# Chaos synchronization in a BEC system using fuzzy logic controller

E. Tosyali  $|$ ${}^{\textsf{\footnotesize{\mbox{?}}}}$, Y. Oniz  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$, F. Aydogmus  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$ Corresponding author:
<EMAIL_ADDRESS>
(Received April 13, 2022, in final form August 31, 2022)

###### Abstract

Since the presence of chaos in Bose-Einstein condensate (BEC) systems plays a
destructive role that can undermine the stability of the condensates,
controlling the chaos is of great importance for the creation of the BEC. In
this paper, a fuzzy logic controller (FLC) to synchronize the chaotic dynamics
of two identical master-slave BEC systems has been proposed. Unlike the
conventional approaches, where expert knowledge is directly used to construct
the fuzzy rules and membership functions, the fuzzy rules have been
constructed using Lyapunov stability theorem ensuring the synchronization
process. The effectiveness of the proposed controller has been demonstrated
numerically.

Key words: fuzzy logic controller, synchronization, chaos, Bose-Einstein
condensate

###### Abstract

Îñêëüêè íàÿâíñòü õàîñó â Áîçå-Åéíøòåéíâñüêîìó êîíäåíñàò (ÁÅÊ) âäãðà
äåñòðóêòèâíó ðîëü òà ìîæå çìåíøóâàòè ñòàáëüíñòü êîíäåíñàòó, êîíòðîëþâàííÿ
õàîcó ìà âåëèåçíå çíàåííÿ äëÿ ñòâîðåííÿ ÁÅÊ. Ó öé ñòàòò çàïðîïîíîâàíèé íåòêèé
ëîãíèé êîíòðîëåð äëÿ ñèíõðîíçàö¿ õàîòèíî¿ äèíàìêè äâîõ äåíòèíèõ êåðóþèõ ðâíÿíü
ÁÅÊ ñèñòåì. Íà âäìíó âä òðàäèöéíèõ ïäõîäâ, äå åêñïåðòí çíàííÿ âèêîðèñòîâóòüñÿ
äëÿ îòðèìàííÿ ïðàâèë íåòêîãî êîíòðîëþ äëÿ âäïîâäíèõ ôóíêöé, ó öé ðîáîò çãàäàí
ïðàâèëà ïîáóäîâàí ç âèêîðèñòàííÿì òåîðåìè ñòéêîñò Ëÿïóíîâà äëÿ çàáåçïååííÿ
ïðîöåñó ñèíõðîíçàö¿. Åôåêòèâíñòü çàïðîïîíîâàíîãî ïðîöåñó êîíòðîëþ
ïðîäåìîíñòðîâàíî èñåëüíî.

Ключов слова: íåòêèé ëîãíèé êîíòðîëåð, ñèíõðîíçàöÿ, õàîñ, Áîçå-Åéíøòåéíâñüêèé
êîíäåíñàò

##  1 Introduction

Bose-Einstein condensation (BEC) is a process, in which the system forms a
single coherent matter wave after the temperature of boson gases is reduced
below a critical level. The theoretical background of BEC was set by Einstein
in [1, 2], with the idea that the boson gases will experience a phase
transition at their critical temperature, whereas the idea was experimentally
verified in 1995 using the dilute atomic vapor of rubidium and sodium [3, 4].

Despite the fact that the temperatures obtained with lasers are quite low, to
be able to form BEC, an additional cooling method is required to enable the
atoms with relatively higher energy to escape from the trap [5]. In this
cooling method, which reduces the kinetic energy of the entire system, the
magneto-optical trap and lasers are turned off while another magnetic field is
activated at the same time. The energy of the atoms at the center of the trap
is considerably smaller than the energy of the atoms at the corners of the
trap. Trapped dilute boson gases interact with each other due to their
physical properties or due to collisions. In the interacting gases, only
weakly interacting states caused by binary collisions ($s$-wave scattering)
are considered, as it is not possible to express the system macroscopically
with a single wave function in non-weak interactions.

Radiofrequency is used to enable atoms with higher energies to escape from the
trap, which provides a change in the spinning direction of the atoms. This
process generates a repulsive force for atoms, where the magnetic field and
the magnetic moment are parallel. An attractive force occurs among the atoms
due to opposite magnetic moments. The repulsive force separates the atomic
cloud as trapped and untrapped, and allows the atoms with more energy standing
at the corners to be thrown out of the trap. The atoms in the trap collide and
transfer their momentum to each other; and they come into equilibrium at a new
low thermal energy called back thermalization. This process is repeated until
the critical temperature is reached [6].

The condensation of weakly interacting boson gases, for which the temperature
is close to zero, is well expressed by the Gross-Pitaevskii equation (GPE).
This equation was derived in 1961 by Gross and Pitaevskii independently and
with different techniques to describe weakly interacting dilute boson gases
[7, 8]. Basically, the GPE is the nonlinear Schrödinger equation derived from
mean-field theory. The GPE gives favorable results in the experiments with
weakly interacting BEC.

It is of great importance to study how to control the chaos of a BEC system in
an optical lattice which exhibits many rich and complex phenomena typical of
nonlinear systems [9, 10, 11, 12, 13]. An important approach to consider is
the synchronization problem from a control theory perspective.

Different control schemes including feedback control [14, 15, 16, 17], sliding
mode control, [18, 19] and fuzzy logic control [20, 21] have been proposed
over the last decade for the synchronization problem of chaotic systems. The
main drawback of the feedback control schemes is that the control signals are
generated relying on the mathematical model of the chaotic system. However, in
many applications the dynamics of the system will be perturbed because of the
uncertainties in the system parameters and external disturbances. Hence, these
controllers may fail to provide reliable results. The sliding mode control
approach can be pointed out among the most effective robust controllers to
handle high-order nonlinear systems. However, this approach inherently suffers
from the chattering problem. On the other hand, fuzzy logic controllers (FLC)
provide an easy but effective way to cope with uncertain and nonlinear system
dynamics, and they were successfully applied in many areas such as control
[22, 23], decision making [24, 25], prediction [26, 27], forecasting [28, 29],
and modelling [30, 31]. Recently, fuzzy logic control of chaotic systems has
become an active research area. In the fuzzy logic control, the output of the
controller is determined using the fuzzy inference. The rules typically rely
on expert knowledge. Although promising results have been reported in the
literature for the use of this conventional scheme in chaos synchronization
[32, 33], the performance of these controllers might significantly degrade if
expert knowledge is incomplete and/or uncertain. To alleviate this issue,
adaptive approaches are commonly preferred in the design of FLCs, in which the
controller parameters are updated to lead the synchronization error to zero
[34, 35]. Despite the fact that the adaptive schemes can provide fairly good
results, their time requirements for the adaptation process might pose a
problem in real-time applications. In the proposed work, the Lyapunov
stability theorem was directly employed to construct the consequent part of
the fuzzy rules such that two identical master-slave BEC systems can be
synchronized. One of the most prominent advantages of this control scheme is
that the stability in the Lyapunov sense of error dynamics of two identical
chaotic BEC motions was ensured. The feasibility and effectiveness of the
proposed controller were demonstrated by numerical simulation results.

##  2 Description of system

GPE including macroscopic wave function can well describe the evolution of the
BEC simultaneously with regard to time and space [7, 8]. One-dimensional (1D)
GPE can be described as below:

$\mathrm{i}\hbar\frac{\partial}{\partial
t}\Psi\left(x,t\right)=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}\Psi\left(x,t\right)+\left[V_{\rm{ext}}\left(x\right)+g_{1D}\left|\Psi\left(x,t\right)\right|^{2}\right]\Psi\left(x,t\right),$
(2.1)

where $m$ stands for the mass of the atoms which constitute the BEC,
$V_{\rm{ext}}$ is the external potential with tilted term trapping from the
BEC, and $g_{1D}$ is the one dimensional interaction term between the atoms
defined as:

$g_{1D}=\frac{g_{3D}}{2\piup a_{r}^{2}}=2a_{s}\hbar\omega_{r},$

with $a_{s}$ being the $s$-wave scattering length between atoms. $s$-wave
scattering length could be positive or negative depending on the interactions
whether it is repulsive or attractive, respectively. In our case, its value is
negative due to attractive interactions. $\omega_{r}$ is the ground state of a
harmonic frequency of the oscillator.

The external trap potential $V_{\rm{ext}}\left(x\right)$ is given as:

$V_{\rm{ext}}\left(x\right)=V_{1}\cos^{2}(\omega_{1}x)+V_{2}\cos^{2}(\omega_{2}x)+Fx.$
(2.2)

$V_{\rm{ext}}$ comprises two parts: while the first part of double well
potential with two frequencies is related to the optical lattice potential,
the second part is related to the tilted potential. Here, $V_{1}$ and $V_{2}$
are the amplitudes, $F$ is the internal force and $Fx$ corresponds to the
tilted potential, which makes the atoms tunnelling out from the potential and
accelerates them in the $x$ direction. 1D Hamiltonian
$\left(H_{F}=H_{0}+Fx\right)$ tends to infinity when
$|x\rvert\rightarrow\infty$ [36]. However, the Hamiltonian is always bounded
if the lattice size is finite $-L\leqslant x\leqslant L$ [37]. The BEC system
of this study is bounded with 100 lattice sites. The number of lattice sites
was determined empirically as $100$, which implies that $L\sim 100\piup
k^{-1}$ ($k={2\piup}/{850}$ nm-1) [38, 39, 40]. To meet the requirements on
the numbers of the lattice sites and thus on the boundary conditions, the
simulated studies were carried out for 1000 steps with a step size of 0.1, and
for 10000 steps with a step size of 0.01. In figure 1, the evolution of the
external potential for parameters set $\nu_{1}=1$, $\nu_{2}=0.8$,
$\omega_{1}=2\piup$, $\omega_{2}=5\piup$, (a) ${F}=0$, (b) ${F}=0.1$ is
illustrated.

(a) (b)

Figure 1: (Colour online) Plot of the bichromatic optical lattice potential
with the parameters $\nu_{1}=1$, $\nu_{2}=0.8$, $\omega_{1}=2\piup$,
$\omega_{2}=5\piup$, (a) ${F}=0$, (b) ${F}=0.1$.

There are different time-dependent ansatzs to solve the GPE [41, 42]. In this
study, the following widely-used form of the time-dependent wave function is
preferred:

$\Psi\left(x,t\right)=\Phi\left(x\right)\mathrm{e}^{{-\mathrm{i}\mu
t}/{\hbar}},$ (2.3)

here, $\mu$ is the chemical potential of the condensate and $\Phi(x)$ is a
real function independent of time. Normalized $\Phi(x)$ gives the total number
of particles in the system, i.e.,

$\int\left|\Phi\left(x\right)\right|^{2}\mathrm{d}x=N,$ (2.4)

where $N$ is the particle number. Substitution of equations (2.2) and (2.3)
into equation (2.1) yields:

$\mu\Phi\left(x\right)=-\frac{\hbar^{2}}{2m}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\Phi\left(x\right)+\left[V_{1}\cos^{2}\left(\omega_{1}x\right)+V_{2}\cos^{2}(\omega_{2}x)+Fx+g_{1D}\left|\Phi\left(x\right)\right|^{2}\right]\Phi\left(x\right).$
(2.5)

Using the dimensionless parameters $\upsilon_{1}={2mV_{1}}/{\hbar^{2}}$,
$\upsilon_{2}={2mV_{2}}/{\hbar^{2}}$, $\gamma={2m\mu}/{\hbar^{2}}$,
$\eta={2mg_{0}}/{\hbar^{2}}$, $\Gamma={2mF}/{\hbar^{2}}$, the equation (2.5)
can be written as:

$\frac{\mathrm{d}^{2}\Phi}{\mathrm{d}x^{2}}=\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+\eta\left|\Phi\right|^{2}\right]\Phi.$ (2.6)

The solution of equation (2.6) has the following form:

$\Phi(x)=\phi(x)\mathrm{e}^{\mathrm{i}\theta(x)},$ (2.7)

where $\phi$ and $\theta$ are real functions of $x$, expressing the amplitude
and the phase, respectively. The first derivative of the phase is proportional
to the velocity field, and the squared amplitude corresponds to the density of
the atoms in the condensate. Substituting equation (2.7) into equation (2.6)
results in two coupled equations, which correspond to the real and imaginary
parts.

$\frac{\mathrm{d}^{2}\phi}{\mathrm{d}x^{2}}=\phi\left(\frac{\mathrm{d}\theta}{\mathrm{d}x}\right)^{2}+\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+\eta\left|\phi\right|^{2}\right]\phi,$ (2.8)
$\frac{\mathrm{d}^{2}\theta}{\mathrm{d}x^{2}}+2\frac{1}{\phi}\frac{\mathrm{d}\theta}{\mathrm{d}x}\frac{\mathrm{d}\phi}{\mathrm{d}x}=0.$
(2.9)

Integration of equation (2.9) gives the relation between the velocity field
and the particle number density,

$J=2\phi^{2}\left(\frac{\mathrm{d}\theta}{\mathrm{d}x}\right),$ (2.10)

where $J$ represents the steady super-fluidity phase in fluid dynamics. Using
$J$ in equation (2.8), the following nonlinear equation can be obtained:

$\frac{\mathrm{d}^{2}\phi}{\mathrm{d}x^{2}}=\frac{J^{2}}{4\phi^{3}}+\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+\eta\left|\phi\right|^{2}\right]\phi.$ (2.11)

The equation (2.11) was solved numerically for $1000$ steps with $100$
eliminated steps, with a step size of $0.1$ to satisfy the 100 lattice site
boundary condition. In the simulation, the following values for the parameters
were assumed: $J=0.4$, $\nu_{1}=1$, $\nu_{2}=0.8$, $\omega_{1}=2\piup$,
$\omega_{2}=5\piup$, $\Gamma=0.1$, $\eta=-0.015$, $\gamma=0.5$ and possible
initial conditions $(x_{1}=1$, $y_{1}=-1)$ by the Runge-Kutta method. The
Lyapunov characteristic exponents (LCEs) of the BEC system given in figure 2
are $\lambda_{1}=0.0046543$, $\lambda_{2}=0$ and $\lambda_{3}=-0.00465435$.
For the given system parameters and initial conditions, the system exhibits
chaotic behaviour due to the presence of positive LCE.

Figure 2: (Colour online) LCEs for paramter sets: $J=0.4$, $\nu_{1}=1$,
$\nu_{2}=0.8$, $\omega_{1}=2\piup$, $\omega_{2}=5\piup$, $\Gamma=0.1$,
$\eta=-0.015$, $\gamma=0.5$ and initial condition: $(x_{1}=1$, $y_{1}=-1)$.

Using the transformations
${\mathrm{d}\phi}/{\mathrm{d}x}={\mathrm{d}x_{1}}/{\mathrm{d}x}$ and
${\mathrm{d}^{2}\phi}/{\mathrm{d}x^{2}}={\mathrm{d}y_{1}}/{\mathrm{d}x}$, the
equation (2.11) can be re-written with the following first-order coupled
equations, which defines the master system as:

$\frac{\mathrm{d}x_{1}}{\mathrm{d}x}=y_{1},$ (2.12)
$\frac{\mathrm{d}y_{1}}{\mathrm{d}x}=\frac{J^{2}}{4x_{1}^{3}}+\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+\eta\left|x_{1}\right|^{2}\right]x_{1},$ (2.13)

and the dynamics of the slave system are given below:

$\frac{\mathrm{d}x_{2}}{\mathrm{d}x}=y_{2}.$ (2.14)
$\displaystyle\frac{\mathrm{d}y_{2}}{\mathrm{d}x}=\frac{J^{2}}{4x_{2}^{3}}+\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+\eta\left|x_{2}\right|^{2}\right]x_{2}+\Delta f(x_{2},y_{2})+u(x),$
(2.15)

where $u$ is the control input, $\Delta f(x_{2},y_{2})$ is the uncertain term
which is assumed bounded, i.e., $\Delta f(x_{2},y_{2})\leqslant\alpha$, where
$\alpha$ is a positive constant. It is also assumed that $\Delta
f(x_{2},y_{2})$ satisfy all the necessary conditions, such as system (2.14),
(2.15) having a unique solution in the spatial evolution $x_{0}+\infty$, for
any given initial condition. In this paper, the control input $u(x)$ is
derived to synchronize the master and slave system.

##  3 Fuzzy design for chaotic synchronization

Fuzzy inference can be defined as a process, in which the given input(s) are
mapped to the output(s) using the fuzzy set theory. Figure 3 depicts the
general structure of a fuzzy inference system, which consists of four main
function blocks:

  1. 1.

The knowledge base comprises expert knowledge and it can be separated into two
parts: database and rule base. In the database, the type and parameters of the
membership functions are stored, whereas the rule base includes the fuzzy “if-
then” rules.

  2. 2.

In the fuzzification stage, the crisp input variables are transformed into
fuzzy sets using the expert knowledge stored in the database. Hence, the
outputs of this block are fuzzy sets.

  3. 3.

In the inference engine, using the fuzzy “if-then” rules stored in the rule
base, rule consequents are computed for each rule and aggregated into a single
fuzzy output set.

  4. 4.

Output fuzzy sets are transformed into crisp output values in the
defuzzification step.

Figure 3: Structure of a fuzzy inference system.

In order to synchronize the master and slave system with the dynamics given in
equations (2.12)–(2.15), the error states are defined as $e_{1}=x_{2}-x_{1}$
and $e_{2}=y_{2}-y_{1}$ by subtracting the master system from the slave
system. The error dynamics can be specified as:

$\frac{\mathrm{d}e_{1}}{\mathrm{d}x}=e_{2},$ (3.1)
$\displaystyle\frac{\mathrm{d}e_{2}}{\mathrm{d}x}=\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}+\Delta f(x_{2},y_{2})+u_{L}.$ (3.2)

The control input $u(x)$ in equation (2.15) can be written as
$u(x)={u}_{eq}+u_{L}$, where
$u_{eq}=-\eta\left|x_{2}\right|^{2}x_{2}+\eta\left|x_{1}\right|^{2}x_{1}$. A
fuzzy logic controller is proposed to construct the control input $u_{L}$,
where the error dynamics in equation (3.1) and (3.2) are utilized as the input
signals of the FLC. These incoming signals, $(e_{1},e_{2})$, are fuzzified
using triangular membership functions shown in figure 4, and associated with
$A_{i}$ and $B_{j}$ fuzzy subsets, respectively, which are defined by their
corresponding membership functions $\mu_{A_{i}}(e_{1})$ and
$\mu_{B_{j}}(e_{2})$ for $i=1,\dots,I$ and $j=1,\dots,J$.

The fuzzy “if-then” rule $R_{ij}$ with the inputs $(e_{1},e_{2})$ and the
output $u_{L}$ can be stated as:

$R_{ij}$: If $e_{1}$ is $A_{i}$ and $e_{2}$ is $B_{j}$, then
$u_{L}=u_{L,ij}(e_{1},e_{2})$,

where $u_{L,ij}$ is an analytical function of $e_{1},e_{2}$ that stabilizes
the error dynamics in equation (3.1) and (3.2). Minimum t-norm is employed to
calculate the firing strengths of each rule, i.e.,
$W_{ij}=\min[\mu_{A_{i}}(e_{1}),\mu_{B_{j}}(e_{2})]$, whereas the centroid
defuzzification method is used to compute the output control signal $u_{L}$:

$u_{L}=\frac{\sum_{i=1}^{I}\sum_{j=1}^{J}{W}_{ij}u_{L,ij}}{\sum_{i=1}^{I}\sum_{j=1}^{J}{W}_{ij}}.$
(3.3)

The fuzzy rules used in this study are presented in table 1, in which $e_{1}$
and $e_{2}$ correspond to the input variables used in the antecedent part of
the rules, and $u_{L_{ij}}$ denotes the output variable of the consequent. As
illustrated in figure 4, three membership functions are used for each input:
$A_{1}$, $A_{2}$ and $A_{3}$ for the input signal $e_{1}$, and $B_{1}$,
$B_{2}$ and $B_{3}$ for the input signal $e_{2}$. The membership functions
$\\{A_{1},B_{1}\\}$, $\\{A_{2},B_{2}\\}$ and $\\{A_{3},B_{3}\\}$ correspond to
positive, zero and negative error dynamics, respectively.

Table 1: Rule Table of FLC. Rule | Antecedent | Consequent
---|---|---
| $e_{1}$ | $e_{2}$ | $u_{L,ij}$
1 | $A_{1}$ | $B_{1}$ | $u_{L,11}$
2 | $A_{1}$ | $B_{2}$ | $u_{L,12}$
3 | $A_{1}$ | $B_{3}$ | $u_{L,13}$
4 | $A_{2}$ | $B_{1}$ | $u_{L,21}$
5 | $A_{2}$ | $B_{2}$ | $u_{L,22}$
6 | $A_{2}$ | $B_{3}$ | $u_{L,23}$
7 | $A_{3}$ | $B_{1}$ | $u_{L,31}$
8 | $A_{3}$ | $B_{2}$ | $u_{L,32}$
9 | $A_{3}$ | $B_{3}$ | $u_{L,33}$

Figure 4: Membership functions.

The following function, which is positive and continuously differentiable, is
selected as the Lyapunov function candidate:

$V=\frac{1}{2}\left(e_{1}^{2}+e_{2}^{2}\right).$ (3.4)

To ensure the Lyapunov stability, the following condition should be met [43]:

$\dot{V}=e_{1}\dot{e}_{1}+e_{2}\dot{e}_{2}<0,$ (3.5)

which requires $\dot{e}_{2}<-{e_{1}\dot{e}_{1}}/{e_{2}}$. For the following
cases, the consequents of the FLC are proposed such that the stability
condition is ensured.

Case 1: $e_{2}<0$

For $e_{2}<0$, the stability condition requires that:

$\dot{e}_{2}>-e_{1}.$ (3.6)

Substituting equation (3.6) into equation (3.1) and (3.2) produces

$\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}+\Delta f+u_{L}>-e_{1},$ (3.7)

then
$-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma-1\right]e_{1}-\Delta f<u_{L}$ should be provided. Assuming that the
uncertainty term $\Delta f$ is bounded, that is $|\Delta f|\leqslant\alpha$,
with $\alpha$ being a positive constant, the following equality can be
derived:

$-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+1\right]e_{1}+\alpha=u_{1}^{*}.$ (3.8)

Case 2: $e_{2}>0$

If $e_{2}>0$, then:

$\dot{e}_{2}<-e_{1},$ (3.9)

should be provided to meet the stability condition. Substituting equation
(3.9) into equation (3.1) and (3.2) results in:

$\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}+\Delta f+u_{L}<-e_{1}.$ (3.10)

Hence, the control input $u_{L}$ should satisfy the following inequality:

$-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma-1\right]e_{1}-\Delta f>u_{L}.$ (3.11)

A new term $u_{2}^{*}$ can be derived as:

$-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+1\right]e_{1}-\alpha=u_{2}^{*}.$ (3.12)

According to table 1, $e_{2}$ is negative for Rules 3, 6 and 9, which
corresponds to the above mentioned Case 1. If the terms in the consequent
parts of these rules are selected as $u_{1}^{*}=u_{L,13}=u_{L,23}=u_{L,33}$,
then the stability condition will be satisfied, i.e., $\dot{V}<0$, and the
error states will be asymptotically driven to zero. Similarly, for Rules 1, 4
and 7, $e_{2}$ is positive, which corresponds to Case 2. Hence, if the
consequent parameters of these rules are selected as
$u_{2}^{*}=u_{L,11}=u_{L,21}=u_{L,31}$, then the stability can be ensured.

Case 3: $e_{1}>0$ and $e_{2}\in 0$

In table 1 for Rule 2, $e_{1}$ is positive, whereas $e_{2}$ is zero. To
satisfy the Lyapunov stability condition stated in equation (3.5), the
following equality should be provided:

$e_{1}+\dot{e}_{2}=-{\rm{sgn}}(e_{2}),$ (3.13)

where

${\rm{sgn}}(e_{2})=\begin{cases}1:e_{2}>0,\\\ -1:e_{2}<0.\end{cases}$

As $e_{1}$ is positive, then

$\dot{e}_{2}<-{\rm{sgn}}(e_{2}),$ (3.14)

should be satisfied. Substituting equation (3.14) into equation (3.1) and
(3.2) yields

$\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}+\Delta f+u_{L}<-{\rm{sgn}}(e_{2}).$ (3.15)

The equation can be written as

$u_{L}<-{\rm{sgn}}(e_{2})-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}-\Delta f,$ (3.16)

which implies that if $u_{L,12}$ is selected as:

$u_{L,12}=-{\rm{sgn}}(e_{2})-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}-\alpha,$ (3.17)

then it will satisfy the stability condition.

Case 4: $e_{1}<0$ and $e_{2}\in 0$

The controller $u_{L,32}$ in Rule 8 can be derived in a way similar to Rule 2,
which requires:

$\dot{e}_{2}>-{\rm{sgn}}(e_{2}).$ (3.18)

Substituting equation (3.18) into equation (3.1) and (3.2) yields

$\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}+\Delta f+u_{L}>-{\rm{sgn}}(e_{2}).$ (3.19)

The equation can be written as

$u_{L}>-{\rm{sgn}}(e_{2})-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}-\Delta f.$ (3.20)

To ensure the stability, the controller $u_{L,32}$ should be selected as:

$u_{L,32}=-{\rm{sgn}}(e_{2})-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}+\alpha.$ (3.21)

Case 5: $e_{1}\in 0$ and $e_{2}\in 0$

For Rule 5 in Table 1, the error states $e_{1}$ and $e_{2}$ are both zero.
Therefore, $u_{L,22}=0$.

The control function of $u_{L_{ij}}$ with $i=1,2,3$ and $j=1,2,3$ depending on
$e_{1},e_{2}$ is summarized below. The controller parameters are derived such
that all of the rules in the FLC can lead to Lyapunov stable subsystems under
the same Lyapunov function (3.4). Therefore, it is ensured that the chaotic
master and slave system will be synchronized.

$\displaystyle u_{L,11}$ $\displaystyle=$ $\displaystyle
u_{L,21}=u_{L,31}=-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+1\right]e_{1}-\alpha,$ $\displaystyle u_{L,13}$ $\displaystyle=$
$\displaystyle
u_{L,23}=u_{L,33}=-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma+1\right]e_{1}+\alpha,$ $\displaystyle u_{L,12}$ $\displaystyle=$
$\displaystyle-{\rm{sgn}}(e_{2})-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}-\alpha,$ $\displaystyle u_{L,32}$ $\displaystyle=$
$\displaystyle-{\rm{sgn}}(e_{2})-\left[\upsilon_{1}\cos^{2}\left(\omega_{1}x\right)+\upsilon_{2}\cos^{2}\left(\omega_{2}x\right)+\Gamma
x-\gamma\right]e_{1}+\alpha,$ $\displaystyle u_{L,22}$ $\displaystyle=$
$\displaystyle 0.$ (3.22)

##  4 Simulations and results

In this section, the FLC is applied to synchronize the two identical master
and slave BEC systems of the form (2.12), (2.13) and (2.14), (2.15). For this
purpose, the equation systems are solved numerically by using fourth order
Runge-Kutta method. The same parameter sets are used in both master and slave
system as $J=0.4$, $\nu_{1}=1$, $\nu_{2}=0.8$, $\omega_{1}=2\piup$,
$\omega_{2}=5\piup$, $\Gamma=0.1$, $\eta=-0.015$, $\gamma=0.5$. Initial
conditions are selected as $(x_{1}(0),y_{1}(0))=(1,-1)$ and
$(x_{2}(0),y_{2}(0))=(0.2,0.3)$ for master and slave systems, respectively.
The system is evaluated for $10000$ steps with a step size of $0.01$ which
fulfills the 1D 100 lattice sites boundary condition. Spatial evaluation and
phase space displays for master and slave BEC systems are given in Figure 5
and 6, respectively. To observe the effectiveness of the proposed control
scheme more clearly, for the first 250 steps, the control input of the fuzzy
logic controller, $u_{L}$, is set to zero, such that the master and slave
systems are not synchronized. Following the $250^{\text{th}}$ step, the
computed value of $u_{L}$ is directly fed to the system, which leads the error
states $e_{1}$ and $e_{2}$, to converge to zero exponentially. Thus, it can be
concluded that two identical BEC systems are synchronized along the flow. The
results of error states are given in figure 7, which are compatible with
spatial evolution and the phase space results given in figure 5 and 6,
respectively. Finally, the graph of the control input $u_{L}$ is given in
figure 8, from which it can be inferred that $u_{L}$ is zero until $x=250$,
and it is computed with regard to equation (3.22) only after this step.

(a) (b)

Figure 5: (Colour online) Master and slave system synchronization for chaotic
BEC sytems: (a) system outputs $x_{1}$ and $x_{2}$, (b) system outputs $y_{1}$
and $y_{2}$.

Figure 6: (Colour online) Synchronized phase space of the master and slave
systems.

Figure 7: (Colour online) Error states for master and slave systems.

Figure 8: (Colour online) The evolution of control input $u$ depending on $x$.

##  5 Conclusion

In this paper, the effectiveness of the fuzzy logic controller method to
synchronize chaotic BEC systems is theoretically and numerically demonstrated.
The chaotic synchronization system consists of the master and slave systems.
In the master slave scheme, the given BEC system is considered as a master
system, and the other identical BEC system is considered as a slave system.
The difference between the states of the master and slave system and its
derivative is selected as the input signals for the fuzzy logic controller.
The consequent part of the fuzzy rules, which consists of analytical functions
of the error and its derivative, is determined using the Lyapunov stability
theorem to ensure the stability of the synchronization process. Furthermore,
this way the dependence of the algorithm on the completeness and accuracy of
the expert knowledge is also overcome. In addition, the numerical simulation
results show that the error dynamics of the identical chaotic BEC
synchronization systems are regulated to zero asymptotically in shorter time
in spite of the overall system undergoing irregularity.

The chaos which has a destructive role for BEC could disrupt the stability in
the condensate. Therefore, controlling the chaos is of great importance for
the creation of BEC. A distinguishing feature of the present work is the first
synchronization of the identical chaotic Bose-Einstein condensate held in a 1D
tilted bichromatical optical lattice potential by using the fuzzy logic
control technique. Consequently, this study will make a significant
contribution in this field.

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