# Two-center problem with harmonic-like interactions: periodic orbits and
integrability

Adrian M. Escobar Ruiz1, Lidia Jiménez–Lara1 and Jaume Llibre2 1 Departamento
de Física, Universidad Autónoma Metropolitana–Iztapalapa, P.O. Box 55–534,
México, D.F., 09340 México<EMAIL_ADDRESS><EMAIL_ADDRESS>2
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193
Bellaterra, Barcelona, Catalonia, Spain<EMAIL_ADDRESS>
###### Abstract.

We study the classical planar two-center problem of a particle $m$ subjected
to harmonic-like interactions with two fixed centers. For convenient values of
the dimensionless parameter of this problem we use the averaging theory for
showing analytically the existence of periodic orbits bifurcating from two of
the three equilibrium points of the Hamiltonian system modeling this problem.
Moreover, it is shown that the system is generically non-integrable in the
sense of Liouville–Arnold. The analytical results are complemented by
numerical computations of the Poincaré sections as well as providing some
explicit periodic orbits.

###### 1991 Mathematics Subject Classification:

Primary: 34C05

##  1\. Introduction and statement of the main results

Nonlinear dynamical systems are central objects in theoretical physics. For
instance, in classical mechanics [1, 2, 3, 4] appears the rich and complex
behaviour of physical relevant systems such as the 3-body problem, dynamical
astronomy, Henon-Heiles Hamiltonian systems, rigid bodies problems, and non-
linear Hamiltonian systems [5, 6, 7].

Among the various trajectories that mechanical systems can exhibit, periodic
solutions play a fundamental role. These trajectories represent a closed
motion in the phase space, offering valuable insights into the underlying
dynamical properties and stability of the system. Needless to say that, in
general, the intricate time evolution of non-linear systems do not admit a
straightforward analysis. There is a lack of universally applicable formulas
for determining periodic trajectories in dynamical systems. Therefore, the
development of asymptotic approximations to the solutions of a nonlinear
differential system as well as their numerical approaches are needed.

In this context the averaging theory formulated in Fatou’s seminal work [8]
offers a systematic approach to extract essential information from complex
dynamical systems. Subsequent contributions in the 1930s by Bogoliubov and
Krylov [9], as referenced by Bogoliubov [10] in 1945, significantly increased
both practical applications and theoretical understanding of the averaging
theory.

Over time, the ideas of averaging theory have undergone refinement and
expansion in various directions, catering to both finite and infinite-
dimensional differentiable systems. For contemporary literature and
developments in averaging theory, we refer the readers to the works of
Sanders, Verhulst, Murdock (see [11] and refereces therein), and Verhulst
[12], among others, which provide modern expositions and present-day results
on the subject.

The fundamental premise of the averaging theory lies in the recognition that
many physical systems exhibit fast and slow motions simultaneously. By
exploiting this timescale separation, averaging techniques aim to construct
simplified models that capture the essential dynamics while filtering out fast
oscillations and transient behavior. This reduction in complexity not only
facilitates analytical tractability but also provides qualitative insights
into the long-term behavior of the system. Concrete applications can be found
in the works [13, 14].

In this paper we aim to investigate the dynamics of a two-center problem with
harmonic interactions using the averaging theory. This system can be viewed as
a limiting case of the 3-body harmonic oscillator, a nine-parameter system
with three arbitrary masses, three rest lengths, and three spring constants.

In the simplest scenario involving equal masses on the plane and equal spring
constants, the 3-body harmonic oscillator exhibits a remarkably diverse
dynamics in function of the energy [15, 16]. This diversity manifests in a
power-law statistics reminiscent of the Levi-walk model [15]. Even when
restricted to the invariant manifold of zero total angular momentum, the
parameter space displays regions of both regular and chaotic dynamics due to
inherent non-linearities stemming from non-zero rest lengths [17]. Upon
setting these rest lengths equal to zero, the system attains
superintegrability.

It is noteworthy that at zero rest lengths, the corresponding classical and
quantum systems become exactly solvable [18]. However, for non-zero rest
lengths, an exact solution does not exist. Consequently the quantum 3-body
harmonic oscillator [19] serves as a practical model for testing theoretical
and numerical methodologies aimed at elucidating the interplay between
classical and quantum mechanics within chaotic systems.

In the case when two bodies are considered infinitely massive, the 3-body
harmonic oscillator reduces to the two-center problem the one investigated in
this paper. Numerical as well as analytical tools based on the averaging
theory are used to explore the dynamics of this system. The main goal is to
find periodic trajectories emanating from the equilibrium points of this
system.

###  1.1. Equations of motion of the two-center problem with harmonic-like
interactions

In the Euclidean space $\mathbb{R}^{2}$ we consider a two-center problem of a
non-relativistic point particle $m$ subjected to harmonic-like interactions
with two fixed centers possessing the same constant of elasticity $k>0$. In
cartesian coordinates $(X,Y)$ the Hamiltonian of the system is of the form:

(1) ${\mathcal{H}}\ =\ \dfrac{1}{2\,m}\left(\,P_{X}^{2}\ +\
P_{Y}^{2}\,\right)\ +\ V(X,Y)\ ,$

where $V(X,Y)$ is the translational-invariant potential

$\begin{split}V(X,Y)\ &=\ \dfrac{1}{2}\,k\,\big{(}\,\left(R_{1}-A\right)^{2}\
+\ \left(R_{2}-A\right)^{2}\,\big{)}\ ,\\\ &=\
\dfrac{1}{2}\,k\left(\sqrt{(X+L)^{2}+Y^{2}}-A\right)^{2}\ +\
\left(\sqrt{(X-L)^{2}+Y^{2}}-A\right)^{2}\,\big{)}\ ,\end{split}$

$R_{1}$, $R_{2}$ are the distances from the mass $m$ to the two fixed centers,
respectively, which we assume are located at $(\pm L,0)$, and the constant
$A\geq 0$ denotes the equilibrium distance from $m$ to each one of the fixed
centers. Hence, the phase space is four-dimensional. First, for the non-
dimensionalization of ${\mathcal{H}}$, we divide the expression (1) by
$m\,L^{2}\omega^{2}$, where $\omega^{2}=k/m$, and define a dimensionless time
$\tau=t\,\omega$. More precisely, we introduce the set of non-dimensional
quantities:

$\begin{split}x&=X/L\ ,\quad y=Y/L\ ,\quad a=A/L\,,\quad r_{i}=R_{i}/L\
,\quad\tau=t\,\omega\ ,\\\ p_{x}&=\dfrac{dx}{d\tau}\ ,\quad
p_{y}=\dfrac{dy}{d\tau}\ ,\quad H={\mathcal{H}}/(m\,L^{2}\omega^{2})\ ,\quad
U=V/(m\,L^{2}\omega^{2})\ .\end{split}$

In these variables the original Hamiltonian (1) and potential are written in
dimensionless form as follows

(2) $H\ =\ \dfrac{1}{2}\left(\,p_{x}^{2}\ +\ p_{y}^{2}\,\right)\ +\ U(x,y)\ ,$
$U(x,y)\ =\
\dfrac{1}{2}\left(\,\left(\sqrt{(x+1)^{2}+y^{2}}\,-\,a\right)^{2}+\left(\sqrt{\left(x-1)^{2}+y^{2}\right)}\,-\,a\right)^{2}\,\right)\
,$

here the only remaining dimensionless parameter is $a$. Below in Fig. 1, the
geometrical settings of the system are presented in detail, and in Fig. 2 we
graph the potential.

In the special case $a=2$, a configuration of equilibrium $r_{1}=r_{2}=a$
corresponds to the equilateral triangle with sides $(2,2,2)$, where the
particle and the two centers mark the vertices.

Figure 1. Planar two-center problem with harmonic-like interactions in
dimensionless variables. The distance between the two fixed centers is 2 and
the only free parameter is $a=A/L$. The reference system with the origin in
the midpoint of the line which connects the two centers is adopted. These
centers are located at $(\pm 1,\,0)$, respectively.

At $a=0$ system (2) coincides with the 2D isotropic harmonic oscillator, a
superintegrable system possessing three algebraically independent first
integrals in the Liouville sense.

Along the line $x=0$, for $0\leq a\leq 1$ the potential (2) possesses a
critical point (a minimum) at $y=0$, $U(0,0)={(a-1)}^{2}$, whilst for $a>1$
this point $(0,0)$ becomes a maximum and two symmetric minima located at
$y_{\pm}=\pm\sqrt{a^{2}-1}$, respectively, occur with $U(0,y_{\pm})=0$.

On the line $y=0$, for $0\leq a\leq 1$ the potential (2) displays a minimum at
$x=0$, whereas for $a>1$ two additional symmetric maxima located at $x=\pm 1$,
respectively, emerge. Also, for any value of $a$ the derivative of the
potential is discontinuous at $x=\pm 1$.

Figure 2. The potential $U(x,y)$ in (2) as a function of the parameter $a$;
(a) $a=\frac{1}{2}$, (b) $a=\frac{3}{2}$, (c) $a=2$. The corresponding level
curves are shown in Figs. (d), (e) and (f), respectively.

For the Hamiltonian (2) the associated Hamilton’s equations of motion are

(3) $\displaystyle\dot{x}$ $\displaystyle=\ p_{x},$ $\displaystyle\dot{y}$
$\displaystyle=\ p_{y},$ $\displaystyle\dot{p}_{x}$ $\displaystyle=\
-\left(2\,x\,-\,a\left(\dfrac{x+1}{\sqrt{(x+1)^{2}+y^{2}}}\ +\
\dfrac{x-1}{\sqrt{(x-1)^{2}+y^{2}}}\right)\right),$ $\displaystyle\dot{p}_{y}$
$\displaystyle=\ -\left(2\,y\,-\,a\,y\left(\dfrac{1}{\sqrt{(x+1)^{2}+y^{2}}}\
+\ \dfrac{1}{\sqrt{(x-1)^{2}+y^{2}}}\right)\right).$

This differential system is the main object of study of the present paper. It
is invariant under the discrete symmetries

$\begin{array}[]{l}\mathcal{S}_{1}:\ (x\rightarrow-x,\,y\rightarrow
y,p_{x}\rightarrow-p_{x},\,p_{y}\rightarrow p_{y}),\\\ \mathcal{S}_{2}:\
(x\rightarrow x,\,y\rightarrow-y,p_{x}\rightarrow p_{x},\,p_{y}\rightarrow-
p_{y}),\end{array}$

and $\mathcal{S}_{1}\circ\mathcal{S}_{2}$. Then the orbits are symmetric by
the planes $(x,0,p_{x},0)$ and $(0,y,0,p_{y})$ as well as from the origin
under the symmetry $\mathcal{S}_{1}\circ\mathcal{S}_{2}$.

###  1.2. Main results

System (3) has three equilibrium points
${\mathbf{x}}_{0}\,\equiv\,(x,y,p_{x},p_{y})$, namely:

$(0,\,0,\,0,\,0),\quad(0,\,-\sqrt{a^{2}-1},\,0,\,0),\quad(0,\,\sqrt{a^{2}-1},\,0,\,0).$

For $a>1$ the eigenvalues of the equilibrium point
$(0,\,\sqrt{a^{2}-1},\,0,\,0)$ are purely imaginary, i.e. they are $\pm
i\,\sqrt{2}/a$ and $\pm i\sqrt{2(a^{2}-1)}/a$, and in the next theorem we
describe the periodic orbit that bifurcate from this equilibrium.

######  Theorem 1.

For $a>1$ in each energy level $H=h$ with $h>0$ sufficiently small, from the
equilibrium point $(0,\,\sqrt{a^{2}-1},\,0,\,0)$ of the Hamiltonian system (3)
can bifurcate one or more periodic orbits
$(x(t,\varepsilon),y(t,\varepsilon),p_{x}(t,\varepsilon),p_{y}(t,\varepsilon))$
with initial conditions
$(x(0,\varepsilon),y(0,\varepsilon),p_{x}(0,\varepsilon),p_{y}(0,\varepsilon))$
of the form

(4)
$\left(\varepsilon\tilde{r},\,\sqrt{a^{2}-1}+\varepsilon\tilde{\rho}\cos\frac{\sqrt{2}g}{a}\tilde{s},\,0,\,-\varepsilon\dfrac{\sqrt{2}g}{a}\tilde{\rho}\sin\frac{\sqrt{2}g}{a}\tilde{s}\right)\
+\ O(\varepsilon^{2}),$

when the determinant of the Jacobian matrix

(5)
$\left.\frac{\partial(f_{1}(\rho,s),f_{2}(\rho,s))}{\partial(\rho,s)}\right|_{\rho=\tilde{\rho},s=\tilde{s}}\neq
0\ .$

Here $\varepsilon>0$ is a small parameter and
$g\equiv\sqrt{a^{2}-1}\in\mathbb{Q}$. The values of the functions
$f_{i}(\rho,s)$ for $i=1,2$, and of the constants $\tilde{r}$, $\tilde{\rho}$,
$\tilde{s}$ are given in the proof of this theorem.

Theorem 1 is proved in section 4. In section 5 for different values of the
parameter $a$ we prove the existence of one or two periodic orbits given by
Theorem 1.

Note that when $\varepsilon\to 0$ the periodic orbit of Theorem 1 bifurcates
from the equilibrium point $(0,\,\sqrt{a^{2}-1},\,0,\,0)$. Due to the
$\mathcal{S}_{2}$ symmetry by studying the periodic orbit bifurcating from the
equilibrium $(0,\,\sqrt{a^{2}-1},\,0,\,0)$ we are also studying the periodic
orbit bifurcating from the symmetric equilibrium
$(0,\,-\sqrt{a^{2}-1},\,0,\,0)$.

If the periodic orbit obtained in Theorem 1 is not invariant under the
$\mathcal{S}_{1}$ symmetry of the differential system (3), there is another
symmetric periodic orbit distinct to the one given in Theorem 1 with the
initial condition $(-x(0,\varepsilon),\allowbreak
y(0,\varepsilon),\allowbreak-p_{x}(0,\varepsilon),\allowbreak
p_{y}(0,\varepsilon))$ which is also near the equilibrium point
$(0,\,\sqrt{a^{2}-1},\,0,\,0)$.

On the other hand, if

$C=\left(\tilde{\rho}\cos\frac{\sqrt{2}g}{a}\tilde{s}\right)^{2}+\left(\dfrac{\sqrt{2}g}{a}\tilde{\rho}\sin\frac{\sqrt{2}g}{a}\tilde{s}\right)^{2}\neq
0\,,$

then the $\mathcal{S}_{2}$ symmetry of the differential system (3) provides
another periodic orbit distinct to the one given by Theorem 1 with initial
conditions $(-x(0,\varepsilon),\allowbreak y(0,\varepsilon),\allowbreak-
p_{x}(0,\varepsilon),\allowbreak p_{y}(0,\varepsilon))$, however this is not
near the original equilibrium point $(0,\sqrt{a^{2}-1},\,0,\,0)$, but near
$(0,-\sqrt{a^{2}-1},\,0,\,0)$.

Finally, if $\tilde{r}C\neq 0$, then the $\mathcal{S}_{1}\circ\mathcal{S}_{2}$
symmetry of the differential system (3) provides another periodic orbit
distinct to the one given in Theorem 1 with the initial condition
$(-x(0,\varepsilon),-y(0,\varepsilon),-p_{x}(0,\varepsilon),-p_{y}(0,\varepsilon))$,
and consequently near the equilibrium point $(0,-\sqrt{a^{2}-1},\,0,\,0)$.

There is a local symmetry which inverts $y$ and $p_{y}$ around the equilibrium
point $(0,\sqrt{a^{2}-1},\,0,\,0)$, valid only when $\varepsilon\rightarrow
0$. In this way, for each periodic orbit bifurcating from the equilibrium
point, there can be four periodic orbits bifurcating simultaneously from it.

In short, we have proved the next corollary.

######  Corollary 2.

Under the assumptions of Theorem 1 and for each periodic orbit bifurcating
from the equilibrium $(0,\,\sqrt{a^{2}-1},\,0,\,0)$, if $\tilde{r}C\neq 0$ and
$a>1$, then at each energy level $H=h$ of the Hamiltonian system (3) with
$h>0$ sufficiently small, there are at least $4$ periodic orbits, $2$ near the
equilibrium $(0,\,\sqrt{a^{2}-1},\,0,\,0)$, and the other two near the
equilibrium $(0,-\sqrt{a^{2}-1},\,0,\,0)$.

Of course, the integrable and non–integrable Hamiltonian systems can have
infinitely many periodic orbits. In general, it is not easy to find explicitly
a whole family of analytical periodic orbits mainly when the Hamiltonian
system is non–integrable. Here we find them in Theorem 1 and in Corollary 2.
Once we have proved that at any positive energy level sufficiently small there
exist analytic periodic orbits, we can use them to prove the next result about
the non–integrability of the Hamiltonian system (3) in the sense of
Liouville–Arnold.

######  Theorem 3.

Suppose that the Hamiltonian system (3) satisfies the hypotheses of Theorem 1
and Corollary 2. Then one of the following two statements hold:

  * (a)

If the determinant of the fundamental matrix associated to some of the
periodic orbits of Corollary 2 is different from $1$, then the Hamiltonian
system is not Liouville-Arnold integrable.

  * (b)

If all the determinants of the fundamental matrices associated to the periodic
orbits of Corollary 2 are $1$, the Hamiltonian system can be Liouville-Arnold
integrable.

The paper is structured as follows. In section 2 we show the Poincaré sections
of the flow of the Hamiltonian system (3) for the values of the parameter
$a=3/2,2,\sqrt{5},5$ and for some fixed values of the energy $H$. These
Poincaré sections provide some information about the global dynamics of the
Hamiltonian system (3).

In section 3 we recall the basic results of the averaging theory of first
order for computing periodic orbits that we shall need for proving Theorem 1.
As we have said in section 4 we prove Theorem 1.

In section 5 we compute the periodic orbit of Theorem 1 for the values of the
parameter $a=\sqrt{29}/2,\sqrt{5},5/4$. The case $a=\sqrt{5}$ which is
excluded in Theorem 1, is addressed in this section, and the periodic orbits
are explicitly exhibited and graphed.

Finally, in section 6 we prove Theorem 3.

##  2\. Poincaré sections

Now for the Hamiltonian (2) we present the Poincaré sections on the
$(y,p_{y})$ plane, considering the values $a=3/2,2,\sqrt{5},5$ as a function
of the energy $E=H$. The configuration of equilibrium $r_{1}=r_{2}=a$ with
$a=2$ corresponds to the equilateral triangle with sides $(2,2,2)$. The
Poincaré sections are determined from the intersection of trajectories,
associated with given initial conditions within the phase space, with a lower-
dimensional subspace ($x=0$, $y$, $p_{x}(y,p_{y},E)$, $p_{y}$). They are
transversal planes to the flow of the Hamiltonian system, and they can be
regarded as a discretized version of the dynamical system retaining relevant
properties of the original continuous system but acting in a reduced phase
space.

For the calculations of the Poincaré sections we take as a reference point of
energy the value $E_{s}$ of the potential (2) evaluated at the saddle point
$(0,0)$, namely $E_{s}\ =\ U(0,0)\ =\ {(a-1)}^{2}$.

The Figures (3), (4), (5), and (6), display the (oriented) Poincaré sections
for the system (2) at $a=3/2,2,\sqrt{5},5$, respectively, as a function of the
energy $E=H$ in units of $E_{s}$. They were obtained using numerical
simulations for 120 random initial conditions with a simulation time of 6000.
Computations were performed in MATLAB utilizing a personal laptop.

(a) $E=\frac{1}{4}E_{s}$

(b) $E=\frac{1}{2}E_{s}$

(c) $E=\frac{3}{4}E_{s}$

(d) $E=\frac{99}{100}E_{s}$

(e) $E=4\,E_{s}$

Figure 3. Poincaré sections on the plane $(y,\,p_{y})$, for the Hamiltonian
$H$ (2) with $a=\frac{3}{2}$, at different values of the energy $H=E$. In this
case the equilibrium points are $(0,\,\pm\frac{1}{2}\sqrt{5},\,0,\,0)$ and
$E_{s}=\frac{1}{4}\,.$

For $E<E_{s}$, the Poincaré sections consist of two symmetric disconnected
regions which smoothly merge at $(y=0,p_{y}=0)$ when $E=E_{s}$. For low
energy, $E\ll E_{s}$, the zones of regular dynamics dominate the accessible
phase space landscape.

At fixed $a$, the presence of chaotic behaviour is prominent at $E\sim E_{s}$.
Interestingly, at fixed energy $E$ (in units of $E_{s}$), additional numerical
experiments indicate that the degree of chaoticity is not a monotonous
function of the parameter $a$ in the interval $a\in(1,5]$ (see, for instance,
the Figures (3)-(6) for the case $E=\frac{1}{4}E_{s}$). Nevertheless, the
presence of chaos tends to decrease as the value of $a\gtrsim 3$ grows.

Notice that even at $E\sim E_{s}$, islands of stability (regular dynamics)
persist. Hence, the coexistence of regularity and chaos exhibits the rich
dynamics of the two-center problem with harmonic-like interactions. In
particular, we highlight the complexity of the structure of the Poincaré
sections for the case $a=\sqrt{5}$, see Fig. 5.

(a) $E=\frac{1}{4}E_{s}$

(b) $E=\frac{1}{2}E_{s}$

(c) $E=\frac{3}{4}E_{s}$

(d) $E=\frac{99}{100}E_{s}$

(e) $E=4\,E_{s}$

Figure 4. Poincaré sections on the plane $(y,\,p_{y})$, for the Hamiltonian
$H$ (2) with $a=2$, at different values of the energy $H=E$. In this case the
equilibrium points are $(0,\,\pm\sqrt{3},\,0,\,0)$ and $E_{s}=1\,.$

(a) $E=\frac{1}{4}E_{s}$

(b) $E=\frac{1}{2}E_{s}$

(c) $E=\frac{3}{4}E_{s}$

(d) $E=\frac{99}{100}E_{s}$

(e) $E=4\,E_{s}$

Figure 5. Poincaré sections on the plane $(y,\,p_{y})$, for the Hamiltonian
$H$ (2) with $a=\sqrt{5}$, at different values of the energy $H=E$. The
equilibrium points are $(0,\,\pm 2,\,0,\,0)$ and $E_{s}={(\sqrt{5}-1)}^{2}\,.$

(a) $E=\frac{1}{4}E_{s}$

(b) $E=\frac{1}{2}E_{s}$

(c) $E=\frac{3}{4}E_{s}$

(d) $E=\frac{99}{100}E_{s}$

(e) $E=4\,E_{s}$

Figure 6. Poincaré sections on the plane $(y,\,p_{y})$, for the Hamiltonian
$H$ (2) with $a=5$, at different values of the energy $H=E$. The equilibrium
points are $(0,\,\pm 2\,\sqrt{6},\,0,\,0)$ and $E_{s}=16\,.$

##  3\. The averaging theory of first order

It is worth recalling the basics of the averaging theory (periodic case) of
first order. This tool will be used to derive the main results of the present
study. Essentially, we deal with the problem of finding $T$-periodic solutions
for a differential system whose vector field depends on a small parameter
$\varepsilon$. For more details about the averaging theory of first order for
finding periodic orbits see [20].

We consider the differential system

(6) ${\dot{\bf x}}(t)\ =\ \varepsilon\,F_{1}(t,{\bf x})\ +\
O(\varepsilon^{2}),$

where $\varepsilon\neq 0$ is a sufficiently small parameter, i.e.
$|\varepsilon|\ll 1$, $F_{1}:\mathbb{R}\times\Omega\rightarrow\mathbb{R}^{n}$
is a continuous function $T$-periodic in the variable $t$, and $\Omega$
denotes an open subset of $\mathbb{R}^{n}$. The above equation often arises by
expansion in the neighborhood of an equilibrium point taking convenient
coordinates.

Now we introduce the averaged function of first order
$f_{1}:\Omega\rightarrow\mathbb{R}^{n}$ as follows

(7) $f_{1}({\bf z})\ =\ \frac{1}{T}\int_{0}^{T}F_{1}(s,\,{\bf z})\,ds\ ,$

and also assume that:

  * •

(i) $F_{1}$ is locally Lipschitz with respect to ${\bf x}$;

  * •

(ii) for ${\bf z}_{0}$ in $\Omega$ with $f_{1}({\bf z}_{0})=0$, there exists a
neighborhood $U$ of ${\bf z}_{0}$ such that $f_{1}({\bf z})\neq 0$ for all
${\bf z}\in U\setminus\\{{\bf z}_{0}\\}$ and $d_{B}(f_{1},\,U,\,{\bf
z}_{0})\neq 0$ (the Brouwer degree of the $f_{1}$ at ${\bf z}_{0}$ is not
zero).

Then for $|\varepsilon|$ sufficiently small, there exists a $T-$periodic
solution ${\bf x}(t,\,\varepsilon)$ of system (6) such that ${\bf
x}(0,\,\varepsilon)\rightarrow{\bf z}_{0}$ as $\varepsilon\rightarrow 0$. That
is, the simple zeros of the averaged function (7) provide the initial
conditions for isolated $T$-periodic solutions of the differential system (6).
Here a simple zero ${\bf z}_{0}$ of the function $f_{1}$ means that the
Jacobian of $f_{1}$ at ${\bf z}_{0}$ is not zero.

We recall that if the Jacobian of $f_{1}$ at ${\bf z}_{0}$ is not zero, then
the Brouwer degree $d_{B}(f_{1},\,U,\,{\bf z}_{0})\neq 0$, for details see
[21].

##  4\. Proof of theorem 1

In this section we address the problem of finding periodic orbits of the
differential system (3) bifurcating from the equilibrium point
${\mathbf{x}}_{0}\,=\,(x,y,p_{x},p_{y})\ =\ (0,\,\sqrt{a^{2}-1},\,0,\,0)$.

As a first step we translate the equilibrium ${\mathbf{x}}_{0}$ to the origin
of coordinates. To this end, we introduce the canonical transformation

(8) $(x,y,p_{x},p_{y})=(X,Y+g,P,Q).$

In these new variables the Hamiltonian (2) writes

$\begin{array}[]{rl}H=&\frac{1}{2}\big{(}\,P^{2}\ +\ Q^{2}\,\big{)}\ +\ X^{2}\
+\ Y^{2}\ +\ 2\,a^{2}\ +\ 2\,Y\,g\\\ &-\
a\,\bigg{(}\sqrt{\left(g+Y\right)^{2}+(X-1)^{2}}\ +\
\sqrt{\left(g+Y\right)^{2}+(X+1)^{2}}\bigg{)},\end{array}$

and its Hamilton’s equations are

(9) $\displaystyle\dot{X}=$ $\displaystyle P,$ $\displaystyle\dot{Y}=$
$\displaystyle Q,$ $\displaystyle\dot{P}=$
$\displaystyle-\frac{(X-1)\left(\sqrt{(g+Y)^{2}+(X-1)^{2}}-a\right)}{\sqrt{(g+Y)^{2}+(X-1)^{2}}}$
$\displaystyle-\frac{(X+1)\left(\sqrt{(g+Y)^{2}+(X+1)^{2}}-a\right)}{\sqrt{(g+Y)^{2}+(X+1)^{2}}},$
$\displaystyle\dot{Q}=$
$\displaystyle(g+Y)\left(a\left(\frac{1}{\sqrt{(g+Y)^{2}+(X+1)^{2}}}+\frac{1}{\sqrt{(g+Y)^{2}+(X-1)^{2}}}\right)-2\right).$

Now it is convenient to perform another change of variables such that the
linear part at the origin of the differential system (9) be in its real Jordan
normal form. Direct calculations lead to the transformations

(10)
$(X,Y,P,Q)=\left(U,W,-\frac{\sqrt{2}}{a}\,V,-\frac{\sqrt{2}\,g}{a}Z\right).$

From (9) and (10) we obtain the differential system

(11) $\displaystyle\dot{U}$ $\displaystyle=-\frac{\sqrt{2}}{a}\,V,$
$\displaystyle\dot{W}$ $\displaystyle=-\frac{\sqrt{2}\,g}{a}\,Z,$
$\displaystyle\dot{V}$ $\displaystyle=\
-\frac{a}{\sqrt{2}}\left[\left[\frac{a\,(U+1)}{\sqrt{(g+W)^{2}+(U+1)^{2}}}+\frac{a\,(U-1)}{\sqrt{(g+W)^{2}+(U-1)^{2}}}\right]-\
2\,U\,\right],$ $\displaystyle\dot{Z}$
$\displaystyle=\frac{a\,(g+W)}{\sqrt{2}g}\,\left[2-a\left[\frac{1}{\sqrt{(g+W)^{2}+(U+1)^{2}}}+\frac{1}{\sqrt{(g+W)^{2}+(U-1)^{2}}}\right]\right].$

which admits from (4) the first integral

(12)
$\begin{array}[]{rl}H=&\dfrac{(V-Z)(V+Z)}{a^{2}}+a^{2}+(g+W)^{2}+U^{2}+Z^{2}+1\vspace{0.2cm}\\\
&-a\left(\sqrt{(g+W)^{2}+(U-1)^{2}}+\sqrt{(g+W)^{2}+(U+1)^{2}}\right).\end{array}$

Next we rescale the variables for rewritting system (11) into a suitable form
for applying the averaging theory. Let $\varepsilon$ be a small parameter, and
we do the rescaling

(13)
$(U,W,V,Z)=(\varepsilon\,u,\varepsilon\,w,\varepsilon\,v,\varepsilon\,z),$

in the differential system (11), and expanding the new differential system in
powers of the small parameter $\varepsilon$ we obtain

(14) $\displaystyle\dot{u}$ $\displaystyle=\ -\frac{\sqrt{2}\,v}{a},$
$\displaystyle\dot{w}$ $\displaystyle=\ -\frac{\sqrt{2}\,g\,z}{a},$
$\displaystyle\dot{v}$ $\displaystyle=\
\frac{\sqrt{2}\,u}{a}+\frac{\sqrt{2}\left(a^{2}-3\right)\,g\,u\,w}{a^{3}}\varepsilon+O(\varepsilon^{2}),$
$\displaystyle\dot{z}$ $\displaystyle=\ \frac{\sqrt{2}\,g\,w}{a}\ +\
\frac{\left(\left(a^{2}-3\right)u^{2}+3w^{2}\right)}{\sqrt{2}\,a^{3}}\,\varepsilon+O(\varepsilon^{2}).$

whereas for (12) the first integral becomes

$H\ =\
\frac{\left(g^{2}\left(w^{2}+z^{2}\right)+u^{2}+v^{2}\right)}{a^{2}}\varepsilon^{2}\
+\ O(\varepsilon^{3}).$

We introduce the following polar variables $(r,\theta,\rho,\,s)$

(15)
$\begin{array}[]{ll}u\,=\,r\,\cos(\sqrt{2}\,\theta/a),&v\,=\,r\,\sin(\sqrt{2}\,\theta/a),\\\
w\,=\,\rho\,\cos(\sqrt{2}\,g\,(\theta+s)/a),&z\,=\,\rho\,\sin(\sqrt{2}\,g\,(\theta+s)/a).\end{array}$

Now we take the angular variable $\theta$ as the new independent variable, in
this way the differential system will be periodic in the variable $\theta$.
So, from (14) and (15) we arrive to the differential system

(16)
$\begin{array}[]{rl}\dfrac{dr}{d\theta}=&\dfrac{\varepsilon}{\sqrt{2}a^{3}}\left(a^{2}-3\right)\,g\,\rho\,r\,\sin\frac{2\sqrt{2}\theta}{a}\cos\frac{\sqrt{2}g(\theta+s)}{a}+O(\varepsilon^{2}),\vspace{0.2cm}\\\
\dfrac{d\rho}{d\theta}=&\dfrac{\varepsilon}{\sqrt{2}a^{3}}\sin\frac{\sqrt{2}g(\theta+s)}{a}\left(\left(a^{2}-3\right)r^{2}\cos^{2}\frac{\sqrt{2}\theta}{a}+3\rho^{2}\cos^{2}\frac{\sqrt{2}g(\theta+s)}{a}\right)+O(\varepsilon^{2}),\vspace{0.2cm}\\\
\dfrac{ds}{d\theta}=&\dfrac{\varepsilon}{4a^{2}g\rho}\cos\frac{\sqrt{2}g(\theta+s)}{a}\left(\left(a^{2}-3\right)\cos\frac{2\sqrt{2}\theta}{a}\left(r^{2}-2g^{2}\rho^{2}\right)+\left(a^{2}-3\right)r^{2}\right.\vspace{0.2cm}\\\
&\left.+\left(8a^{2}-2a^{4}-3\right)\rho^{2}+3\rho^{2}\cos\frac{2\sqrt{2}g(\theta+s)}{a}\right)+O(\varepsilon^{2}),\end{array}$

possessing the first integral

$\displaystyle H\ =\
\dfrac{\left(g^{2}\,\rho^{2}\,+\,r^{2}\right)}{a^{2}}\,\varepsilon^{2}\ +\
O(\varepsilon^{3}).$

Since in the Hamiltonian systems generically the periodic orbits appear in
cylinders of periodic orbits parametrized by the values of the Hamiltonian
$H$, and the averaging theory only can detect periodic orbits that are
isolated, we restrict the above differential system (16) to the energy level
$H=\varepsilon^{2}\,h$ with $h>0$. We impose this restriction computing $r$ as
a function of $\rho$ and $s$ in the energy level $H=\varepsilon^{2}\,h$,
namely

$r\ =\ \sqrt{a^{2}\left(h-\rho^{2}\right)\,+\,\rho^{2}}+O(\varepsilon).$

Therefore, up to first order in $\varepsilon$ we obtain from (16) the
differential system

(17)
$\begin{array}[]{rl}\dfrac{d\rho}{d\theta}=&\dfrac{\varepsilon}{\sqrt{2}a^{3}}\sin\frac{\sqrt{2}g(\theta+s)}{a}\left(\left(a^{2}-3\right)\left(a^{2}\left(h-\rho^{2}\right)+\rho^{2}\right)\cos^{2}\frac{\sqrt{2}\theta}{a}\right.\vspace{0.2cm}\\\
&\left.+3\rho^{2}\cos^{2}\frac{\sqrt{2}g(\theta+s)}{a}\right)+O(\varepsilon^{2})\vspace{0.2cm}\\\
=&F_{11}(\theta,\rho,s)+O(\varepsilon^{2}),\vspace{0.2cm}\\\
\dfrac{ds}{d\theta}=&\dfrac{\varepsilon}{4a^{2}g\rho}\cos\frac{\sqrt{2}g(\theta+s)}{a}\left(\left(a^{2}-3\right)\left(a^{2}\left(h-3\rho^{2}\right)+3\rho^{2}\right)\cos\frac{2\sqrt{2}\theta}{a}\right.\vspace{0.2cm}\\\
&\left.+\left(a^{2}-3\right)a^{2}h-3\left(a^{4}-4a^{2}+2\right)\rho^{2}+3\rho^{2}\cos\frac{2\sqrt{2}g(\theta+s)}{a}\right)+O(\varepsilon^{2})\vspace{0.2cm}\\\
=&F_{12}(\theta,\rho,s)+O(\varepsilon^{2}).\end{array}$

Assuming $a\notin\\{0,1,\sqrt{3},\sqrt{5}\\}$, by direct integration we
compute the first averaged function $f(\rho,s)=(f_{1}(\rho,s),f_{2}(\rho,s))$,
i.e.

(18)
$\begin{array}[]{rl}f_{1}(\rho,\,s)=&\dfrac{1}{2\pi}\displaystyle\int_{0}^{2\pi}F_{11}(\theta,\rho,s)d\theta\vspace{0.2cm}\\\
=&\dfrac{1}{16\pi
a^{2}(a^{2}-5)g}\Bigg{[}\bigg{[}4(a^{3}-3a)^{2}h+(21-57a^{2}+28a^{4}-4a^{6})\rho^{2}\bigg{]}\cos\frac{\sqrt{2}gs}{a}\vspace{0.2cm}\\\
&+\left(a^{2}-5\right)\rho^{2}\,\cos\frac{3\,\sqrt{2}\,g\,s}{a}-\
\left(a^{2}-5\right)\rho^{2}\cos\frac{3\,\sqrt{2}\,g(s+2\pi)}{a}\vspace{0.2cm}\\\
&+\left(a^{2}-5\right)\big{(}\left(2a^{4}-8a^{2}+3\right)\rho^{2}-2a^{2}\left(a^{2}-3\right)h\big{)}\cos\frac{\sqrt{2}g(s+2\pi)}{a}\vspace{0.2cm}\\\
&-\left(a^{2}-3\right)\left(a^{2}+2g-1\right)\left(a^{2}\left(h-\rho^{2}\right)+\rho^{2}\right)\,\cos\frac{\sqrt{2}\,(g\,s+2\pi(g-2))}{a}\vspace{0.2cm}\\\
&-\big{(}\left(a^{2}-3\right)\left(a^{2}-2g-1\right)\left(a^{2}\left(h-\rho^{2}\right)+\rho^{2}\right)\big{)}\cos\frac{\sqrt{2}\,(g\,s+2\pi(g+2))}{a}\Bigg{]},\vspace{0.2cm}\end{array}$
$\begin{array}[]{rl}f_{2}(\rho,\,s)=&\dfrac{1}{2\pi}\displaystyle\int_{0}^{2\pi}F_{12}(\theta,\rho,s)d\theta\vspace{0.2cm}\\\
=&-\dfrac{1}{16\sqrt{2}\pi a(a+1)(a-1)(a^{2}-5)\rho}\vspace{0.2cm}\\\
&\Bigg{[}\bigg{(}4(a^{3}-3a)^{2}h+3(21-57a^{2}+28a^{4}-4a^{6})\rho^{2}\bigg{)}\sin\frac{\sqrt{2}gs}{a}\vspace{0.2cm}\\\
&+\left(a^{2}-5\right)\rho^{2}\,\sin\frac{3\,\sqrt{2}\,g\,s}{a}-\left(a^{2}-5\right)\rho^{2}\sin\frac{3\,\sqrt{2}\,g(s+2\pi)}{a}\vspace{0.2cm}\\\
&+\
\left(a^{2}-5\right)\big{(}3\left(2a^{4}-8a^{2}+3\right)\rho^{2}-2a^{2}\left(a^{2}-3\right)h\big{)}\sin\frac{\sqrt{2}g(s+2\pi)}{a}\vspace{0.2cm}\\\
&-\
\left(a^{2}-3\right)\left(a^{2}+2g-1\right)\left(a^{2}\left(h-3\rho^{2}\right)+3\rho^{2}\right)\,\sin\frac{\sqrt{2}\,(g\,s+2\pi(g-2))}{a}\vspace{0.2cm}\\\
&-\
\big{(}(a^{2}-3)(a^{2}-2g-1)(a^{2}(h-3\rho^{2})+3\rho^{2})\big{)}\sin\frac{\sqrt{2}(g\,s+2\pi(g+2))}{a}\Bigg{]}.\end{array}$

We are interested in the zeros
$(\tilde{\rho},\tilde{s})=(\tilde{\rho}(a,\,h),\tilde{s}(a,\,h))$ of the
function $f(\rho,\,s)$. From the averaging theory described in section 3 such
solutions must satisfy the requirements:

(19) $\tilde{\rho}>0,\quad
0\leq\tilde{s}<2\pi,\quad\tilde{r}=\sqrt{a^{2}(h-\tilde{\rho}^{2})+\tilde{\rho}^{2}}>0,\quad\dfrac{\partial(f_{1},\,f_{2})}{\partial(\rho,\,s)}|_{\rho=\tilde{\rho},s=\tilde{s}}\neq
0.$

Since the variable $\rho^{2}$ appears in (18) linearly, we solve
$f_{1}(\rho,s)=0$ for $\rho^{2}$ and then substitute its expression into the
equation $f_{2}(\rho,s)=0$. So we arrive to a quadratic equation in the
variable $\xi^{2}$ being $\xi=\cos\left(\sqrt{2}\,g\,s/a\right)$. Thus we have
the equation

(20) $h\,(a^{2}-3)\,\big{[}\,4\,P_{1}^{2}\,\xi^{2}(\xi^{2}-1)\ +\
{(\,P_{2}\,+\,P_{3}\,\xi^{2}\,)}^{2}\,\big{]}\ =\ 0,$

where the constants $P_{1},P_{2},P_{3}$ depend on $a$ but not on the parameter
$h$. These constants are given in the Appendix.

From (20) we must compute the solutions $s=\tilde{s}$. If the solutions $\xi$
of the biquadratic equation (20) are complex or real outside the interval
$[-1,1]$, then equation (20) has no solutions in the variable $s$.

Assume that equation (20) has some solutions $\tilde{s_{i}}$ in the variable
$s$. Then for each one of these solutions substituting it into equation (18)
we obtain $\tilde{\rho}_{i}$. Substituting $\tilde{\rho}_{i}$ in equation (19)
we obtain $\tilde{r}_{i}$. So, from the averaging theory of section 3, we have
the initial conditions for a periodic orbit of the differential system (17) if
the determinant of the matrix

$\left.\frac{\partial(f_{1}(\rho,s),f_{2}(\rho,s))}{\partial(\rho,s)}\right|_{\rho=\tilde{\rho}_{i},s=\tilde{s}_{i}}\neq
0.$

Going back from polar coordinates (15), the scale factor $\varepsilon$ (13),
the change of coordinates (10) and the translation transformations (8), we
obtain for $\varepsilon>0$ sufficiently small the initial conditions
$(\,x(0,\varepsilon),y(0,\varepsilon),p_{x}(0,\varepsilon),p_{y}(0,\varepsilon))$
for a periodic orbit of the Hamiltonian system (3) in an energy level of
$H=\tilde{h}>0$ sufficiently small because
$\tilde{h}=h\varepsilon^{2}+O(\varepsilon^{3})$. More precisely, the initial
conditions of such periodic orbit are

(21)
$\left(\varepsilon\tilde{r}_{i},\varepsilon\tilde{\rho}_{i}\cos\frac{\sqrt{2}g}{a}\tilde{s}_{i}+\sqrt{a^{2}-1},0,\varepsilon\dfrac{\sqrt{2}g}{a}\tilde{\rho}_{i}\sin\frac{\sqrt{2}g}{a}\tilde{s}_{i}\right),$

with an error of $O(\varepsilon^{2})$. Theorem 1 is proved.

##  5\. Numerical examples

In this section we present some examples of periodic orbits with initial
conditions calculated by the averaging theory discussed in the previous
section. The equilibrium point from which they bifurcate is ${\bf x}_{0}$, and
their frequencies in the $x$ and $y$ directions are $\omega_{x}=\sqrt{2}/a$
and $\omega_{y}=\sqrt{2}g/a$, respectively. The frequency ratio of the
periodic motion in the $(x,p_{x})$ and $(y,p_{y})$ planes,
$\omega_{x}/\omega_{y}=1/g=1/\sqrt{a^{2}-1}$, is determined by the parameter
$a$. Although we have found periodic solutions on both planes for $h>0$
sufficiently small, the motion in the phase space can be either periodic or
quasi-periodic, depending on whether the frequency ratio
$\omega_{x}/\omega_{y}=1/\sqrt{a^{2}-1}\ $ is a rational or an irrational
number. If the commensurability condition $\omega_{x}/\omega_{y}=l/m\
\in\mathbb{Q}$ holds, with $l$, $m\,\in\mathbb{Z}$ relative primes, the orbit
will be periodic for $h>0$ sufficiently small, with $l$ oscillations in the
$x$ direction and $m$ oscillations in the $y$ direction. In order to guarantee
the periodicity of the solutions constructed from the averaging method, $a$
must be selected such that

(22) $a\ =\ \sqrt{\left(\frac{m}{l}\right)^{2}\,+\,1}\,,\qquad
l\,,m\in\mathbb{Z}$

###  5.1. Case $a=\sqrt{29}/2$

For any value of $h>0$ sufficiently small, we obtain one zero of the function
(18) such that
$\dfrac{\partial(f_{1},\,f_{2})}{\partial(\rho,\,s)}|_{\rho=\tilde{\rho},s=\tilde{s}}\,\neq\,0$,
i.e.

$(\tilde{\rho}_{1},\tilde{s}_{1})=(0.6270241608257199..,1.7084698955985402..).$

(a) $x$ vs $y$ (b) $x$ vs $y$

Figure 7. Case $a=\frac{\sqrt{29}}{2}$. (a) Periodic orbit and its initial
condition (23) marked by a black dot, obtained by averaging theory. (b) The
four periodic orbits bifurcating from the equilibrium point
$(0,\sqrt{a^{2}-1},0,0)$ and their initial conditions marked by black dots.

By taking $\varepsilon=10^{-2}$ the above values provide the following initial
conditions (21):

(23)
$\begin{split}&(x(0,\varepsilon),y(0,\varepsilon),p_{x}(0,\varepsilon),p_{y}(0,\varepsilon))=\\\
&\quad(0.02189236027905628...,2.496093823665088...,0,-0.00644040535596368...),\end{split}$

which correspond to the periodic orbit displayed in Fig. 7. In this case
$\omega_{x}/\omega_{y}=1/\sqrt{a^{2}-1}=2/5$, and the periodic orbit has 2
oscillations in the $x$ direction by 5 oscillations in the $y$ direction. The
three periodic orbits related to (23) by reflection near the equilibrium point
have the initial conditions

$\begin{split}&(-0.021892360279056278...,2.496093823665088...,0,-0.00644040535596368...)\,,\\\
&(0.021892360279056278...,2.503906176334912...,0,0.00644040535596368...)\,,\\\
&(-0.021892360279056278...,2.503906176334912...,0,0.00644040535596368...)\,,\end{split}$

###  5.2. Case $a=\sqrt{5}$

This case was excluded from the proof of Theorem 1, and we address this
special value here. The averaged function (18) reduces to

(24)
$\begin{array}[]{rl}f_{1}(\rho,\,s)=&\dfrac{1}{2\pi}\displaystyle\int_{0}^{2\pi}F_{11}(\theta,\rho,s)d\theta\vspace{0.2cm}\\\
=&\frac{1}{800\pi}\Bigg{[}5\left(25h-17\rho^{2}\right)\cos\left(2\sqrt{\frac{2}{5}}s\right)\vspace{0.2cm}\\\
&+5\left(4\rho^{2}-5h\right)\cos\left(2\sqrt{\frac{2}{5}}(s+4\pi)\right)+40\pi\sqrt{10}h\sin\left(2\sqrt{\frac{2}{5}}s\right)\vspace{0.2cm}\\\
&-100h\cos\left(2\sqrt{\frac{2}{5}}(s+2\pi)\right)-32\sqrt{10}\pi\rho^{2}\sin\left(2\sqrt{\frac{2}{5}}s\right)\vspace{0.2cm}\\\
&+5\rho^{2}\cos\left(6\sqrt{\frac{2}{5}}s\right)+65\rho^{2}\cos\left(2\sqrt{\frac{2}{5}}(s+2\pi)\right)\vspace{0.2cm}\\\
&-5\rho^{2}\cos\left(6\sqrt{\frac{2}{5}}(s+2\pi)\right)\,\Bigg{]}\vspace{0.2cm}\end{array}$
$\begin{array}[]{rl}f_{2}(\rho,\,s)=&\dfrac{1}{2\pi}\displaystyle\int_{0}^{2\pi}F_{12}(\theta,\rho,s)d\theta\vspace{0.2cm}\\\
=&\dfrac{1}{640\,\pi\,\rho}\Bigg{[}16\pi\left(5h-12\rho^{2}\right)\cos\left(2\sqrt{\frac{2}{5}}s\right)\vspace{0.2cm}\\\
&+\sqrt{10}\,\Bigg{(}\left(51\rho^{2}-25h\right)\sin\left(2\sqrt{\frac{2}{5}}s\right)+\left(5h-12\rho^{2}\right)\sin\left(2\sqrt{\frac{2}{5}}(s+4\pi)\right)\vspace{0.2cm}\\\
&+2\sin\left(2\sqrt{\frac{2}{5}}(s+2\pi)\right)\left(10h-19\rho^{2}+\rho^{2}\cos\left(4\sqrt{\frac{2}{5}}(s+2\pi)\right)\right)\vspace{0.2cm}\\\
&-\
\rho^{2}\,\sin\left(6\sqrt{\frac{2}{5}}s\right)\Bigg{)}\Bigg{]},\end{array}$

Eliminating the variable $\rho^{2}$ in the system of equations, $f_{1}=0$ and
$f_{2}=0$, we arrive to the equation

$\begin{array}[]{l}400\sin\left(4\sqrt{\frac{2}{5}}s\right)\ +\
24\left(5\sin\left(4\sqrt{\frac{2}{5}}\pi\right)-4\sqrt{10}\pi\right)\sin^{2}\left(2\sqrt{\frac{2}{5}}\pi\right)\vspace{0.2cm}\\\
+8\pi\Bigg{[}\sqrt{10}\left(29\cos\left(4\sqrt{\frac{2}{5}}(s+\pi)\right)-38\cos\left(4\sqrt{\frac{2}{5}}s\right)+8\cos\left(4\sqrt{\frac{2}{5}}(s+2\pi)\right)\right.\vspace{0.2cm}\\\
\left.+\cos\left(4\sqrt{\frac{2}{5}}(s+3\pi)\right)\right)-64\pi\sin\left(4\sqrt{\frac{2}{5}}s\right)\Bigg{]}+5\Bigg{[}\sin\left(4\sqrt{\frac{2}{5}}(s-2\pi)\right)\vspace{0.2cm}\\\
+4\sin\left(4\sqrt{\frac{2}{5}}(s-\pi)\right)-134\sin\left(4\sqrt{\frac{2}{5}}(s+\pi)\right)+11\,\sin\left(4\sqrt{\frac{2}{5}}(s+2\pi)\right)\vspace{0.2cm}\\\
+34\sin\left(4\sqrt{\frac{2}{5}}(s+3\pi)\right)+4\sin\left(4\sqrt{\frac{2}{5}}(s+4\pi)\right)\Bigg{]}=0.\end{array}$

(a) $x$ vs $p_{x}$ (b) $p_{x}$ vs $p_{y}$ (c) $x$ vs $y$

Figure 8. Case $a=\sqrt{5}$. (a) and (b) Periodic orbit and its initial
conditions (25) obtained using the averaging theory on the $(x,p_{x})$ and
$(p_{x},p_{y})$ planes, marked by black dots. (c) The two periodic orbits that
bifurcate from the equilibrium point $(0,\sqrt{a^{2}-1},0,0)$. The initial
conditions are marked by black dots

For $a=\sqrt{5}$ and for any value of $h>0$ we only have the zero

$(\tilde{\rho},\tilde{s})=(0.66138231965282354...,2.21971887978497651...),$

of the averaged function (24). By taking $\varepsilon=10^{-2}$ this provides
the following initial conditions (21)

(25)
$\begin{split}&(x(0,\varepsilon),y(0,\varepsilon),p_{x}(0,\varepsilon),p_{y}(0,\varepsilon))=\\\
&\quad(0.01802857096112335...,1.9937513313786317...,0,-0.002741327484343618...),\end{split}$

for the periodic orbit displayed in Fig. 8. In this case
$\omega_{x}/\omega_{y}=1/\sqrt{a^{2}-1}=1/2$, and the periodic orbit exhibits
1 oscillation in the $x$ direction for every 2 oscillations in the $y$
direction. In this special case, only two periodic orbits bifurcate from the
equilibrium point $(0,\sqrt{a^{2}-1},0,0)$, with the second orbit given by

(26)
$\begin{split}&(x(0,\varepsilon),y(0,\varepsilon),p_{x}(0,\varepsilon),p_{y}(0,\varepsilon))=\\\
&\quad(0.01802857096112335...,2.0062486686213683...,0,0.002741327484343618...),\end{split}$

This is due to the fact that the periodic orbit obtained from averaging (25)
is invariant under the symmetry $\mathcal{S}_{1}$. Of course, with the
symmetry $\mathcal{S}_{2}$ we obtain two additional periodic orbits from
averaging theory, bringing the total to four.

###  5.3. Case $a=5/4$

In this case there exist two zeros of the averaged function (24) such that
$\dfrac{\partial(f_{1},\,f_{2})}{\partial(\rho,\,s)}|_{\rho=\tilde{\rho},x=\tilde{s}}\,\neq\,0$.
They are

$\begin{array}[]{l}(\tilde{\rho}_{1},\tilde{s}_{1})=(0.78394053965364..,2.85274914632239..),\\\
(\tilde{\rho}_{2},\tilde{s}_{2})=(1.13713501897674..,2.22877470926716..),\end{array}$

which provide the initial conditions

$\begin{array}[]{l}(x_{1}(0,\varepsilon),y_{1}(0,\varepsilon),p_{x_{1}}(0,\varepsilon),p_{y_{1}}(0,\varepsilon))=\\\
(0.011030904051965..,0.744111227990272..,0,-0.004390970468570..),\\\
(x_{2}(0,\varepsilon),y_{2}(0,\varepsilon),p_{x_{2}}(0,\varepsilon),p_{y_{2}}(0,\varepsilon))=\\\
(0.009138625285550..,0.7464188333578204..,0,-0.009157928392703...),\end{array}$

for two periodic orbits, respectively. These periodic orbits have 4
oscillations in the $x$ direction and 3 in the $y$. They are displayed in Fig.
9.

(a) $x$ vs $y$ (b) $x$ vs $y$

Figure 9. Case $a=\frac{5}{4}=1.25$, two periodic orbits and their associated
initial conditions (5.3) (marked by black dots), obtained using the averaging
theory. The frequency ratio is $\omega_{1}/\omega_{2}=4/3$.

As we have mentioned, these orbits provide symmetrical periodic orbits by
applying $\mathcal{S}_{1}$ to the previous initial conditions.

Of course, all the periodic orbits obtained in this section have Jacobian
determinant (5) different from 0.

##  6\. On the non-integrability

We consider the autonomous differential system

(27) $\dot{x}=f(x),$

where $f\colon U\to\mathbb{R}^{n}$ is $C^{2}$, $U$ is an open subset of
$\mathbb{R}^{n}$ and the dot denotes the derivative with respect to the time
$t$. Let $x(t,x_{0})$ be a periodic solution of the differential system (27)
of period $T$ such that $x(0,x_{0})=x_{0}$.

The variational equation associated to the $T$-periodic solution $x(t,x_{0})$
is

(28) $\dot{M}=\bigg{(}\frac{\partial f(x)}{\partial
x}\Big{|}_{x=x(t,x_{0})}\bigg{)}M,$

where $M$ is an $n\times n$ matrix. Of course $\partial f(x)/\partial x$
denotes the Jacobian matrix of $f$ with respect to $x$.

The _monodromy matrix_ associated to the $T$-periodic solution $x(t,x_{0})$ is
the solution $M(T,x_{0})$ of (28) satisfying that $M(0,x_{0})$ is the identity
matrix. The eigenvalues of the monodromy matrix associated to the periodic
solution $x(t,x_{0})$ are called the _multipliers_ of the periodic orbit.

We recall an important theorem due to Poincaré [23], (see also [22] p. 36 and
[24]), on the integrability of a Hamiltonian system with two degrees of
freedom.

Poincaré Theorem. If a Hamiltonian system with two degrees of freedom and
Hamiltonian $H$ is Liouville–Arnold integrable, and $C$ is a second first
integral such that the differentials of $H$ and $C$ are linearly independent
at each point of a periodic orbit of the system, then all the multipliers of
this periodic orbit are equal to $1$.

###### Proof of Theorem 3.

Under the hypotheses of Theorem 1 and Corollary 2, assume that the determinant
of the fundamental matrix associated to some of the periodic orbits of
Corollary 2 is different from $1$, then the Hamiltonian system is not
Liouville-Arnold integrable because some multiplier is distinct from $1$,

If all the determinants of the fundamental matrices associated to the periodic
orbits of Corollary 2 are $1$, the Hamiltonian system can be Liouville-Arnold
integrable. ∎

##  7\. Conclusions

We have used the averaging theory for studying the periodic orbits of the
Hamiltonian system modeling the two-center problem with harmonic-like
interactions in some of their fixed Hamiltonian levels, see Theorem 1 and
Corollary 2. This tool can be applied to Hamiltonian systems with an arbitrary
degrees of freedom.

Using a result due to Poincaré we have analyzed the non–integrability in the
sense of Liouville–Arnol’d of the mentioned Hamiltonian systems, see Theorem
3. Again this tool can be applied to Hamiltonian systems with an arbitrary
number of degrees of freedom.

We must remark that these two tools are possible to apply if we have analytic
information on the periodic orbits of the Hamiltonian system. Here this is the
case thanks to the averaging theory for computing periodic orbits.

### Credit authorship contribution statement

All the authors have contributed equally to this paper.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

### Data availability

No data was used for the research described in the article.

##  8\. Appendix

Here, we present the constants $P_{1},P_{2}$ and $P_{3}$ appearing in (20):

$P_{1}\ =\
a^{8}\left(8c_{g-2}+16c_{g}-6c_{2g}+8c_{g+2}-c_{2g-4}-4\left(c_{2g-2}+c_{2g+2}+4\right)-c_{2g+4}\right)+2a^{6}\left(8(g-4)c_{g-2}-96c_{g}+38c_{2g}-8(g+4)c_{g+2}-2gc_{2g-4}+c_{2g-4}-4gc_{2g-2}+20c_{2g-2}+4gc_{2g+2}+20c_{2g+2}+(2g+1)c_{2g+4}+80\right)-4a^{4}\left((28g-43)c_{g-2}-195c_{g}+82c_{2g}+c_{3g}-(28g+43)c_{g+2}-5gc_{2g-4}-2c_{2g-4}-18gc_{2g-2}+31c_{2g-2}+18gc_{2g+2}+31c_{2g+2}+(5g-2)c_{2g+4}+140\right)+2a^{2}\left(4(29g-21)c_{g-2}-584c_{g}+258c_{2g}+16c_{3g}-4(29g+21)c_{g+2}-14gc_{2g-4}-9c_{2g-4}-90gc_{2g-2}+64c_{2g-2}+90gc_{2g+2}+64c_{2g+2}+(14g-9)c_{2g+4}+368\right)-104gc_{g-2}+52c_{g-2}+340c_{g}-130c_{2g}-60c_{3g}+104gc_{g+2}+52c_{g+2}+12gc_{2g-4}+9c_{2g-4}+100gc_{2g-2}-40c_{2g-2}-100gc_{2g+2}-40c_{2g+2}-12\,g\,c_{2g+4}\
+\ 9\,c_{2g+4}\ -\ 192\ .$

$P_{2}\ =\
-16a^{8}b_{g}+6a^{8}b_{2g}-8a^{8}b_{g+2}+a^{8}b_{2g-4}+4a^{8}b_{2g-2}+4a^{8}b_{2g+2}+192a^{6}b_{g}-76a^{6}b_{2g}+16a^{6}gb_{g+2}+64a^{6}b_{g+2}+4a^{6}gb_{2g-4}-2a^{6}b_{2g-4}+8a^{6}gb_{2g-2}-40a^{6}b_{2g-2}-8a^{6}gb_{2g+2}-40a^{6}b_{2g+2}-796a^{4}b_{g}+328a^{4}b_{2g}+4a^{4}b_{3g}-112a^{4}gb_{g+2}-168a^{4}b_{g+2}-20a^{4}gb_{2g-4}-8a^{4}b_{2g-4}-72a^{4}gb_{2g-2}+124a^{4}b_{2g-2}+72a^{4}gb_{2g+2}+124a^{4}b_{2g+2}+1280a^{2}b_{g}-516a^{2}b_{2g}-32a^{2}b_{3g}+224a^{2}gb_{g+2}+144a^{2}b_{g+2}+28a^{2}gb_{2g-4}+18a^{2}b_{2g-4}+180a^{2}gb_{2g-2}-128a^{2}b_{2g-2}-180a^{2}gb_{2g+2}-128a^{2}b_{2g+2}+24\left(a^{2}-5\right)b_{2}g+\left(a^{2}-3\right)\left(a^{2}-1\right)^{2}\left(a^{2}-4g+3\right)b_{2g+4}-8\left(a^{6}-7a^{4}+14a^{2}-4\right)\left(a^{2}+2g-1\right)b_{g-2}-500b_{g}+130b_{2g}+60b_{3g}-64gb_{g+2}-32b_{g+2}-12gb_{2g-4}-9b_{2g-4}-100gb_{2g-2}+40b_{2g-2}+\
100\,g\,b_{2g+2}\ +\ 40\,b_{2g+2}\ .$

$P_{3}\ =\
-6a^{8}b_{2g}+8a^{8}b_{g+2}-a^{8}b_{2g-4}-4a^{8}b_{2g-2}-4a^{8}b_{2g+2}+76a^{6}b_{2g}-16a^{6}gb_{g+2}-64a^{6}b_{g+2}-4a^{6}gb_{2g-4}+2a^{6}b_{2g-4}-8a^{6}gb_{2g-2}+40a^{6}b_{2g-2}+8a^{6}gb_{2g+2}+40a^{6}b_{2g+2}-328a^{4}b_{2g}-4a^{4}b_{3g}+112a^{4}gb_{g+2}+174a^{4}b_{g+2}+20a^{4}gb_{2g-4}+8a^{4}b_{2g-4}+72a^{4}gb_{2g-2}-124a^{4}b_{2g-2}-72a^{4}gb_{2g+2}-124a^{4}b_{2g+2}+516a^{2}b_{2g}+32a^{2}b_{3g}-236a^{2}gb_{g+2}-180a^{2}b_{g+2}-28a^{2}gb_{2g-4}-18a^{2}b_{2g-4}-180a^{2}gb_{2g-2}+128a^{2}b_{2g-2}+180a^{2}gb_{2g+2}+128a^{2}b_{2g+2}-\left(a^{2}-3\right)\left(a^{2}-1\right)^{2}\left(a^{2}-4g+3\right)b_{2g+4}+8\left(a^{2}-5\right)\left(2a^{2}\left(a^{4}-7a^{2}+14\right)-11\right)b_{g}+2\left(4a^{6}-28a^{4}+59a^{2}-31\right)\left(a^{2}+2g-1\right)b_{g-2}-130b_{2g}-60b_{3g}+124gb_{g+2}+62b_{g+2}+12gb_{2g-4}+9b_{2g-4}+100gb_{2g-2}-40b_{2g-2}-100gb_{2g+2}-40b_{2g+2}\
,$

where

$c_{x}\ \equiv\ \cos\left(\dfrac{2\,\sqrt{2}\,x\,\pi}{a}\right)\qquad;\qquad
b_{x}\ \equiv\ \sin\left(\dfrac{2\,\sqrt{2}\,x\,\pi}{a}\right).$

## Acknowledgments

A.M. Escobar Ruiz would like to thank the support from Consejo Nacional de
Humanidades, Ciencias y Tecnologías (CONAHCyT) of Mexico under Grant
CF-2023-I-1496 and from UAM research grant 2024-CPIR-0.

The third author is partially supported by the Agencia Estatal de
Investigación of Spain grant PID2022-136613NB-100, AGAUR (Generalitat de
Catalunya) grant 2021SGR00113, and by the Reial Acadèmia de Ciències i Arts de
Barcelona.

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∎

11institutetext: Kartick Sutradhar 22institutetext: Indian Institute of
Technology (ISM) Dhanbad
Mobile: +91-7602621359
22email<EMAIL_ADDRESS>33institutetext: Hari Om 44institutetext:
Indian Institute of Technology (ISM) Dhanbad
44email<EMAIL_ADDRESS>
# An Efficient Simulation of Quantum Secret Sharing

Kartick Sutradhar  Hari Om

(Received: date / Accepted: date)

###### Abstract

In quantum cryptography, quantum secret sharing $(QSS)$ is a fundamental
primitive. $QSS$ can be used to create complex and secure multiparty quantum
protocols. Existing $QSS$ protocols are either at the $(n,n)$ threshold $2$
level or at the $(t,n)$ threshold $d$ level with a trusted player, where $n$
denotes the number of players and $t$ denotes the threshold number of players.
Here, we propose a secure $d$-level $QSS$ protocol for sharing a secret with
efficient simulation. This protocol is more secure, flexible, and practical as
compared to the existing $QSS$ protocols: $(n,n)$ threshold $2$-level and
$(t,n)$ threshold $d$-level with a trusted player. Further, it does not
disclose any information about the secret to players. Its security analysis
shows that the intercept-resend, intercept, entangle-measure, forgery,
collision and collusion attacks are not possible in this protocol.

###### Keywords:

Secure Computation Quantum Cryptography Information Security Quantum Secret
Sharing

##  1 Introduction

A dealer shares a secret with $n$ players in secret sharing $(SS)$, and when
the secret needs to be reconstructed, the threshold number of players can do
so collaboratively. The quantum secret sharing hillery1999quantum ;
bao2009threshold ; yang2013secret ; Gang4 ; lu2018verifiable ; lau2013quantum
; Hao ; mashhadi2016fairly ; dehkordi2019proactive ; mashhadi2017provably ;
mashhadi2016share ; mashhadi2016analysis ; mashhadi2020toward ;
mashhadi2020csa ; mashhadi2017new ; karimifard2016semiquantum ;
charoghchi2021three ; mashhadi2020improvement ; shi2010quantum ;
run2010efficient ; shi2011multi ; gyongyosi2019quantum ;
sutradhar2020efficient  is a fundamental primitive protocol for sharing a
secret in quantum cryptography, which may be considered as an extension of
secret sharing. The $QSS$ protocol can be used to create complex multiparty
quantum computing protocols that are secure. In the $(n,n)$ threshold $QSS$, a
dealer shares a secret with $n$ players by dividing it into $n$ bits, known as
shares, which are distributed among $n$ players, each of whom has only his
share. The secret can be reconstructed by the $n$ players working together.
Similarly, in the $(t,n)$ threshold $QSS$, a dealer shares a secret with $n$
players by dividing it into $n$ bits and distributing them to the $n$ players.
The $t$ players will work together to solve the mystery. Because it protects
the quantum threshold and secure quantum multiparty computation, the $QSS$ is
commonly used in quantum threshold cryptography and secure quantum multiparty
computation.
Here, we propose a secure $d$-level $QSS$ protocol for sharing a secret, where
$t$ players can reconstruct the secret without a trusted player. In our
protocol, each player knows only his share, even the reconstructor knows only
his share. In this protocol, we use some basic operations i.e., protocol-I of
Shi et al. shi2016secure , $CNOT$ gate nielsen2002quantum , secure
communication Gang1 ; Gang3 ; shi2017quantum ; shi2016efficient ;
sun2020toward ; shi2018efficient ; peng2018novel ; zhang2018economic ;
luo2018novel ; xu2017nearest ; sutradhar2020hybrid ; sutradhar2020generalized
; sutradhar2021efficient , entangle state Gang2 ; shi2016comment ;
shi2016quantum ; dan2016efficient ; shi2016data ; shi2015quantum ;
shi2015comments ; shi2013multi ; shi2012multiparty ; shi2012novel ;
shi2011efficient ; run2011novel ; shi2011multi ; shi2011asymmetric , Quantum
Fourier Transform $(QFT)$ Nielsen2002  and Inverse Quantum Fourier Transform
$(QFT^{-1})$ Nielsen2002 , to transform the particles. We use a quantum
approach in classical secret sharing to combine the benefits of both classical
and quantum secret sharing, preventing attacks such as Intercept-Resend (IR),
Intercept, Entangle-Measure (EM), Forgery, Collision, and Collusion.

##  2 Related Work

There are numerous $QSS$ protocols for secret sharing in quantum cryptography
Mashhadi2019 ; mashhadi2012novel ; hillery1999quantum ; bao2009threshold ;
mashhadi2012analysis ; yang2013secret ; Gang4 ; lu2018verifiable ;
lau2013quantum ; Hao ; dehkordi2008new ; dehkordi2008efficient ;
mashhadi2015two ; dehkordi2008verifiable ; mashhadi2017secure ;
mashhadi2015computationally ; mashhadi2013novel . In 1999, Hillery et al.
discussed the first $QSS$ protocol hillery1999quantum  based on the
Greenberger-Home-Zeilinger $(GHZ)$ state. In 2009, Li et al. introduced a
$QSS$ protocol bao2009threshold  of secure direct communication. This protocol
is $(t,n)$ threshold scheme but $2$ level. In 2013, Yang et al. introduced a
$QSS$ protocol yang2013secret  based on the $QFT$. This protocol is $d$-level
$(t,n)$ threshold scheme but it is not secure because each player broadcasts
the results of the measurement at the last step. Because the measurement
results contain information about the secret, if an attacker intercepts the
measurement results, he may expose the secret or execute an intercept-resend
attack. In 2015, Qin et al. discussed a $QSS$ protocol qin2015t  based on the
phase shift operation, which is $2$-level $(t,n)$ threshold scheme. The
protocols bao2009threshold  and qin2015t  are not secure because the unitary
operation transforms the private information of player $P_{e-1}$ and then the
transformed information is transmitted to player $P_{e}$. So, the players
$P_{e-1}$ and $P_{e+1}$ collaboratively can retrieve the private information
of player $P_{e}$. In 2017, Song et al. discussed a $(t,n)$ threshold
$d$-level $QSS$ protocol song2017t  based on some basic operations, i.e.,
$d$-level $CNOT$ gate, $QFT$, generalized Pauli operator, and $QFT^{-1}$. In
that protocol, Alice (dealer) selects $Bob_{1}$ as a trusted reconstructor
from the set of participants $\mathbb{B}=\\{Bob_{1},Bob_{2},\dots,Bob_{n}\\}$
and then selects a hash function $SHA1$ eastlake2001us  to compute the hash
value of the secret (which is to be shared) and sends this hash value to the
trusted reconstructor $Bob_{1}$. Here, $Bob_{1}$ can perform collision attack
to reveal the secret. So, the security of this protocol is dependent on the
trusted reconstructor $Bob_{1}$. The main problem of Song et al.’s protocol is
that the reconstructor $Bob_{1}$ cannot recover the original secret because
$QFT^{-1}$ cannot be summed up over all the states CommentKao2018 . In other
words, the reconstructor $Bob_{1}$ needs the secret information of other
players to reconstruct the original secret. In 2018, Qin et al.
Qin2018Multidimensional  discussed a $QSS$ protocol which can efficiently
share a secret by using the $QFT$ and Pauli operator, but it is a $(n,n)$
threshold scheme.

In our protocol, any $t$ players can reconstruct the secret without a trusted
player and each player knows only his share, nothing else. Furthermore, the
reconstructor is unable to perform the collision attack because the secret’s
hash value is shared among the players.

##  3 Preliminaries

The $QFT$, $QFT-1$, Control-NOT $(CNOT)$ gate, and Shamir’s Secret Sharing,
which will be used in the proposed $QSS$ protocol, are all introduced here.

###  3.1 Quantum Fourier Transform

The $QFT$ Nielsen2002 , a unitary transform, is based on the quantum
phenomenon and expansion of the standard discrete Fourier transform. For
$s\in\\{0,1,\dots d-1\\}$, the $QFT$ of $d$-level quantum system is defined as
follows:

$QFT:\ket{s}\rightarrow\frac{1}{\sqrt{d}}\sum_{q=0}^{d-1}e^{2\pi
i\frac{s}{d}q}\ket{q}.$ (1)

The $QFT^{-1}$ is defined by

$QFT^{-1}:\ket{q}\rightarrow\frac{1}{\sqrt{d}}\sum_{s=0}^{d-1}e^{-2\pi
i\frac{q}{d}s}\ket{s}.$ (2)

Further,

$\sum_{q=0}^{d-1}e^{2\pi i\frac{s}{d}q}=\begin{cases}0\leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ s\neq 0\leavevmode\nobreak\
mod\leavevmode\nobreak\ d\\\ d\leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ s=0\leavevmode\nobreak\ mod\leavevmode\nobreak\
d\end{cases}$ (3)

So,

$\begin{split}QFT^{-1}\Bigg{(}\frac{1}{\sqrt{d}}\sum_{q=0}^{d-1}e^{2\pi
i\frac{s}{d}q}\ket{q}\Bigg{)}&=\frac{1}{\sqrt{d}}\sum_{q=0}^{d-1}e^{2\pi
i\frac{s}{d}q}QFT^{-1}\ket{q}\\\
&=\frac{1}{d}\sum_{q=0}^{d-1}\ket{s}+\frac{1}{d}\sum_{k=0\wedge k\neq
s}^{d-1}0.\ket{k}=\ket{s}\end{split}$ (4)

That is,

$QFT^{-1}(QFT\ket{s})=\ket{s}.$ (5)

###  3.2 Control-NOT $(CNOT)$ gate

The $CNOT$ gate nielsen2002quantum  is a two-qubit gate, one is control qubit
and other is target qubit. If the control bit of $CNOT$ gate is set to
$\ket{0}$, then the $NOT$ gate would not be applied to the target bit. If the
control bit of the $CNOT$ gate is set to $\ket{1}$, then the $NOT$ gate would
be applied to the target bit.

###  3.3 Shamir’s Secret Sharing

In the Shamir’s secret sharing shamir1979share , there are a dealer
$\mathbb{D}$ and $n$ players $\mathcal{P}=\\{P_{1},P_{2},\dots P_{n}\\}$. The
Shamir’s secret sharing consists of two phases:

####  3.3.1 Secret Sharing Phase

In this phase, the dealer selects a polynomial
$f(x)=S+a_{1}x+a_{2}x^{2}+\dots+a_{t-1}x^{t-1}$ of degree $(t-1)$, where $S$
is a secret and $a_{1},a_{2},\dots,a_{t-1}$ are coefficients of the polynomial
$f(x)$. The dealer computes $n$ shares and distributes them among $n$ players,
each player $P_{i}$ only knows $f(x_{i})$, where $i=1,2,\dots,n$.

####  3.3.2 Secret Reconstruction Phase

Using $t$ shares of the secret and the Lagrange interpolation formula, $t$
players will jointly reconstruct the secret in this phase.

$f(x)=\sum_{r=1}^{t}f(x_{r})\prod_{1\leq j\leq t,j\neq
r}\frac{x-x_{j}}{x_{r}-x_{j}}$ (6)

To calculate the polynomial at $x=0$, Eq.(6) can be simplified as

$\begin{split}f(0)&=\sum_{r=1}^{t}f(x_{r})\prod_{1\leq j\leq t,j\neq
r}\frac{x_{j}}{x_{j}-x_{r}}\end{split}$ (7)

##  4 Proposed Method

We present a $d$-level $QSS$ protocol for sharing a secret that allows $t$
players to reconstruct the secret without the help of a trusted player. In
comparison to the existing $QSS$ protocols, such as the $(n,n)$ threshold
$2$-level and $(t,n)$ threshold $d$-level, which both require a trusted
player, this protocol is more secure, versatile, and practical.Furthermore, no
information about the secret is revealed to any of the players. There are two
stages to the $QSS$ protocol: secret sharing and secret reconstruction.

###  4.1 Secret Sharing Phase

In this phase, the dealer $\mathbb{D}$ shares the secret among players
$\mathcal{P}=\\{P_{1},P_{2},\dots P_{n}\\}$. Initially, the dealer
$\mathbb{D}$ selects a prime $d$ such that $2\leq d\leq 2n$ and sets a finite
field $Z_{d}$. Then, the dealer $\mathbb{D}$ selects a polynomial
$f(x)=S+a_{1}x+a_{2}x^{2}+\dots+a_{t-1}x^{t-1}$ of degree of $(t-1)$, where
$S$ is secret, $a_{1},a_{2},\dots,a_{t-1}$ are coefficients of polynomial
$f(x)\in Z_{d}$ and the symbol ${}^{\prime}+^{\prime}$ is defined as addition
modulo $d$. The dealer computes the classical shares $f(xi)$ and uses the BB84
protocol to encode these classical shares $f(xi)$ in a qubit string
bennett1984update . The qubit string of $f(x_{i})$ is distributed among $n$
players, player $P_{i}$ only knows the share $f(x_{i})$. In addition, the
dealer $\mathbb{D}$ selects the $SHA1$ hash function to compute the hash value
$\mathcal{H}(S)$ eastlake2001us  and shares it among $n$ players using the
polynomial $g(x)=\mathcal{H}(S)+b_{1}x+b_{2}x^{2}+\dots+b_{t-1}x^{t-1}$.
Player $P_{i}$ only knows the share $g(x_{i})$, where $i=1,2,\dots,n$.

###  4.2 Secret Reconstruction Phase

Suppose $\mathcal{Q}=\\{P_{1},P_{2}\dots P_{t}\\}$ is a qualified subset from
all the qualified subsets, where the number of players in each qualified
subset is $t$. The dealer $\mathbb{D}$ selects a player from the qualified
subset $\mathcal{Q}=\\{P_{1},P_{2}\dots P_{t}\\}$ as a reconstructor. Here,
the dealer $\mathbb{D}$ selects player $P_{1}$ from the qualified subset
$\mathcal{Q}=\\{P_{1},P_{2}\dots P_{t}\\}$ as a reconstructor. The
reconstructor $P_{1}$ only knows his share, nothing else. This reconstructor
$P_{1}$ reconstructs the secret and hash value. The process of reconstruction
is given as follows:
Step 1: Player $P_{r}$, $r=1,2,\dots,t$, calculates the shadow $(s_{r})$ of
the share as follows.

$s_{r}=f(x_{r})\prod_{1\leq j\leq t,j\neq r}\frac{x_{j}}{x_{j}-x_{r}}\mod d$
(8)

Step 2: Player $P_{1}$ (reconstructor) makes basis state $\ket{s_{1}}_{H}$,
where size of the basis state is $c$-qubit, $s_{1}$ is his private shadow of
the share and $c=\lceil\log_{2}^{d}\rceil$. Then, player $P_{1}$ applies $QFT$
on the state $\ket{s_{1}}_{H}$ and the resultant state $\ket{\varphi_{1}}$ is
calculated as follows:

$\begin{split}\ket{\varphi_{1}}&=(QFT\ket{s_{1}}_{H})\\\
&=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{2\pi
i\frac{s_{1}}{d}k}\ket{k}_{H}\end{split}$ (9)

Step 3: Player $P_{1}$ again makes ancillary state $\ket{0}_{T}$, where size
of the ancillary state is $c$-qubit and $c=\lceil\log_{2}^{d}\rceil$, and then
executes $CNOT^{\otimes c}$ operations on the combined state
$\ket{\varphi_{1}}\ket{0}_{T}$, where the first $c$-qubits is control qubit
and second $c$-qubits is target qubit. After performing $CNOT^{\otimes c}$
operations, the state $\ket{\varphi_{1}}$ evolves as an entangled state
$\ket{\varphi_{2}}$, where subscript $H$ or $T$ represents home state (non-
transmitted state) or transmitted state.

$\begin{split}\ket{\varphi_{2}}&=CNOT^{\otimes
c}\ket{\varphi_{1}}\ket{0}_{T}\\\ &=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{2\pi
i\frac{s_{1}}{d}k}\ket{k}_{H}\ket{k}_{T}\end{split}$ (10)

Step 4: Player $P_{1}$ communicates with player $P_{2}$ using the
authenticated quantum channel to send the ancillary state $\ket{k}_{T}$ (i.e.,
second $c$-qubits).

Step 5: Player $P_{2}$ applies an oracle operator $C_{k}$ on
$\ket{k}_{T}\ket{s_{2}}$, where $C_{k}$ is given by

$C_{k}:\ket{k}_{T}\ket{s_{2}}\rightarrow\ket{k}_{T}U^{k}\ket{s_{2}}$ (11)

with

$U\ket{s_{2}}=e^{2\pi i\frac{s_{2}}{d}}\ket{s_{2}}$ (12)

where, $\ket{s_{2}}$ is an eigenvector of $U$ with eigenvalue $e^{2\pi
i\frac{s_{2}}{d}}$. The combined quantum system of $P_{1}$ and $P_{2}$ is
shown as follows.

$\begin{split}\ket{\varphi_{3}}&=C_{k}\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{2\pi
i\frac{s_{1}}{d}k}\ket{k}_{H}\ket{k}_{T}\ket{s_{2}}\\\
&=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{2\pi
i\frac{s_{1}+s_{2}}{d}k}\ket{k}_{H}\ket{k}_{T}\ket{s_{2}}\end{split}$ (13)

Step 6: Player $P_{2}$ communicates with player $P_{3}$ through an
authenticated quantum channel to send the ancillary state $\ket{k}_{T}$ and
keeps $\ket{s_{2}}$ as secret. Player $P_{3}$ performs $t-1$ times similar
process as done by $P_{2}$. If $t$ players honestly perform the protocol, then
the combined quantum state is obtained as shown below.

$\ket{\varphi_{4}}=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{2\pi
i\big{(}\frac{\sum_{r=1}^{t}s_{r}}{d}\big{)}k}\ket{k}_{H}\ket{k}_{T}\ket{s_{2}}\dots\ket{s_{t}}.$
(14)

Step 7: The ancillary state $\ket{k}_{T}$ is sent by $P_{t}$ back to $P_{1}$
through an authenticated quantum channel. Player $P_{1}$ again performs
$CNOT^{\otimes c}$ operation on his $2c$ qubits, where the first $c$-qubits is
control qubit and second $c$-qubits is target qubit. The output state is shown
as below.

$\begin{split}\ket{\varphi_{5}}&=CNOT^{\otimes
c}\ket{\varphi_{4}}=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{2\pi
i\big{(}\frac{\sum_{r=1}^{t}s_{r}}{d}\big{)}k}\ket{k}_{H}\ket{0}_{T}\ket{s_{2}}\dots\ket{s_{t}}\end{split}$
(15)

Step 8: The second $c$-qubits (i.e., ancillary state $\ket{0}_{T}$) is
measured by player $P_{1}$ in computational basis. If the output of the
measurement is $\ket{0}$, then player $P_{1}$ continues the process;
otherwise, he believes that the protocol executes with at least one corrupted
player and ends the protocol.

Step 9: Player $P_{1}$ applies $QFT^{-1}$ on the first $c$-qubits and measures
the output to get the secret $f(0)^{\prime}=\sum_{r=1}^{t}\leavevmode\nobreak\
s_{r}\leavevmode\nobreak\ mod\leavevmode\nobreak\ d$.

Step 10: Finally, $t$ players perform all the above nine steps again to get
the hash value of the secret and player $P_{1}$ gets the hash value of the
secret $g(0)^{\prime}=\sum_{r=1}^{t}\leavevmode\nobreak\
h_{r}\leavevmode\nobreak\ mod\leavevmode\nobreak\ d$, where $h_{r}$ is the
shadow of hash value shares. Player $P_{1}$ uses the hash function $SHA1$ to
compute the hash value $\mathcal{H}(f(0)^{\prime})$ and compares it with the
hash value $g(0)^{\prime}$. If $(\mathcal{H}(f(0)^{\prime})=g(0)^{\prime})$,
then player $P_{1}$ realizes that all $t$ players have performed the
reconstruction phase honestly; otherwise, player $P_{1}$ believes that there
is at least one corrupted player.

##  5 Correctness Proof of $(t,n)$ threshold $d$-level $QSS$

Here, we prove the correctness of the proposed $(t,n)$ threshold $d$-level
$QSS$. We mainly focus on the correctness proof of Step $9$ of secret
reconstruction phase.

###### Lemma 1

If $QFT^{-1}$ (as given in Equation 2) is applied to the first $c$-qubits,
then the measurement of the output is secret $(f(0)^{\prime})$.

###### Proof

Applying $QFT^{-1}$ to the first $c$-qubits provides the process of secret
recovery as given below:
The original secret $f(0)^{\prime}$ can be calculated using the Lagrange
interpolation and Equation 7 as follows.

$\begin{split}f(0)^{\prime}&=f(x_{1})\prod_{1\leq j\leq t,j\neq
1}\frac{x_{j}}{x_{j}-x_{1}}+\dots+f(x_{t})\prod_{1\leq j\leq t,j\neq
t}\frac{x_{j}}{x_{j}-x_{t}}\mod d\\\ &=(s_{1}+\dots+s_{t})\mod d\\\
&=(\sum_{r=1}^{t}s_{r}\mod d)\end{split}$ (16)

Player $P_{1}$ applies $QFT^{-1}$ to the first $c$-qubits.

$\begin{split}QFT^{-1}\Bigg{(}\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{2\pi
i\big{(}\frac{\sum_{r=1}^{t}s_{r}}{d}\big{)}k}\ket{k}_{H}\Bigg{)}&=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{2\pi
i\big{(}\frac{\sum_{r=1}^{t}s_{r}}{d}\big{)}k}QFT^{-1}\ket{k}_{H}\\\
&=\Bigg{|}\sum_{r=1}^{t}s_{r}\leavevmode\nobreak\ mod\leavevmode\nobreak\
d\Bigg{>}_{H}+\frac{1}{d}\sum_{l=0}^{d-1}0.\ket{l}_{H}\\\
&=\Bigg{|}\sum_{r=1}^{t}s_{r}\leavevmode\nobreak\ mod\leavevmode\nobreak\
d\Bigg{>}_{H}=\ket{f(0)^{\prime}}_{H}\end{split}$ (17)

Therefore, if this protocol honestly is executed by $t$ players, the
reconstructor $P_{1}$ will get the original secret.

##  6 Simulation Results

In this protocol, the initiator $P_{1}$ applies $QFT$ on the $q$-qubit state
and executes $CNOT$ gate. Then, the ancillary qubit is sent to player $P_{2}$,
who applies the oracle operator on the ancillary qubit. Thereafter, player
$P_{2}$ sends the ancillary qubit to $P_{3}$ and player $P_{3}$ performs
similar process. This process is performed $(t-1)$ times. After that, the
ancillary qubit is sent back to player $P_{1}$ by $P_{t}$. The player $P_{1}$
performs $CNOT$ and $QFT^{-1}$ to get the multiplication. In the secure
multiparty quantum multiplication, the Hadamard gate is taken to be the $QFT$.
After this, the initiator $P_{1}$ performs $QFT$ on the $c$-qubit state and
executes $CNOT$ gate. Then, $\ket{k}_{T}$ is sent to player $P_{2}$ who
applies the oracle operator on $\ket{k}_{T}$. Thereafter, player $P_{2}$ sends
$\ket{k}_{T}$ to $P_{3}$, who performs similar process. This process is
performed $(t-1)$ times. After that, the ancillary state $\ket{k}_{T}$ is sent
back to player $P_{1}$ by $P_{t}$. Player $P_{1}$ performs $CNOT$ and
$QFT^{-1}$ to get the multiplication. We have executed this quantum protocol
for $(t,n)$ threshold secure multiparty multiplication using the following
number of players and qubits:

  * •

In simulations $1-3$, we have considered three players with one qubit, three
players with two qubits, and three players with three qubits, respectively,
and got efficient result after taking $8192$ number of average shots.

  * •

In simulations $4-6$, we have considered four players with one qubit, four
players with two qubits, and four players with three qubits, respectively, and
got efficient result after taking $8192$ number of average shots.

  * •

In simulations $7-9$, we have considered fifteen players with one qubit,
fifteen players with two qubits, and fifteen players with three qubits,
respectively, and got efficient result after taking $8192$ number of average
shots.

We got efficient results of multiplication after taking $8192$ number of
average shots.

##  7 Results and Discussion

In this section, we discuss the security and performance analysis of the
proposed $(t,n)$ threshold $QSS$ protocol based on some properties.

###  7.1 Security Analysis

Here, we analyze the outside (i.e., outside eavesdropper wants to steal the
private information of all players) and participant (i.e., attack from one or
more dishonest players) attacks. We discuss four types of outside attacks
(i.e., Intercept-Resend $(IR)$, Intercept, Entangle-Measure $(EM)$ and
Forgery) and two types of participant attacks (i.e., Collision and Collusion)
cai2019cryptanalysis ; ting2009participant ; Wang2008 ; wang2011security ;
wang2013cryptanalysis ; wang2017security  .

####  7.1.1 Outside Attack

In this type of attack, an outside eavesdropper wants to steal the private
information of all players. We discuss the Intercept-Resend $(IR)$, Intercept,
Entangle-Measure $(EM)$ and Forgery attacks as follows.

##### Intercept-Resend $(IR)$ Attack:

In intercept-resend attack, a player measures the quantum state, which is sent
by another player and replaces this state with his own state and then sends
the replacement state to other players. In our proposed protocol, player
$P_{1}$ sends the ancillary state $\ket{k}_{T}$ to dishonest player $P_{2}$
through an authenticated quantum channel and player $P_{2}$ wants to eavesdrop
$P_{1}$’s shadow of the share $s_{1}$. If the ancillary state measured by
dishonest player $P_{2}$ in computational basis
${\ket{0},\ket{1},\dots,\ket{d-1}}$. The dishonest player $P_{2}$ can succeed
to get $\ket{l}_{T}$ with the probability of $1/d$, but the output of the
measurement $k$ is totally independent of $P_{1}^{\prime}s$ share $s_{1}$.
Further, player $P_{2}$ sends the state $\ket{k}_{T}$ to player $P_{3}$.
Unfortunately, $k$ does not possess any partial information about $P_{1}$’s
shadow of the share $s_{1}$. The dishonest player $P_{2}$ cannot get any
information from the intercepted state, and similarly the dishonest player
$P_{3}$ cannot get any information from the transmitted state $\ket{k}_{T}$.
So, the intercept-resend attack is infeasible.

##### Intercept Attack:

In this attack, the dishonest player $P_{2}$ wants to eavesdrop $P_{1}$’s
shadow of the share $s_{1}$. The dishonest player $P_{2}$ can measure the
output of the unitary operator (transformed state) because, based on $QFT$,
player $P_{2}$ knows that player $P_{1}$’s shadow of the share state
$\ket{s_{1}}$ has evolved as the ancillary state $\ket{k}_{T}$. So, $QFT^{-1}$
can be performed on the ancillary state $\ket{k}_{T}$ by dishonest player
$P_{2}$ to reveal $s_{1}$. If the ancillary state measured by dishonest player
$P_{2}$ in computational basis ${\ket{0},\ket{1},\dots,\ket{d-1}}$, then
$P_{2}$ can succeed to get $\ket{l}_{T}$ with the probability of $1/d$, but
$P_{2}$ cannot get $P_{1}$’s shadow of the share, because the global
information cannot be extracted from the limited number of qubits. The
entangled systems cannot be disentangled by the limited number of qubits. So,
the attacker cannot get any information about $P_{1}$’s shadow of the share.

##### Entangle-Measure $(EM)$ Attack:

The dishonest player $P_{2}$ performs a more complicated entangle-measure
attack. Player $P_{2}$ prepares an ancillary state $\ket{0}_{P_{2}}$ that gets
entangled with the transmitted state $\ket{k}_{T}$ using the local unitary
operations. Then, player $P_{2}$ measures the entangle state to get the
partial information about player $P_{1}$’s shadow of the share. After
successful completion of honesty test, it can easily be deduced that
$\eta_{k}=1$. After performing $\bar{U}_{TP_{2}}$, $P_{2}$ sends $\ket{k}_{T}$
back to $P_{1}$ and measures the ancillary system after execution of
$CNOT^{\otimes c}$ operation by player $P_{1}$. If player $P_{2}$ measures the
ancillary state $\ket{\phi(k)}_{P_{2}}$, $P_{2}$ cannot get any information
about $P_{1}$’s shadow of the share $s_{1}$ because of entanglement of
$\ket{k}_{H}$ and $\ket{\phi(k)}_{P_{2}}$. So, this attack is also infeasible.

##### Forgery Attack:

In forgery attack, the participants can execute the protocol with the fake
shares. The proposed $QSS$ protocol can prevent the forgery attack, which is
one of the important issues, where the participants can provide the fake
shares. If any dishonest player performs the Pauli operator with the fake
shadow, the original secret cannot be reconstructed correctly. In the proposed
protocol, player $P_{1}$ uses the hash function $SHA1$ to compute the hash
value $H(f(0)^{\prime})$ and compares it with the hash value $g(0)^{\prime}$.
If $(H(f(0)^{\prime})=g(0)^{\prime})$, then $P_{1}$ shares the secret with
other $t-1$ players; otherwise, $P_{1}$ realizes that at least one player
performs the reconstruction phase dishonestly and terminates the
reconstruction phase. So, the forgery attack is not possible in our quantum
($t,n$) threshold $QSS$ protocol.

####  7.1.2 Participant Attack

This type of attack is performed by one or more dishonest players to reveal
the secret information.

##### Collision (attack from one) Attack:

In collision attack, the attacker performs an attack on the hash function,
where the hash function produces the same hash value for two different inputs.
Many existing $QSS$ protocols cannot prevent the collision attack. In
song2017t , Alice (dealer) selects $Bob_{1}$ as a trusted reconstructor from
the set of participants $\mathbb{B}=\\{Bob_{1},Bob_{2},\dots,Bob_{n}\\}$ and
then selects a hash function $SHA1$ to compute the hash value of the secret
(which is to be shared). Then, Alice sends this hash value to the trusted
reconstructor $Bob_{1}$. At this point, $Bob_{1}$ can perform collision attack
to reveal the secret. So, the security of their protocol is dependent on the
trusted reconstructor $Bob_{1}$. In our protocol, the dealer $\mathbb{D}$
computes the hash value $H(S)$ using the $SHA1$ hash function and shares it
among $n$ players. Therefore, the reconstructor $P_{1}$ does not have any
information about the hash value and he cannot perform the collision attack.

##### Collusion (attack from more than one dishonest players) Attack:

In collusion attack, some players can collude together to get the shadow of
the share of other player. In order to get the private information of $P_{e}$
, players $P_{e-1}$ and $P_{e+1}$ perform the protocol dishonestly. In our
proposed protocol, players $P_{e-1}$ and $P_{e+1}$ cannot perform the
collusion attack because the unitary operation is performed by each
participant with his private information. Moreover, this private information
is not transmitted through a quantum channel.

###  7.2 Performance Analysis

We analyze and compare the performance of the proposed $QSS$ protocol with the
existing $QSS$ protocols, i.e., Li et al.’s $QSS$ bao2009threshold , Yang et
al.’s $QSS$ yang2013secret , Qin et al.’s $QSS$ qin2015t , Song et al.’s $QSS$
song2017t , and Qin et al.’s $QSS$ Qin2018Multidimensional  in terms of three
parameters: universality, cost, and attack. Li et al.’s $QSS$ protocol
bao2009threshold  is ($t,n$) threshold scheme, but it is not for $d$-level
particle. Yang et al.’s $QSS$ protocol yang2013secret  is for $d$-level
particle, but this protocol is ($n,n$) threshold scheme. Qin et al.’s $QSS$
protocol qin2015t  is ($t,n$) threshold scheme, but it is not for $d$-level
particle. Song et al.’s $QSS$ protocol song2017t  is for $d$-level particle,
and also it is ($t,n$) threshold scheme. Song et al.’s $QSS$ protocol
song2017t  can prevent the $IR$, $EM$, forgery attacks, but it cannot prevent
the collision attack. Qin et al.’s $QSS$ protocol Qin2018Multidimensional  is
for $d$-level particle and $cn$ qubits, but this protocol is ($n,n$) threshold
scheme. Here, we have compared the proposed $QSS$ protocol with six existing
protocols in terms of the communication cost and computation cost.The
communication cost can be computed based on the transmitted particles, i.e.,
message particles and decoy particles. The computation cost can be computed
based on five parameters: $QFT$, $U$ operation, $QFT^{-1}$, measure operation,
and hash operation. The Li et al.’s $QSS$ protocol bao2009threshold  needs to
perform $t(2t-1)$ number of $U$ operations, $t$ number of measure operations,
needs to transmit $t(t+1)$ number of messages, and $z(t+1)$ number of decoy
particles, where $z$ is the number of decoy particles. The Yang et al.’s
protocol yang2013secret  needs to perform $n$ number of $QFT$, $n$ number of
$U$ operations, $n$ number of measure operations, and needs to transmit
$(n-1)$ number of message particles. The Qin et al.’s protocol qin2015t  needs
to perform $t(t+1)$ number of $U$ operations, needs to transmit $t(t+1)$
number of messages, and $z(t+1)$ number of decoy particles. The Song et al.’s
protocol song2017t  needs to perform $1$ number of $QFT$, $t$ number of $U$
operations, $1$ number of $QFT^{-1}$, $1$ number of measure operations, $2$
number of hash operations and needs to transmit $(t-1)$ number of message
particles. The Qin et al.’s protocol Qin2018Multidimensional  needs to perform
$1$ number of $QFT$, $t(t+1)+n$ number of $U$ operations, $1$ number of
$QFT^{-1}$, $n$ number of measure operations, and needs to transmit $n$ number
of message particles as well as $z(t+1)$ number of decoy particles. Our
proposed protocol needs to perform $1$ number of $QFT$, $(t-1)$ number of $U$
operations, $1$ number of $QFT^{-1}$, $2$ number of hash operations, and needs
to transmit $t$ number of decoy particles. So, the complexity of our proposed
protocol is less as compared to the existing $QSS$ protocols.

##  8 Conclusion

In this paper, we discussed a secret-sharing protocol in which $t$ players can
reconstruct the secret without the help of a trusted player. In comparison to
existing $QSS$ protocols, our protocol is more secure, versatile, and
practical. The reconstructor $P1$ only knows his share and nothing else; even
the secret’s hash value is unknown to him. Since the reconstructor $P_{1}$
only knows his share, it cannot perform the collision attack.

## Ethical Statement

This article does not contain any studies with human or animal subjects
performed by the any of the authors. The manuscript has been prepared
following the instructions provided in the Authors Guidelines of the journal.

## Conflict of Interest

The authors declare that they have no conflict of interest.

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