10 components as in Table 1 containing a total of 2,245 systems;

  * •

one component containing 282,123 systems;

  * •

one component containing 56 systems;

  * •

one component containing 11,084,590,315 systems.

###  3.5 Switching 12-cycles

By Lemma 2, every component of $\operatorname{S}_{12}(19)$ is a subset of some
component of $\operatorname{S}_{4}(19)$. There are 2,727 systems with no
12-cycle, again in agreement with [1]. There are a further 147 small
components identified at step 3 of the algorithm, as pictured in Figure 4.

Component types (c) and (d) are interesting. There are a total of 126 such
components; the systems involved are precisely the 126 pairs of twins which
appear as components in the 4-cycle switching graph. Note that although the
systems in these components have two or three 12-cycles, each has only a
single 4-cycle; this situation is explained by the remark in Section 1, since
a 4-cycle may be complementary to up to three 12-cycles.

Components (k) and (l) are subsets of the component containing 16 systems in
$\overrightarrow{\operatorname{S}_{4}}(19)$. In Figure 4, the vertices in
these components are identified with the corresponding labels from Figure 2.

At step 4, the remaining systems all form a single component. We summarise
these results in the following theorem.

######  Theorem 8.

The 2,825 connected components of $\operatorname{S}_{12}(19)$ are as follows.

  * •

2,727 12-cycle free systems;

  * •

147 components as in Figure 4 (b) to (l) containing a total of 320 systems;

  * •

one component containing the remaining 11,084,871,782 systems.

Component type |  Number
---|---
|  |
(a) |  |  2,727
(b) |  |  6
(c) |  |  42
(d) |  |  84
(e) |  |  1
(f) |  |  1
(g) |  |  2
(h) |  |  7
(i) |  |  1
(j) |  |  1
(k) |  $B3$$A3$$B1$$B0$$A0$$A1$$B2$$A2$ |  1
(l) |  $A5$$B5$$A7$$A6$$B4$$B7$$A4$$B6$ |  1

Type | Example representative system
---|---
(a) | 2468acegibfegchdif6d9cehi7idhcgheicgdaigfabfdhighgfefhihi
(b) | 2468acegig5aebhfih7b8dcfiaec9ifidcfghdfiehhfdggbihigegihh
(c) | 2468acegic5igdhbf6dc9bghi8eifghfaicehfehigbhgdehifegdihfi
(d) | 2468acegifbcig9dha67iegdh8beghiicfdhghfeg9dcfhbhigheififi
(e) | 2468acegiefcdhaig578fcdhi9ibfghhg9deiagdifebhicfgeihhgihf
(f) | 2468acegi68eaigdh9gb7hcfifebdih7afheidchi9ifghgdhiigfehfg
(g) | 2468acegihfceabig9c7adfih8gefiddihgbefheibhdfgifecihgihgh
(h) | 2468acegi6dge9hif7gi9hcdfafehgi7ecbihdcfhibfdhgbhfeihiggi
(i) | 2468acegi59gafheid7cebfhi9hcfigdhgbfigihefcadiiefehgdighh
(j) | 2468acegiceh7gfbii7adbcfhg8aefhchfigde9ifbifghgdidhigehhi
(k) | 2468acegi6d8bhcfid5hiafge7cibghiahgefdfighgecdfehfbgiihhi
(l) | 2468acegicha89dfi9gbieahffhabig7f9eiddighgdcefihfihgehhgi

Figure 4: Small connected components in
$\overrightarrow{\operatorname{S}_{12}}(19)$

## References

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dt\,\,dt^{\prime}\chi_{R_{1}}(t)K_{11}(t,t^{\prime})\chi_{R_{1}}(t^{\prime})$
(B.10)

where the kernel can be seen to be the time ordered correlator
$\langle{\cal{O}}_{R_{1}}(t){\cal{O}}_{R_{1}}(t^{\prime})\rangle$, i.e.

$\displaystyle\langle{\cal{O}}_{R_{1}}(t){\cal{O}}_{R_{1}}(t^{\prime})\rangle$
$\displaystyle=-iK_{11}(t,t^{\prime})$ $\displaystyle=-ir^{\Delta}\int_{{\cal
F}}\frac{d\omega}{2\pi}e^{-i\omega(t-t^{\prime})}\left(r\partial_{r}q_{1,\omega(r)}\right)_{r\rightarrow\infty}$
$\displaystyle=-i\frac{\Delta}{2}\int_{{\cal{F}}}d\omega\;e^{-i\omega(t-t^{\prime})}\times$
(B.11)
$\displaystyle\qquad\qquad\qquad\times\frac{\Gamma\left(\frac{1}{2}-\Delta\right)\left(\csc(\pi\omega)\csc(\pi(\Delta+\omega))-e^{i\pi\omega}(\cot(\pi\omega)+i)\csc(\pi(\Delta-\omega))\right)}{4^{\Delta}e^{i\pi(\Delta+\omega)}\Gamma\left(\Delta-\frac{1}{2}\right)\Gamma(1-\Delta-\omega)\Gamma(1-\Delta+\omega)}$
$\displaystyle=\frac{2^{-\Delta}\Gamma(\Delta+1)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\frac{1}{\left(\cos((t-t^{\prime})(1-i\epsilon))-1\right)^{\Delta}}\;.$
(B.12)

In the last line we noted that the integral closes differently for
$(t-t^{\prime})\gtrless 0$, capturing the residues at $\omega=\pm(\Delta+n)$
respectively. A similar analysis to the one around eq. (3.42) shows that the
integral expression is equivalent the Feynman order correlator, as written in
the last line.

To give an example, we can explicitly carry the integral for
$(t-t^{\prime})<0$ capturing the $\omega=+(\Delta+n)$,

$\displaystyle\langle{\cal{O}}_{R_{1}}(t){\cal{O}}_{R_{1}}(t^{\prime})\rangle$
$\displaystyle=\frac{\Delta(\cot(\pi\Delta)+i)\Gamma\left(\frac{1}{2}-\Delta\right)}{4^{\Delta}\Gamma\left(\Delta-\frac{1}{2}\right)}\sum_{n=0}^{\infty}\frac{(-1)^{n}e^{-i(nt+\Delta(t+\pi))}}{\Gamma(n+1)\Gamma(1-n-2\Delta)}$
(B.13)
$\displaystyle=\frac{2^{-\Delta}\Gamma(\Delta+1)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\frac{1}{\left(\cos(t-t^{\prime})-1\right)^{\Delta}}\;.$
(B.14)

The correlators
$\langle{\cal{O}}_{R_{2}}(t){\cal{O}}_{R_{2}}(t^{\prime})\rangle$ on
$r\to-\infty$ can be seen to be also time ordered and identical to
$\langle{\cal{O}}_{R_{1}}(t){\cal{O}}_{R_{1}}(t^{\prime})\rangle$ by virtue of
the $r\to-r$ symmetry in the metric and EOMs.

The correlator between boundaries
$\langle{\cal{O}}_{R_{1}}(t){\cal{O}}_{R_{2}}(t^{\prime})\rangle$ requires a
bulk solution with sources on both sides turned on, i.e.

$\chi(t,r)=\int
dt^{\prime}\int_{{\cal{F}}}\frac{d\omega}{2\pi}e^{-i\omega(t-t^{\prime})}(q_{1,\omega}(-r)\chi_{L}(t^{\prime})+q_{1,\omega}(r)\chi_{R}(t^{\prime}))$
(B.15)

whose on shell action results, see (B.10),

$I^{0}_{R_{1}+R_{2}}=I^{0}_{11}-i\int
dt\,dt^{\prime}\chi_{R_{1}}(t)K_{12}(t,t^{\prime})\chi_{R_{2}}(t^{\prime})+I^{0}_{22}$
(B.16)

leading to

$\displaystyle\langle{\cal{O}}_{R_{1}}(t){\cal{O}}_{R_{2}}(t^{\prime})\rangle=-iK_{12}(t,t^{\prime})=\frac{2^{-\Delta}\Gamma(\Delta+1)}{\sqrt{\pi}\Gamma\left(\Delta-\frac{1}{2}\right)}\frac{1}{\left(\cos((t-t^{\prime})(1-i\epsilon))+1\right)^{\Delta}}\;.$
(B.17)

that correctly reproduces a time ordered correlator with no singularities at
$t=t^{\prime}$.

###  B.2 Expectation values

We now compute expectation values of the KG probe field $\chi$ for general
initial and final holographic excited states introduces in Sec. 2.2. To do
this we can reduce the problem to an pure Euclidean set-up and look for the
configuration at the moment of time reflection symmetry $t=\tau=0$. The
Euclidean metric is obtained by a Wick rotation $t\to-i\tau$ on (B.2), i.e.

$ds^{2}=(r^{2}+1)d\tau^{2}+\frac{dr^{2}}{r^{2}+1},\qquad
r,\tau\in(-\infty,\infty)\;$ (B.18)

Notice that this foliation suggests the existence of two disconnected
boundaries in the pure Euclidean AdS2 scenario which is not true, see Fig. 5.
We insist on this foliation here for the two boundaries will become actually
disconnected in the Thermal AdS2 scenario which is the one relevant for JT/SYK
holography and covered in Sec. 4.3.1.

As explained in Sec. 2.2, to describe an initial holographic excited state, we
should put a general Euclidean source in the Euclidean past region $\tau<0$.
Our coordinates force us to further split this into 2 pieces that we can call
$\chi_{I_{1}}$ and $\chi_{I_{2}}$, in the sense that they both belong to the
same past Euclidean segment, $\chi_{I_{1/2}}(\tau)=0$ for $\tau>0$, but are
defined only on $r\to\pm\infty$ respectively. We will show the positive energy
modes that arise from sources turned on only on the past Euclidean region.
Negative energy modes arise from sources put in the Euclidean future and these
can be obtained by $\tau\to-\tau$ reflection symmetry. We refer to [28] for a
detailed description of this construction.

The Euclidean problem analogous to (B.3) on the metric (B.18) with sources in
the Euclidean past can be written as,

$\displaystyle\chi_{I}(\tau,r)$
$\displaystyle=\int_{-\infty}^{0}d\tau^{\prime}\int_{\cal
F}\frac{d\omega}{2\pi}e^{-i\omega(-i\tau+i\tau^{\prime})}\left(\chi_{I_{1}}(\tau^{\prime})q_{1,\omega}(r)+\chi_{I_{2}}(\tau^{\prime})q_{1,\omega}(-r)\right)$
(B.19) $\displaystyle=\int_{\cal
F}\frac{d\omega}{2\pi}e^{-\omega\tau}\left(\bar{\chi}_{I_{1}}(\omega)q_{1,\omega}(r)+\bar{\chi}_{I_{2}}(\omega)q_{1,\omega}(-r)\right)\;.$
(B.20)

We have written the solution in terms of the Lorentzian solutions found above
for convenience. The $\bar{\chi}_{I_{1/2}}(\omega)$ stand for the Laplace
transform of the asymptotic sources. To be precise, the solution meets

$\displaystyle\chi_{I}(\tau,r)$ $\displaystyle\sim
r^{\Delta-1}\chi_{I_{1}}(\tau)\qquad\qquad r\to+\infty$ (B.21)
$\displaystyle\chi_{I}(\tau,r)$ $\displaystyle\sim
r^{\Delta-1}\chi_{I_{2}}(\tau)\qquad\qquad r\to-\infty\;.$ (B.22)

The physical propagating modes can be read by considering first $\tau\sim 0$
but $\tau>\tau^{\prime}$, i.e. we stand in the Euclidean future of all
asymptotic sources, such that one can use the Residue theorem on the $\omega$
integral. One then takes $\tau\to it$, resulting in

$\displaystyle\chi_{I}(t,r)$
$\displaystyle=\sum_{n=0}^{\infty}e^{-i(\Delta+n)t}\left(\bar{\chi}_{I_{1}}(\Delta+n)+\bar{\chi}_{I_{2}}(\Delta+n)\right)\mathop{\mathrm{Res}}_{\omega=\Delta+n}q_{1,\omega}(r)$
(B.23)

Since we have only excited the Euclidean past, both integrals closed upwards
and only positive energy modes $\omega=+(\Delta+n)$, $n\geq 0$ appear. The
negative frequency modes $\omega=-(\Delta+n)$ should appear when sources on
the Euclidean future $\tau^{\prime}>0$ are turned on. The corresponding
solution yields

$\displaystyle\chi_{F}(t,r)$
$\displaystyle=\sum_{n=0}^{\infty}e^{+i(\Delta+n)t}\left(\bar{\chi}_{F_{1}}(\Delta+n)+\bar{\chi}_{F_{2}}(\Delta+n)\right)\mathop{\mathrm{Res}}_{\omega=-(\Delta+n)}q_{1,\omega}(r)$
(B.24)

Notice that both $\chi_{I}(t,r)$ and $\chi_{F}(t,r)$ solutions consist of N
modes (B.9). The expectation value of boundary operators ${\cal O}_{R/L}$ can
be obtained via the BDHM dictionary, see (3.14), see also [28]

$\displaystyle\langle\chi_{F}|{\cal O}_{R}(t)|\chi_{I}\rangle$
$\displaystyle\equiv(2\Delta-1)\lim_{r\to+\infty}r^{\Delta}\left(\chi_{I}(t,r)+\chi_{F}(t,r)\right)$
(B.25)
$\displaystyle=\sum_{n=0}^{\infty}\left(e^{-i(\Delta+n)t}\left(\bar{\chi}_{I_{1}}(\Delta+n)+\bar{\chi}_{I_{2}}(\Delta+n)\right)+e^{i(\Delta+n)t}\left(\bar{\chi}_{F_{1}}(\Delta+n)+\bar{\chi}_{F_{2}}(\Delta+n)\right)\right)F_{\Delta,n}$
$\langle\chi_{F}|{\cal
O}_{L}(t)|\chi_{I}\rangle\equiv(2\Delta-1)\lim_{r\to-\infty}r^{\Delta}\left(\chi_{I}(t,r)+\chi_{F}(t,r)\right)=\langle\chi_{F}|{\cal
O}_{R}(t)|\chi_{I}\rangle$ (B.26)

The inherited normalization from AdS is,

$\displaystyle F_{\Delta,n}$
$\displaystyle=\frac{\Gamma\left(\frac{3}{2}-\Delta\right)\Gamma(n+2\Delta)}{4^{\Delta-1}\pi\Gamma\left(\Delta-\frac{1}{2}\right)\Gamma(n+1)}$
(B.27)

As a final comment, it has been noticed in the literature that this inherited
operator normalization coming from holography is consistent between the GKPW
and BDHM [47, 28, 42]. This amounts to a non-trivial match since the GKPW
prescription relies only on semiclassical bulk computations and the BDHM
prescription requires to build an orthonormal set of modes to canonically
quantize the fields. Such an identity was proven useful to analytically
compute the correct normalization for these orthonormal modes [41] as well as
to compute Bogoliubov coefficients between different quantizations [64].

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