Chaos synchronization in a BEC system using fuzzy logic controller

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Condensed Matter Physics, 2022, Vol. 25, No. 3, 33501: 1–12

DOI: 10.5488/CMP.25.33501

http://www.icmp.lviv.ua/journal

Chaos synchronization in a BEC system using fuzzy

logic controller

E. Tosyali 1∗

, Y. Oniz 2, F. Aydogmus 3

1Opticianry, Vocational School of Health Services, Istanbul Bilgi University, Kustepe, Sisli, Istanbul, 34387, Turkey

2Faculty of Engineering and Natural Sciences, Department of Mechatronics Engineering, Istanbul Bilgi

University, Eyup, Istanbul, 34060, Turkey

3Faculty of Science, Physics, Istanbul University, Vezneciler, Istanbul, 34134, Turkey

Received April 13, 2022, in ﬁnal form August 31, 2022

Since the presence of chaos in Bose-Einstein condensate (BEC) systems plays a destructive role that can under-

mine 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

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].

∗Corresponding author: eren.tosyali@bilgi.edu.tr.

This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution

of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

33501-1

arXiv:2210.00970v1 [cond-mat.quant-gas] 3 Oct 2022

E. Tosyali, Y. Oniz, F. Aydogmus

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 [9–13]. An important approach

to consider is the synchronization problem from a control theory perspective.

Diﬀerent control schemes including feedback control [14–17], 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:

iℏ𝜕

𝜕𝑡 Ψ(𝑥, 𝑡)=−ℏ2

2𝑚

𝜕2

𝜕𝑥2Ψ(𝑥, 𝑡)+𝑉ext (𝑥)+𝑔1𝐷|Ψ(𝑥, 𝑡)|2Ψ(𝑥, 𝑡),(2.1)

where 𝑚stands for the mass of the atoms which constitute the BEC, 𝑉ext is the external potential with

tilted term trapping from the BEC, and 𝑔1𝐷is the one dimensional interaction term between the atoms

deﬁned as:

𝑔1𝐷=𝑔3𝐷

2π𝑎2

𝑟

=2𝑎𝑠ℏ𝜔𝑟,

with 𝑎𝑠being 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. 𝜔𝑟is the ground state of a harmonic frequency of the

oscillator.

The external trap potential 𝑉ext (𝑥)is given as:

𝑉ext (𝑥)=𝑉1cos2(𝜔1𝑥) + 𝑉2cos2(𝜔2𝑥) + 𝐹𝑥. (2.2)

33501-2

Chaos sync. in a BEC system using FLC

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

Vext(x)

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

2.5

Vext(x)

(a) (b)

Figure 1. (Colour online) Plot of the bichromatic optical lattice potential with the parameters 𝜈1=1,

𝜈2=0.8,𝜔1=2π,𝜔2=5π, (a) 𝐹=0, (b) 𝐹=0.1.

𝑉ext 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,𝑉1and 𝑉2are the amplitudes,

𝐹is the internal force and 𝐹𝑥 corresponds to the tilted potential, which makes the atoms tunnelling out

from the potential and accelerates them in the 𝑥direction. 1D Hamiltonian (𝐻𝐹=𝐻0+𝐹𝑥)tends to

inﬁnity when |𝑥| → ∞ [36]. However, the Hamiltonian is always bounded if the lattice size is ﬁnite

−𝐿6𝑥6𝐿[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∼100π𝑘−1(𝑘=2π/850 nm−1) [38–40].

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 𝜈1=1,𝜈2=0.8,

𝜔1=2π,𝜔2=5π, (a) 𝐹=0, (b) 𝐹=0.1is illustrated.

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:

Ψ(𝑥, 𝑡)= Φ (𝑥)e−i𝜇𝑡/ℏ,(2.3)

here, 𝜇is the chemical potential of the condensate and Φ(𝑥)is a real function independent of time.

Normalized Φ(𝑥)gives the total number of particles in the system, i.e.,

∫|Φ(𝑥)|2d𝑥=𝑁, (2.4)

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

𝜇Φ(𝑥)=−ℏ2

2𝑚

d𝑥2Φ(𝑥)+𝑉1cos2(𝜔1𝑥)+𝑉2cos2(𝜔2𝑥) + 𝐹𝑥 +𝑔1𝐷|Φ(𝑥)|2Φ(𝑥).(2.5)

Using the dimensionless parameters 𝜐1=2𝑚𝑉1/ℏ2,𝜐2=2𝑚𝑉2/ℏ2,𝛾=2𝑚𝜇/ℏ2,𝜂=2𝑚𝑔0/ℏ2,

Γ = 2𝑚𝐹/ℏ2, the equation (2.5) can be written as:

d2Φ

d𝑥2=𝜐1cos2(𝜔1𝑥)+𝜐2cos2(𝜔2𝑥)+Γ𝑥−𝛾+𝜂|Φ|2Φ.(2.6)

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

Φ(𝑥)=𝜙(𝑥)ei𝜃(𝑥),(2.7)

where 𝜙and 𝜃are 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

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CondensedMatterPhysics,2022,Vol.25,No.3,33501:112DOI:10.5488/CMP.25.33501http://www.icmp.lviv.ua/journalChaossynchronizationinaBECsystemusingfuzzylogiccontrollerE.Tosyali1*,Y.Oniz2,F.Aydogmus31Opticianry,VocationalSchoolofHealthServices,IstanbulBilgiUniversity,Kustepe,Sisli,Istanbul,34387,Turkey2Fa...

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Chaos synchronization in a BEC system using fuzzy logic controller

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