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Chaotic control problem of BEC system based on Hartree–Fock mean field theory

  • Yang Shen EMAIL logo and Meng Xu
Published/Copyright: March 1, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

Due to the difficulty of studying nonlinear quantum systems and the unique composition of Bose–Einstein condensate (BEC) systems, BECs face significant difficulties in solving dynamic analysis and chaotic control problems. Therefore, Hartree–Fock mean field theory is introduced to study the chaotic characteristics, control, and synchronization issues of BEC systems loaded on optical lattices. First, the stability and chaos of BECs in optical lattices were analyzed. Subsequently, constant shift method and activation control were introduced based on the Gross–Pitaevskii equation to achieve control and synchronization of the BEC system. Second, based on the Lyapunov exponent theory, offset parameters are added to BEC chaotic control to achieve control of particle density. Finally, based on the stability theory of linear systems, a control term is introduced to achieve variable analysis of the system’s drive–response system, ensuring that chaotic systems with different initial conditions can still achieve good synchronization and anti-synchronization control. The chaotic problem of BEC system was analyzed using numerical and theoretical methods in the experiment. The effect of adjusting the parameters of the BEC system under the constant shift method is significant. The system exhibits a chaotic state under the Lyapunov exponent, which is mainly concentrated between [3.4, 4.5], demonstrating good system stability. When the offset constant range is [4.21, 5.67], the maximum Lyapunov exponent value is below 0. In the problem of chaotic synchronization, adding activation control causes the system’s time series to exhibit anti-synchronization with spatiotemporal variable variation, while adding control terms leads the system to tend towards synchronization and anti-synchronization with time evolution. The analysis of chaotic control problems in BEC systems can provide reference value and theoretical basis for the dynamic research of quantum physics and related nonlinear systems.

Keywords: Hartree–Fock mean field theory; BEC system; chaotic control; constant shift method; Lyapunov index; control response system

1 Introduction

Bose–Einstein condensate (BEC) is different from conventional object morphology. It is composed of a group of bosons present in condensed form and exhibits quantum measurement phenomena at low temperatures, such as condensed states of superfluidity and coherence. These quantum particles approach absolute zero at extremely low temperatures, making them exhibit statistical behavior and volatility. When all particles collectively occupy a quantum state, they can form a macroscopic condensed state [1]. The formation of this condensed state received extensive experimental and theoretical research in the late 1990s. The BEC system has important applications and significance in physics, and it has been extensively studied in fields such as quantum theory, condensed matter physics, superfluidity, and optics. The special cooling technology of the BEC system often requires the use of laser technology to complete, so with the development of laser technology and the continuous improvement of experimental facilities, the research content of BEC has been expanded to a certain extent. It is very important to analyze the phase transition behavior, stability, coherence, and other aspects of BEC bulk systems using chaotic control ideas and methods to expand the understanding of quantum systems. Chaos theory is a significant area of research in nonlinear science. Its objective is to control the system’s behavior, moving it from a chaotic state to an ordered state or achieving specific dynamic behavior by adjusting system parameters or external inputs [2,3]. Among them, optical lattices are tools formed by coherent superposition of laser beams and have long spatial periods of strength and weakness. They are often used in BEC research, and changes in parameters, as well as adjustments in beam wavelength and intensity, can affect the performance of optical lattices. Therefore, the study of BEC atomic physics phenomena in optical lattices has important value [4]. The stability and dynamics of BEC in optical lattices are influenced by various factors, including particle interactions (attraction or repulsion), the geometric structure of the optical lattice, and external control parameters (such as the depth and spacing of the optical lattice potential well). The stability and instability of BEC are related to factors such as collapse, explosion and chaos, dynamic superfluidity, and matter–wave Ming solitons. However, it is important to note that the stability of BEC in a superlattice is limited to certain parameter regions. Additionally, the chemical potential of the optical lattice is related to the trapped particles, while the interaction strength is related to the intensity of the optical lattice potential. By adjusting these parameters, the system can achieve a stable state. In chaotic analysis, BEC often exhibits features such as interaction, coupling, and parameter-driven behavior. Specifically, the interaction of bosons is often described using nonlinear terms and appears in the form of the Schrödinger equation. In the optical lattice, the local minimum values of each potential well may contain condensed atoms. The coupling between potential wells can affect the stability of the entire condensed state, leading to synchronous or unstable fluctuations between condensed states. When the system is in a low-energy state, the BEC in the optical lattice may also face dynamic instability, causing atoms to “spill” from the condensed state into the uncondensed state, thereby disrupting the integrity of the condensed state. Due to the nonlinear macroscopic quantum properties inherent in BEC systems, it is difficult to solve the Schrödinger equation for such multi-particle systems in traditional methods. Therefore, it is necessary to simplify the mathematical model with the help of theory to better analyze the dynamic behavior of BEC systems. This study utilizes the Hartree–Fock mean field theory to study the chaotic characteristics and control problems of BEC and investigates its chaotic synchronization and anti-synchronization problems during the control process. The innovation of this study lies in the analysis of the control and synchronization problems of BEC systems based on the Hartree–Fock mean field theory and linear theory. This study investigates the chaotic control problem of BEC systems from four aspects. First, a literature review was conducted on the research ideas and content of the current BEC system. Second, the chaotic characteristics, control problems, and synchronization problems of BEC were studied. The third part analyzes the numerical simulation results of the BEC system. Finally, there is a summary of the entire article and prospects for the future.

2 Related works

BEC, as a noninteracting condensation, exhibits certain superfluidity and superconductivity. The development of laser cooling and atomic capture technology has expanded the exploration field of BEC atomic gases, and the tools of atomic gas BEC have also promoted the development of experimental physics [5]. Among them, the vortex bright soliton serves as the soliton solution of the BEC system under the high-dimensional Gross–Pitaevskii (G-P) equation. Tan et al. analyzed the anisotropic bright solitons present in two-dimensional optical lattices. These results confirmed that optical crystals and even-order interactions can stabilize anisotropic solitons. And the depth of the optical lattice and the initial distance of the kicked vortex bright soliton will affect its component transformation and stability [6]. Dos Santos and Cardoso conducted numerical simulation analysis on linearly coupled binary BEC and observed the induced asymmetry of the companion field based on the linearly coupled Schrödinger equation. To some extent, the Rabi coupling of dual field components had a promoting effect on studying atomic equilibrium problems [7]. Wang et al. conducted research on two-dimensional nonlinear BEC and proposed a coupling density equation to describe the conversion of light molecules. The condensed-phase energy of this system changed, and there was a certain relationship between its decay and the coupling system [8]. Scholars such as Zhu et al. conducted a theoretical analysis of the plane traveling wave characteristics of defect lattice BEC under periodic modulation and obtained the critical conditions for superfluidity of the system using dual-mode ansatz and tight binding approximation. These results confirmed that the adjustment of plane waves can change the conditions under which the system operates in the superfluid region. And its vibration frequency and lattice strength can affect the propagation and dynamic characteristics of its BEC system under certain conditions [9]. It was difficult to achieve good accuracy in the phase transition solution of BEC with the help of order, so Znojil made the proportional partition correction to the linear Bose–Hubbard model. This method played an important role in the classification of BEC models and ensuring the accuracy of numerical model solutions [10]. Due to the tendency of quasi-particles to oscillate and hybridize in light matter, exciton polarons, as boson quasi particles, had strong coupling and can effectively achieve the formation of BEC [11]. Liu et al. conducted modulation amplitude wave analysis on a one-dimensional BEC system based on external periodic potential. He utilized the averaging method and canonical perturbation to process the region of action for two-dimensional BEC and discussed the components in different regions [12]. Aveline et al. analyzed the generation of rubidium atom BEC bodies in cold atoms and observed that the microgravity environment is conducive to conducting cold atom experiments. It played an important role in the application of conventional BEC systems and the study of trap topology [13].

The application of nonlinear systems in quantum mechanics and communication has attracted the attention of most scholars. Ghasemian and Tavassoly analyzed the chaotic dynamics of the BEC system and the quantized radiation field and numerically solved the nonlinear dynamic equations using coupling strength and semi-classical methods. This method demonstrated the chaotic properties in quantum electrodynamics, which are always within a bounded region of the phase space, and modulation of frequency depth can enhance this chaotic behavior [14]. Scholars such as Sadgrove et al. proposed using momentum splitting for optical standing wave analysis of BEC bodies. They analyzed the average energy change of their quantum phase and analyzed the application of kicking atoms through experimental analysis [15]. Ruban numerically simulated the trapped BEC mixture under different mass cold atoms by coupling the G-P equation. He believed that parameter adjustment has a significant impact on the dynamic characteristics of eddy currents and has found a parameter value threshold that can achieve long lifespan in unstable systems [16]. Niu et al. analyzed the two-dimensional optical lattice BEC with rotational symmetry under the G-P equation using numerical calculations. In an external harmonic potential well, material wave interference exhibited properties that vary with the lattice period. Moreover, under different atomic interactions, BEC exhibited localization characteristics and exhibited different performance properties under different rotational modes [17]. Zhao and Zhao analyzed Shannon information entropy in rotating BEC using the mean field G-P equation. These results confirmed that increasing the depth of the optical lattice changes the momentum space and total information entropy of the system, and the critical point of the optical lattice is related to the rotational frequency [18]. Hartree–Fock–Bogoliubov-type coupled system had an interaction potential. Grillakis and Machedon analyzed its temporal dynamics using the conservation theorem of the system of equations. This can effectively achieve spatial estimation and provide a technical tool for approximating the quasi-free state evolution of boson systems [19]. Qu et al. analyzed the space pure neutron matter using the relativistic Brueckner–Hartree–Fock theory [20]. In comparison, the range results of this method and the multi-perturbation theory confirmed consistency, and the deformation calculation of neutron star material complied with standard constraints. Oztas used numerical analysis to localize the BEC system under the action conditions in the case of optical lattice coupling. These results confirmed that spin orbit coupling can effectively achieve localization, and the presence of localized states plays an important role in studying BEC systems [21]. Rasooli Berardehi et al. proposed a non-dynamic T-S fuzzy sliding mode control method to synchronize the unexpected behavior of fractional-order chaotic systems and reduce chattering. This method can achieve good synchronization in chaotic power systems of power grids and has high application effectiveness [22]. Roohi et al. utilized fuzzy control technology and Lyapunov stability theory to stabilize a four-dimensional chaotic fractional-order laser system. They proposed control strategies for finite time to reduce chaotic behavior of the system by replacing the sign function of the control input with a fractional derivative. They proposed control strategies for finite time to reduce chaotic behavior of the system by replacing the sign function of the control input with a fractional derivative. This improved method better reduced the system’s oscillations. The technology demonstrated high application efficiency and potential in two digital scenarios [23]. Roohi et al. achieved synchronization of chaotic delayed fractional neural network systems using non-dynamic adaptive sliding mode control. They also combined frequency distribution models with Lyapunov stability theory. The results indicated that this control method demonstrated high synchronization in neural network chaotic systems and had high application scalability [24]. Table 1 presents a comparison of research ideas from various literature sources.

Table 1

Analysis of the focus of different literature research ideas

Literature Method and approach Key research objects or issues Result
[6] Theoretical analysis of the G-P equation Anisotropic bright solitons in two-dimensional optical lattices The depth of the optical lattice and the initial distance of the kicked vortex bright soliton can have an impact on the fractional shift as well as stability
[7] Numerical simulations and linear coupled Schrödinger equations Equilibrium problems for linearly coupled binary BECs Rabi coupling of two-field components contributes to the study of atomic equilibrium problems
[8] Description of photomolecular transitions and their relation to the system in terms of coupled density equations Two-dimensional nonlinear BEC The condensed-phase energy of the BEC system changes and its decay is correlated with the coupled system
[9] Theoretical analysis, determination of critical conditions for system superfluidity Theoretical analysis of planar wave characteristics in defect lattice BEC Plane wave tuning can change the conditions in the superfluid region of the system, and the vibrational frequency and lattice strength can affect the propagation characteristics and dynamics of the BEC system
[10] Perform proportional partitioning correction on the linearity of the Bose–Hubbard model The problem of solving phase transition in BEC process This method plays an important role in the classification of BEC models and in ensuring the accuracy of the numerical model solution
[12] External periodic potential theory, averaging method, canonical perturbation Analysis of modulation amplitude waves in one-dimensional BEC systems and analysis of two-dimensional BEC effects Only the second component of a strong particle has a matter cycle
[13] Laboratory content analysis The production of rubidium atomic BEC The microgravity environment is favorable for cold atom experiments
[14] Numerical solution of nonlinear dynamic equations using coupling strength and semi-classical methods Chaotic dynamics of BEC systems and quantized radiation fields The chaotic nature in quantum electrodynamics is always within a bounded region of phase space and modulation of the frequency depth enhances this chaotic behavior
[15] Optical standing wave analysis in momentum splitting form, average energy analysis of quantum phase, experimental analysis BEC The average energy growth of the BEC is suppressed or enhanced at Talbot time
[16] Coupled G-P equation, numerical simulation Trapped BEC mixture under cold atoms The tuning of the parameters for different masses of cold atoms has an important effect on the dynamical properties of the vortex, and a threshold of parameter values is found for which long lifetimes of unsteady systems can be achieved
[17] Numerical calculation of G-P equation Two-dimensional optical lattice BEC In the outer simple harmonic potential well, matter–wave interference is associated with lattice period changes and the BEC is localized
[18] G-P equation for mean field Shannon information entropy in BEC An increase in the optical lattice depth changes the momentum space and the total information entropy of the system, and the critical point of the optical lattice is related to the rotation frequency
[19] The conservation theorem of a system of equations Hartree–Fock–Bogoliubov-type coupled system Technical tools for quasi-free-state evolution of approximate bosonic systems
[20] The Brueckner–Hartree–Fock theory of relativity Space pure neutron matter The deformation of neutron star matter is calculated to be consistent with standard constraints
[21] Numerical analysis BEC system under optical lattice coupling Spin–orbit coupling enables better localization
[22] Nondynamic T-S fuzzy sliding-mode control method Fractional chaotic systems The method can perform better synchronization in grid chaotic dynamical systems
[23] Fuzzy control technology replaces the sign function of the control input with a fractional derivative Four-dimensional chaotic fractional-order laser system The chattering situation of chaotic systems has been improved
[24] The combination of nondynamic adaptive sliding-mode control, frequency distribution model, and Lyapunov stability theory Synchronization of chaotic delayed fractional-order neural network systems This control method exhibits high synchronization in neural network chaotic systems

From the aforementioned analysis, as a special condensed structure, the BEC system has been analyzed by most scholars on the bright solitons, atomic equilibrium problems, decay situations, phase transition structures, etc. that exist in two-dimensional optical lattices. The coupling density equation, Bose–Hubbard model, G-P equation, Brueckner–Hartree–Fock theory, and other methods are often applied in analysis. Therefore, using mathematical models for dynamic analysis of BEC systems is effective. Based on this, research is conducted on BEC chaotic analysis based on the mean field theory. Starting from the issues of chaotic synchronization and anti-synchronization that have been rarely studied by scholars in the past, chaotic control research on BEC systems is conducted to better provide the reference value for theoretical research on BEC systems.

3 BEC chaotic control based on Hartree–Fock mean field theory

BEC system exhibits a complex dynamic behavior. Therefore, when studying the chaotic control problem of BEC systems, attention needs to be paid to how to use external control means or methods to control the chaotic behavior of BEC systems. This study utilizes mean field theory to study the mixing characteristics and control problems of BEC systems and proposes the use of constant shift to achieve the chaotic control of BEC. Subsequently, in response to the synchronization and anti-synchronization issues of BEC system control, activation control was proposed for analysis, and a system-driven response system with control terms was constructed. The study of BEC typically involves the G-P equation, which is a nonlinear Schrödinger equation that describes BEC. The G-P equation considers the interactions between particles and can describe the macroscopic quantum behavior of condensed matter. This equation predicts the evolution of the wave function in BEC over time, including the dynamic behavior under nonlinear effects. The constant shift method involves adding a control function to the G-P equation to affect the nonlinear terms in the equation. This, in turn, affects the dynamic behavior of the system, suppressing or controlling chaotic behavior, and stabilizing the system or achieving a specific dynamic state. Activation control analyzes the synchronization situation of chaotic problems and transforms nonlinear systems into linear systems by selecting appropriate control terms. State variables are then set for the driving system and response system, respectively. The G-P equation is used in the BEC system as a theoretical framework. The key parameters of the equation are adjusted through the constant shift method and activation control, which affect the dynamic behavior of the system. This allows for precise control of the BEC system dynamics and reduces instability and chaos in the BEC system.

3.1 Chaotic control characteristics of BEC based on mean field theory

As a Bose gas system, BEC faces certain difficulties in solving the nonlinear Schrödinger equation due to the interaction between a large number of bosons and the differences in their spatial dimensions. Moreover, the nonlinear Schrödinger equation requires the use of numerical calculations or approximation methods to simplify the process in order to solve high-dimensional and large-scale systems [25]. Hartree–Fock mean field theory is a widely used approximation method in quantum mechanics for constructing the initial state of quantum systems. It describes the average behavior of multi-body quantum systems by treating the action of the outer upper bound on the object as an average action. Essentially, it is a high-dimensional low-order transformation. In Hartree–Fock theory, the interaction between electrons is considered through the mean field approximation, which assumes that the electric field felt by each electron is generated by the average of other electrons. The use of optical lattices to trap BEC atoms increases the interaction between atoms, thereby providing convenience for observing the nonlinear dynamic behavior of BEC [26]. There is a common phenomenon of chaos in BEC quantum systems in nonlinear systems. Therefore, this study uses the dynamic expression of the mean field approximation method to describe the BEC system under optical lattices and analyze its chaotic characteristics. In the following equation, the dynamic equation of BEC system can be expressed using the G-P equation:

(1) i h ∂ Ψ ( r → , t ) ∂ t = − h 2 2 m ∇ 2 + V trap + g 0 ∣ Ψ ( r → , t ) ∣ 2 Ψ ( r → , t ) ,

where i represents the imaginary unit, m represents the mass of the bose atom, V trap is the outfield confinement potential, g 0 represents the coupling constant of the action potential, Ψ ( r → , t ) is the macroscopic wave function, t represents the time, and h is the Planck constant. BEC is mostly based on atomic potential. The proposed model mainly considers the situation where BEC atoms are placed in optical lattices, and the following equation can be used to perform the three-dimensional joint transformation of Eq. (1):

(2) V ( x , y , z , t ) = V 0 cos 2 ( k ξ ) + m ( ω x 2 x 2 + ω y 2 y 2 + ω z 2 z 2 ) / 2 ,

where m ( ω x 2 x 2 + ω y 2 y 2 + ω z 2 z 2 ) / 2 represents the harmonic oscillator potential under magnetic field confinement; ω x , ω y , and ω z are the frequencies in the corresponding dimension direction; x , y , and z represent the dimension directions of the atom; and k is the optical lattice wave vector. The acoustooptic modulation technology can adjust the motion speed of the optical lattice. When BEC is in the central part of the oscillator potential, the harmonic oscillator potential can be approximately ignored, and the trapping frequencies in three dimensions can be expressed in the following equation:

(3) ω x = 2 ω y = 2 ω z .

If the optical lattice potential is significantly greater than the harmonic oscillator potential, then the BEC system at this time can be approximated using a quasi-one-dimensional optical lattice. And due to the temperature effect and the incoherence of atomic exchange, multiple BECs will generate damping effects. Therefore, a quasi-one-dimensional G-P equation can be used to describe the system in the following equation:

(4) i h ( 1 − i γ ) ∂ Ψ ∂ t = − h 2 2 m ∂ 2 Ψ ∂ x 2 + g 0 ∣ Ψ ∣ 2 Ψ + V 0 cos 2 ( k ξ ) Ψ ,

where γ represents the damping effect, ξ represents the spatiotemporal variable, and k is the optical lattice wave vector.

To better analyze the parameters of the BEC system and the chaotic region, equation analysis was conducted using Denschlag parameters to analyze the power situation of the BEC system under different damping coefficients in Figure 1 [27].

Figure 1 Power spectrum of the BEC system with different damping coefficients: (a) damping coefficient is 0.05 and (b) damping coefficient is 0.35.
Figure 1

Power spectrum of the BEC system with different damping coefficients: (a) damping coefficient is 0.05 and (b) damping coefficient is 0.35.

If the damping coefficient in the BEC system increases, the power spectrum exhibited by the system exhibits an increase in wave peaks, indicating that the system is in a chaotic state. The adjustment of the BEC system using optical lattice can be achieved through parameter adjustment. When performing perturbation chaotic analysis, the common Melnikov function’s cross-sectional manifold can lead to intersection problems. And the variation of the damping coefficient value can affect the intensity of the optical lattice potential in chaotic situations. Therefore, Melnikov function is difficult to perform dynamic analysis well. Therefore, this study analyzed the chaotic behavior of BEC using numerical methods. Lyapunov index quantitative analysis has a significant effect on the instability of chaotic systems, which is judged by the degree of divergence or convergence of the average method of adjacent trajectories on the complex plane in the following equation:

(5) λ i = lim t → ∞ 1 t ln ∥ δ x i ( x 0 , t ) ∥ ∥ δ x ( x 0 , t ) ∥ ,

where ∥ δ x i ( x 0 , t ) ∥ represents the half-axis length of the coordinate axis, x 0 is the initial value, x is the trajectory, and δ is the deviation. If Lyapunov exponent is positive or negative, it indicates that adjacent orbits are in a divergent or stable state. If this exponent is zero, it indicates that the motion trajectory of the system is in a critical state. In practical applications, the determination of system state only requires solving for its maximum exponent to understand the chaotic situation. Chaotic systems typically have at least one positive Lyapunov exponent, indicating that the system’s orbit diverges over time in a specific direction and displays highly sensitive behavior to initial conditions. The motion of a chaotic system is deterministic in its current state, but unpredictable. It is important to note that this unpredictability is not due to randomness, but rather to the complexity of the system. Hyperchaotic systems exhibit higher-dimensional chaotic behavior, with multiple positive Lyapunov exponents in multiple directions, resulting in complex dynamic and nonlinear characteristics. The identification of chaotic and hyperchaotic behaviors can be achieved by the maximum Lyapunov exponent. Since BEC systems are composed of multiple quantum systems and influenced by various interactions, they have rich nonlinear characteristics. In BEC systems, achieving synchronization for chaotic systems is typically straightforward, as the system has sensitive dependencies in only one direction and can be stabilized through appropriate feedback control. However, for hyperchaotic systems, synchronization is more challenging due to the sensitive dependencies in multiple directions, which require more complex multidimensional control methods to handle. However, when simulating optical lattices, Lyapunov exponent is prone to intermittent chaotic states in the system under changes in lattice strength and damping coefficient, and its overall regional variation exhibits significant fluctuations. And the two-dimensional graphic bifurcation diagram can display the dynamic changes of the system, and each point corresponds to the stable state. If there are countless chaotic and similar positions of points in the bifurcation diagram, it indicates that the system is in a chaotic state. If the variable in the initial condition is 0, the step size is 0.0001, and the motion speed and damping coefficient are 2.05 and 0.06, respectively, then the output of BEC system is displayed in Figure 2.

Figure 2 Relationship between the output state of the BEC system and optical intensity and damping coefficient: (a) BEC output state under changes in lattice intensity and (b) BEC output state under changes in damping coefficient.
Figure 2

Relationship between the output state of the BEC system and optical intensity and damping coefficient: (a) BEC output state under changes in lattice intensity and (b) BEC output state under changes in damping coefficient.

The bifurcation diagram shows that the BEC system exhibits different periodic situations under different optical lattice strengths and damping coefficients, and there are sudden transitions and periodic situations when the system reaches a chaotic state [28,29,30,31]. Under the approximation of mean field theory, the dynamics of BEC systems in one-dimensional optical lattices can be expressed in the following equation:

(6) i h ∂ Ψ ( x , t ) ∂ t = − h 2 2 m ∂ 2 Ψ ( x , t ) ∂ x 2 + [ V trab ( x ) + g 0 ′ ∣ Ψ ( x , t ) ∣ 2 ] Ψ ( x , t ) V trab ( x ) = V 1 cos ( k 1 x ) + V 2 ( k 2 x ) + F x ,

where g 0 ′ represents the intensity of atomic interaction; ∥ δ x i ( x 0 , t ) ∥ represents the wave function; V trab is the optical lattice potential well; V 1 and V 2 represent the amplitudes of the optical lattice; k 1 and k 2 are the wave vectors of laser beam; and F x is the inclination and inclination factor of the optical lattice. The problem can also be simplified using wave functions containing perturbations to obtain the following equation:

(7) Ψ ( x , t ) = ϕ ( x ) e − i μ t h ,

where μ represents the chemical potential, ϕ ( x ) is the normalized wave function, and e is the real number. By substituting Eq. (7) into Eq. (6), the partial differential equation can be transformed into an ordinary differential equation, resulting in the following equation:

(8) d ϕ ( x ) d x = y ( x ) , d ϕ ' ( x ) d x = [ A 1 cos ( k 1 x ) + A 2 ( k 2 x ) + B x − L ] ϕ ( x ) + γ ' ϕ 3 ( x ) ,

where A 1 and A 2 represent the strengths of the lattice potential, B is the inclination of the lattice, L is related to chemical potential, and γ ' is the interaction factor.

3.2 Chaotic control and synchronization of BEC systems

BEC, as a quantum phenomenon, is a nonlinear dynamic system that, due to its unique characteristics, makes it difficult to determine the initial value and predict long-term processes when analyzing its physical processes. The dual forms of chaotic control information make it have two aspects: control and anti-control. In BEC systems, chaotic and anti-chaotic phenomena are important for studying the properties of condensed matter. Chaos refers to the behavior of a system that is extremely sensitive to small changes in initial conditions, resulting in unpredictable results that can be induced by changing external conditions or perturbing the system. Anti-chaos refers to the ability of a system’s behavior to resist perturbations, which can be achieved by applying external correction forces. The study of chaotic and anti-chaotic phenomena contributes to the analysis of the dynamics of BEC and the stability of the system, and targeted control of chaotic systems needs to be based on their own characteristics. In the previous analysis, adjusting the parameters of the optical lattice can, to some extent, affect the state of the BEC system. Therefore, when analyzing the chaos of the BEC system using optical lattice, the control strategy design should mainly focus on the properties of the optical lattice. In any existing nonlinear system, controllability is mainly achieved through feedback, which can be expressed in the following equation:

(9) Y ̇ = f ( Y , t ) + U ( Y , t ) ,

where Y represents the system state vector, f is the nonlinear function vector, and U is the control item. Considering the BEC system in an optical lattice, and when the spatiotemporal variable and phase satisfy a certain linear relationship, its state equation can be transformed into the following equation:

(10) d y 1 d η = y 2 d y 2 d η = 0.25 v 2 y 1 + g y 1 3 + I 0 cos 2 ( η ) y 1 − γ v y 2 ,

where η represents the spatiotemporal variable, and y 1 and y 2 are the state variables. The BEC-state equation after being controlled can be expressed in the following equation:

(11) d y 1 d η = y 2 + u 1 , d y 2 d η = 0.25 v 2 y 1 + g y 1 3 + I 0 cos 2 ( η ) y 1 − γ v y 2 + u 2 ,

where u 1 and u 2 are the control functions, g represents the interaction strength, and I 0 represents the optical lattice potential strength. The chaotic control of the BEC system by constant shift is similar to the control of particle density by applying an external field, and it has operability in experiments. In lattice analysis, by adding a parameter q to the migration method, the following equation can be obtained, where u 1 and u 2 in the control function are q1 and 0, respectively:

(12) d y 1 d η = y 2 + q , d y 2 d η = 0.25 v 2 y 1 + g y 1 3 + I 0 cos 2 ( η ) y 1 − γ v y 2 .

Using the fourth-order method to numerically calculate the equation, the maximum Lyapunov exponent can be solved. Numerical methods often rely on ordinary differential equations or partial differential equations while solving. In complex systems, an increase in algorithm order means an exponential increase in computational complexity. This study utilizes the fourth-order RK algorithm for calculation in the following equation:

(13) Z n + 1 = z n + 1 6 τ [ f ( Z 1 ) + 2 f ( Z 2 ) + 2 f ( Z 3 ) + f ( Z 4 ) ] ,

where n represents the number of orders. In most cases, suppressing chaos can effectively ensure the stability of the system. However, in secure communication, chaotic sequences themselves have practical value and cannot be blindly suppressed. Chaotic synchronization is an important aspect of nonlinear control research, and previous scholars have rarely analyzed the synchronization and anti-synchronization issues of chaotic problems. The study proposes the use of activation control method to achieve the synchronization of chaotic problems. This method is based on linear system stability theory and achieves the transformation of nonlinear systems by selecting appropriate control terms. Due to the extreme sensitivity of chaotic systems to initial values, the motion trajectories of two chaotic systems cannot achieve synchronous changes when they have different initial conditions. The problem of chaotic synchronization and anti-synchronization of BEC systems through activation control is mainly based on the stability theory of linear systems. By selecting appropriate control terms, the nonlinear system can be transformed into a linear system to achieve synchronization. State variables for the driving and response systems should be set separately. The system’s dynamic behavior should be described using differential equations, and reasonable control functions should be established to ensure synchronization and anti-synchronization of the chaotic system.

If the state variables representing the driving system and the corresponding system are set to y 1 , y 2 , x 1 , x 2 , the dynamic behavior of both can be represented using Eqs. (10) and (11) in the previous text. By using appropriate control methods, the BEC system can achieve synchronization and anti-synchronization.

When there is a difference in the initial conditions of two chaotic systems, their motion trajectories will not undergo synchronous changes, and the response system will track the driving system under the coupling term until it reaches a state consistent with its signal. This study mainly analyzes the complete synchronization of BEC systems with the same structure [32,33]. When the chaotic anti-synchronization situation is that the sum of the corresponding variables of the system approaches 0, the error variable of the drive–response system can be expressed in the following equation:

(14) e 1 = x 1 + y 1 , e 2 = x 2 + y 2 ,

where e 1 and e 2 represent the state variables in the error system. If the appropriate control term can make the error system exhibit a stable state, then the eigenvalues related to the matrix in the drive–response problem are negative or negative real parts. In this case, these two chaotic systems need to satisfy the following equation when their initial values are different:

(15) lim η → ∞ ∥ e i ( η ) ∥ = lim η → ∞ ∥ x i + y i ∥ = 0 , i = 1 , 2 .

If the activation control function is defined as U = [ u 1 , u 2 ] T , the error system can be expressed in the following equation:

(16) e ̇ 1 = e 2 + V 1 , e ̇ 2 = − [ 0.25 v 2 + I 0 cos 2 ( η ) ] − e 1 − v r e 2 + V 2 .

Eq. (16) transforms the problem of error system variables into a control problem. By defining the input part of this system as a control term related to the error variable, the transformation of the nonlinear system can be achieved. The error value can converge in the region under the adjustment of the feedback control term, which can achieve the anti-synchronization of this system. By selecting a function with control input as the error variable and introducing a matrix of 2 × 2, the equation obtained is as follows:

(17) V 1 V 2 = − 1 − 1 − [ 0.25 v 2 + I 0 cos 2 ( η ) ] v γ − 1 e 1 e 2 .

The error system at this point can be expressed in the following equation:

(18) e ̇ 1 = − e 1 , e ̇ 2 = − e 2 .

Eq. (18) shows that when the error variable approaches zero, the two BEC systems can achieve chaotic anti-synchronization. It is important to note that the principle of chaotic synchronization and anti-synchronization is the same, but if the direction is not consistent, the error variable of the drive–response system under synchronization can be defined as shown in the following equation:

(19) e 1 ′ = x 1 − y 1 , e 2 ′ = x 2 − y 2 .

If Eq. (19) satisfies the conditions of Eqs. (15–18), it indicates that the two chaotic BEC systems have achieved synchronization. The adjustment of optical lattice parameters can affect the state of the BEC system. “Offset parameters” mainly adjust system parameters, such as external potential energy and particle interactions, by adding parameters. This suppresses chaotic behavior of the system and stabilizes its orbit. The inclusion of parameters can effectively regulate the behavior of BEC chaotic systems, preventing transient chaos and facilitating the transition from chaotic to periodic operation. This control mechanism remains effective even when the initial state of the system is different. The analysis of the system’s drive–response is primarily based on linear system theory, with appropriate control terms selected to achieve synchronization and anti-synchronization of the system. If there is a difference in the initial conditions of two chaotic systems, their motion trajectories will not undergo synchronous changes. The response system, with a specific control term, will track the driving system under the coupling term until it reaches a state consistent with its signal. The control of state variables can guide the system trajectory, ensuring synchronization regardless of the starting point. The feedback control term can adjust the error value of state variables to make the region converge. Anti-synchronization can be achieved when the error variable of the drive–response system is minimized. The precise control of the BEC system can be achieved through the comprehensive application of offset parameters and activation control. This enhances the system’s robustness to external disturbances and provides the possibility of synchronous or anti-synchronous control state.

4 Application analysis of chaotic control problem in BEC system

Chaos refers to the nonperiodic and highly sensitive behavior of a dynamic system, and chaotic motion represents the motion whose trajectories never coincide. Due to its system variables being less constrained by other factors during the change process, it has significant uncertainty and is difficult to predict. At the same time, chaotic phenomena are relatively complex, exhibiting characteristics such as extreme sensitivity to initial values, randomness, irregular orderliness, and boundedness. This research aims to examine and analyze the chaotic characteristics of the BEC system’s chaotic control problem. It will also test the control performance and investigate synchronization and anti-synchronization issues. The findings will provide valuable references for the application prospects of the BEC system. Due to the nonlinear dynamic characteristics exhibited by the BEC system, based on the optical lattice characteristics, the chaotic motion behavior of the system is analyzed through parameter changes using MATLAB numerical analysis tools. Quantitative analysis was conducted to explore the chaotic control effect of BEC systems. The analysis used indicators such as time series diagrams, Lyapunov exponents, and lattice shifts to demonstrate the dynamic behavior of the system and the effectiveness of chaotic control. The time series diagram is a graphical representation of the system state over time, which reflects the changes in particle density and phase distribution in the BEC system. The system’s behavior can be observed to determine whether it exhibits chaotic behavior, characterized by irregular and unpredictable fluctuations, or more ordered and predictable dynamics, characterized by periodic fluctuations or steady states. The maximum Lyapunov exponent is used to describe the speed of separation between adjacent trajectories in the system. Based on the Lyapunov stability theory, chaotic control can be achieved by selecting different values for the control parameter, as long as the corresponding maximum Lyapunov exponent is less than zero. This can be observed from the Lie exponent diagram and bifurcation diagram. In regional systems, achieving a periodic control state can be described. Lattice displacement is frequently caused by the adjustment of lattice parameters. It can aid in comprehending the chaotic behavior and control effect of BEC systems. First, the parameter conditions of the system equation were set. The driving state variables (y1 and y2) under the initial conditions are 0. Other parameters such as motion speed, quantum mass, lattice intensity, and damping coefficient are 2.05, −0.50, 5.5, and 0.05, respectively. At this point, Figure 3 shows the BEC system status.

Figure 3 Chaotic region diagram of the intensity, wavelength vector, and motion velocity of the optical lattice BEC system: (a) the relationship between optical intensity and lattice wavelength vector and (b) the relationship between optical intensity and lattice motion velocity.
Figure 3

Chaotic region diagram of the intensity, wavelength vector, and motion velocity of the optical lattice BEC system: (a) the relationship between optical intensity and lattice wavelength vector and (b) the relationship between optical intensity and lattice motion velocity.

Figure 3 shows the relationship between the atomic number density and the damping coefficient. In Figure 3(a), the area enclosed above the curve with a larger damping coefficient is smaller than that of the curve with a smaller damping coefficient as the wavelength vector of the light increases and the lattice motion speed remains constant. This suggests a decrease in the chaotic region and an increase in the periodic region. In Figure 3(b), as the velocity of lattice motion increases, the curve with a higher damping coefficient approaches the horizontal axis, and the chaotic state of the lattice intensity requires a higher incident light wavelength and motion velocity. When the velocity of the light grid motion is below the threshold, the system can be in a chaotic state. The constant offset method stabilizes the operating state of the BEC system by adding a control function to its chaotic control. Similarly, the constant shift method controls the particle density of the BEC system by introducing an external field. It is important to note that without external signal control, the system will exhibit chaotic behavior, as evidenced by its attractor, time series, and power spectrum. The criterion for determining whether a system is in a chaotic state is the maximum Lyapunov exponent. If the value is less than zero with the help of constant shift, it indicates that the system is in a stable periodic orbit. In the BEC system of an optical lattice, the chaotic behavior of the system can be affected by the depth, spacing, geometric layout, or intensity of particle interactions of the lattice. Subsequently, the chaotic situation of BEC system before and after the application of the research method is shown in Figure 4.

Figure 4 Chaos of the BEC system before and after parameter adjustment: (a) before application and (b) after application.
Figure 4

Chaos of the BEC system before and after parameter adjustment: (a) before application and (b) after application.

In Figure 4, the chaotic effect of the BEC system without using the constant shift method is more obvious, and there is no obvious periodic motion pattern in its time series diagram. After adjusting the parameters, the BEC system experienced a state transition of Qin Guang, and the system was in a motion state of cycle 1. The change in coefficient can effectively affect the state of BEC system. As mentioned earlier, Lyapunov index can be used to determine the chaotic state of the system. This study first tests the effectiveness of this index in making BEC system judgments in Figure 5.

Figure 5 BEC system with exponential changes in different light grid intensities: (a) Melnikov function and (b) Lyapunov exponent.
Figure 5

BEC system with exponential changes in different light grid intensities: (a) Melnikov function and (b) Lyapunov exponent.

In Figure 5, Lyapunov exponent is less than 0 in the range of light lattice intensity between [0, 0.7], [1.3, 1.5], [2, 3], and [4, 5], respectively. The intensity change of its optical lattice causes continuous chaos in the BEC system. When using Lyapunov exponent for system analysis, its chaotic state is mainly concentrated between [3.4, 4.5], demonstrating good system stability. Subsequently, numerical calculations were conducted on the dynamic equations of the BEC system using RK4 to obtain the maximum Lyapunov exponent shift in Figure 6.

Figure 6 Maximum Lyapunov exponential shift change in the BEC system: (a) change in the maximum Lyapunov index, (b) citation diagram of the BEC system, and (c) time series plot of the BEC system.
Figure 6

Maximum Lyapunov exponential shift change in the BEC system: (a) change in the maximum Lyapunov index, (b) citation diagram of the BEC system, and (c) time series plot of the BEC system.

Figure 6 shows the variation of the maximum Lyapunov exponent, the attractor of the plane, and the time series, respectively. When the range of the offset constant is [4.21, 5.67], the value of the maximum Lyapunov exponent is lower than 0, indicating that the system is in a chaotic state and the overall change is relatively stable. When the constant is 3.6, the period of the BEC system stabilizes at 1, which can result in a periodic motion state. Figure 7 shows the BEC system with added control functions.

Figure 7 Maximum Lyapunov exponential shift of the BEC system with increasing control function: (a) variation of the maximum Lyapunov exponent with control function, (b) lemma plot of the BEC system under the control function, and (c) time series plot of the BEC system under the control function.
Figure 7

Maximum Lyapunov exponential shift of the BEC system with increasing control function: (a) variation of the maximum Lyapunov exponent with control function, (b) lemma plot of the BEC system under the control function, and (c) time series plot of the BEC system under the control function.

In Figure 7, the maximum Lyapunov exponent plot reflects that when the offset constant value is in the later stage, its value is basically less than 0, indicating a relatively stable periodic operation. And when the step size is 0.01, the system period under the maximum Lyapunov exponent is 2, which can achieve the effect of chaotic control. In the case of constant deviation, the control effect of the system is more abundant, which can effectively provide theoretical support for the research of system control problems. The chaotic situation of the lattice under the proposed offset method was analyzed, and the difference between k1 and k2 in the system parameters was set to 3.147. Subsequently, the spatial orbit results of chaotic control are shown in Figure 8.

Figure 8 Chaotic control of the BEC system with different offsets: (a) offset is 5, (b) offset is 10, (c) offset is 15, and (d) offset is 20.
Figure 8

Chaotic control of the BEC system with different offsets: (a) offset is 5, (b) offset is 10, (c) offset is 15, and (d) offset is 20.

In Figure 8, the increase in offset significantly suppresses the curve of the phase space orbit, and the larger the offset, the more obvious the chaotic characteristics of the control system. As the tilt of the lattice increases, its offset needs to be correspondingly increased to achieve the chaotic control of the system. Subsequently, the step size was set to 0.005, and the chaotic synchronization process of the system was analyzed using the RK4 method. The initial conditions for the parameters of these two BEC systems were determined to be the same as above, with variables y1, x1 and y2, x2 being −4, 1, and 2, −7, respectively, for the driving and response systems. Figure 9 shows the system variable situation.

Figure 9 Time series of drive–response system variables without activation control: (a) variation of variables y1 and x1 under the drive–response system and (b) variation of variables y2 and x2 under the drive–response system.
Figure 9

Time series of drive–response system variables without activation control: (a) variation of variables y1 and x1 under the drive–response system and (b) variation of variables y2 and x2 under the drive–response system.

In Figure 9, when the initial value in the error variable is set, there is a significant asynchronous situation in the time series of BEC system without activation control. Specifically, there is a varying degree of mismatch between the variables of the driving system and the variables of the response system. The overall deviation of Y1 and X1 is greater than 1.25%, and the maximum deviation reaches the opposite value between the spatiotemporal variables [20,28]. When the numerical range of Y2 and X2 is less than 20, the fluctuations are more pronounced, with a numerical range of ± 8. After the range exceeds 20, there is occasional synchronization between the two, but the overall average error still varies by 1.65%. Figure 10 shows the state variables of both systems.

Figure 10 Visualization results of the BEC system state variables without activation control: (a) driving system and (b) response system.
Figure 10

Visualization results of the BEC system state variables without activation control: (a) driving system and (b) response system.

In Figure 10, the system variables without activation control exhibit a chaotic state, and there is no synchronization of the variables. Figure 11 shows the time series of the system under activation control.

Figure 11 Time series of the drive–response system variables under activation control: (a) variation of variables y1 and x1 under the drive–response system and (b) variation of variables y2 and x2 under the drive–response system.
Figure 11

Time series of the drive–response system variables under activation control: (a) variation of variables y1 and x1 under the drive–response system and (b) variation of variables y2 and x2 under the drive–response system.

In Figure 11, the addition of activation control causes the time series of the system to exhibit anti-synchronization with changes in spatiotemporal variables. Unlike the randomness in Figure 11, the regularity between the overall system variables is more prominent. Subsequently, the synchronization and anti-synchronization evolution of the system under activation control are shown in Figure 12.

Figure 12 Chaotic synchronization and anti-synchronization of the BEC system under activation control: (a) synchronization of state variables in drive systems, (b) synchronization of state variables in response systems, (c) desynchronization of state variables in drive systems, and (d) state variable desynchronization in response systems.
Figure 12

Chaotic synchronization and anti-synchronization of the BEC system under activation control: (a) synchronization of state variables in drive systems, (b) synchronization of state variables in response systems, (c) desynchronization of state variables in drive systems, and (d) state variable desynchronization in response systems.

In Figure 12, the addition of control items causes the system to synchronize over time. The synchronization error eventually approaches zero. And under the control of the activation function, the two BEC systems can achieve anti-synchronization. The state variables in the error system exhibit different evolutionary trends in two ways, and activation control can effectively achieve variable synchronization. Figure 13 shows the error variables with and without activation control.

Figure 13 Error variables with or without activation control: (a) changes in state variables in error systems under activation-free control, (b) anti-synchronous evolution of state variables in error systems under activation control, and (c) synchronous evolution of state variables in error systems under activation control.
Figure 13

Error variables with or without activation control: (a) changes in state variables in error systems under activation-free control, (b) anti-synchronous evolution of state variables in error systems under activation control, and (c) synchronous evolution of state variables in error systems under activation control.

When studying the chaotic problem of two BEC systems, the values of their error variables are not the same. By analyzing the error system with or without activation control, it is found that activation control can effectively achieve synchronization and anti-synchronization of the system. This is manifested as the poor difference in curve direction between e1 and e2 in Figure 13(a) under different spatiotemporal variable values, with e1 varying within ±6 and e2 between [−12, 10]. The error variable curves reflected in Figure 13(b) and (c) tend to converge when the spatiotemporal variables are 8 and 6. In chaotic dynamics, activation control typically involves selectively activating or modulating system parameters through external signals or internal mechanisms to guide system behavior toward the desired dynamic state. In the problem of chaotic synchronization or anti-synchronization of BEC systems, spatiotemporal variables refer to the spatial distribution of time and control parameters. The values of these variables are related to the effectiveness and stability of control strategies. When the error results tend to balance or converge to a lower value, it indicates that the strategy can effectively enable the BEC system to reach a state of synchronization or anti-synchronization. This means that multiple subsystems in the system can achieve more coordinated dynamic behavior, improving the controllability of the quantum system. This research provides theoretical implications for understanding the chaotic control of BEC systems. It demonstrates that activation control can effectively handle synchronization and anti-synchronization issues of system chaotic control. It explains the inherent mechanism of chaotic behavior and the necessary control conditions. This research expands the theoretical content of nonlinear quantum system dynamics knowledge and corresponding control methods. Subsequently, the system time changes under synchronous and asynchronous conditions are shown in Figure 14.

Figure 14 Time variation of the BEC system under chaotic synchronization and anti-synchronization conditions: (a) desynchronization–e1–η, (b) desynchronization–e2–η, (c) synchronization–e1–η, and (d) synchronization–e2–η.
Figure 14

Time variation of the BEC system under chaotic synchronization and anti-synchronization conditions: (a) desynchronization–e1–η, (b) desynchronization–e2–η, (c) synchronization–e1–η, and (d) synchronization–e2–η.

In Figure 14, under synchronous control, the state variable of the error system gradually converges with the increase of the spatiotemporal variable value. The positive and negative values of the variable value are related to the variable value. When the state variable is positive, the error value finally tends to 0 when the spatiotemporal variable is 3, and the system is in a chaotic state. And in the anti-synchronization results, the initial value of the error variable will not affect the final chaotic situation of the system. The aforementioned results indicate that using activation control to process chaos in the system results in good synchronization and anti-synchronization results.

5 Conclusion

The study of chaos theory has important value and significance for understanding the material world, and the differences in control objects and objectives lead to differences in the methods used for chaotic control. This study conducted chaotic analysis of BEC systems using mean field theory and tested their effectiveness through numerical simulations. These results confirm that the intensity of the light grid is chaotic and requires a higher length and velocity of the incident light wave. The system is in a chaotic state when the velocity of the light grid is less than the threshold. The chaotic behavior of the BEC system can be effectively suppressed using the constant offset method. When the offset constant range is [4.21, 5.67], the maximum Lyapunov exponent value is less than 0. When the constant is 3.6, the BEC system exhibits periodic motion. The variables of the BEC system under the driving and response systems have varying degrees of mismatch. The overall deviation of Y1 and X1 is greater than 1.25%, and the maximum deviation occurs between the spatiotemporal variables [20,25]. The inclusion of activation control results in the time series of the system displaying anti-synchronization with the variation of spatiotemporal variables. The synchronization errors approach zero, and the state variables of the error system gradually converge with the increase of spatiotemporal variable values. The addition of control terms makes the system tend to synchronize and anti-synchronize with the evolution of time, and the error variable curve tends to converge when the spatiotemporal variables are 8 and 6. These aforementioned results confirm that the constant shift method can effectively analyze the chaotic situation of the optical lattice BEC system, and it can achieve the modulation of BEC system according to actual needs. And activation control can effectively analyze the chaotic synchronization problem of the BEC system. BEC, a “macroscopic” quantum system, can be described using coherent wave functions, and its quantum properties can be demonstrated at large scales. Therefore, it has a broad application value in technical and applied sciences, and its coherence properties can be used in the research of atomic lasers for generation and amplification. The BEC system is a nonlinear quantum system. Studying its chaotic dynamics can provide a deeper understanding of quantum computing and simulation. Studying the chaotic behavior and synchronization problem of BEC systems based on the framework of mean-field theory can effectively provide new technical approaches and theoretical foundations for controlling similar chaotic behavior in other quantum systems, improve the accuracy of quantum system operation, and achieve effective control of quantum system dynamic behavior by precisely adjusting system parameters. Controlling chaotic behavior in BEC systems can provide new approaches for quantum information encoding, transmission, and processing, particularly in the fields of quantum networks and communication. The research also demonstrates the application of theoretical physics in solving practical problems, providing new directions for experimental design, and promoting the development of experimental physics. Not only can it expand the application of nonlinear system theory in the quantum field, but it may also provide theoretical guidance and new perspectives for the control of chaotic phenomena in other complex systems (such as biological systems and climate systems). The challenges encountered in analyzing and controlling the dynamics of nonlinear quantum systems and BEC systems are mainly due to dynamic complexity, quantum effects, and technical limitations. Nonlinear effects in quantum systems make the system highly sensitive to initial conditions, making long-term behavior difficult to predict. Additionally, nonlinear systems often exhibit chaotic behavior, and small differences in initial values can lead to different behaviors. In addition, the fundamental properties of quantum entanglement and coherence render the BEC chaotic system unstable. It is crucial to address quantum-classical boundary problems in the macroscopic analysis of BEC, and phenomena such as superfluidity and phase transition can further complicate system control. In BEC systems, atomic behavior undergoes significant changes near absolute zero, and temperature affects the dynamic analysis of the system under extreme conditions. Theoretical models cannot fully capture all the complexities of quantum systems, especially in situations where nonlinear characteristics and chaotic effects are significant. The effective understanding and control of quantum systems and BEC systems require the comprehensive application of interdisciplinary theoretical knowledge and innovative experimental methods. Current research on chaotic problems primarily relies on numerical theoretical methods for description, which may be considered relatively outdated. The Lyapunov function has limitations when transforming chaotic synchronization problems. These include the need for sufficient understanding of the system in practical applications, the need to deal with existence conditions, and the initial analysis process of Lyapunov function in chaotic systems. It is difficult to provide globally stable information on its long-term behavior, and there are difficulties in solving complex differential equations or system dynamics, especially in high-dimensional systems containing multiple state variables and parameters. The study proposes chaotic control and synchronization solutions while considering perturbations. However, controlling the effects of large noise or disturbances remains a challenge, and addressing parameter uncertainty is a major challenge for the future. Additionally, it is important to consider the practical application and generalization of chaotic control effects, as well as the dimensionality of BEC systems, in future research.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: Yang Shen put forward the research experiment: difficulty of studying nonlinear quantum systems and the unique composition of BEC systems. Meng Xu analyzed the data and helped with the constructive discussion. Yang Shen and Meng Xu made great contributions to manuscript preparation. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on a reasonable request.

References

[1] Meng Z, Wang L, Han W, Liu F, Wen K, Gao C, et al. Atomic Bose-Einstein condensate in twisted-bilayer optical lattices. Nat. 2023;615(7951):231–6.10.1038/s41586-023-05695-4Search in Google Scholar PubMed

[2] Yao KX, Zhang Z, Chin C. Domain-wall dynamics in Bose-Einstein condensates with synthetic gauge fields. Nat. 2022;602(7895):68–72.10.1038/s41586-021-04250-3Search in Google Scholar PubMed

[3] Kun QZ, Zi FY, Ji MG, Ai XZ, Hong HPX, Xue JK. Periodically modulated interaction effect on transport of Bose-Einstein condensates in lattice with local defects. Chin Phys B. 2019;28(1):245–50.Search in Google Scholar

[4] Koh YW. Correspondence between excited energy eigenstates and local minima of the energy landscape in quantum spin systems. Phys Rev B. 2023;107(22):224203.10.1103/PhysRevB.107.224203Search in Google Scholar

[5] Unnikrishnan CS. Bose-Einstein condensates as universal quantum matter. Sci Cult. 2019;85(1-2):35–43.Search in Google Scholar

[6] Tan Z, Gong H, Zhu B, Zhong HH, Hu SF. Two-dimensional anisotropic vortex-bright soliton and its dynamics in dipolar Bose-Einstein condensates in optical lattice. Nonlinear Dyn. 2023;111(10):9467–76.10.1007/s11071-023-08335-8Search in Google Scholar

[7] dos Santos MCP, Cardoso WB. Spontaneous symmetry breaking induced by interaction in linearly coupled binary Bose-Einstein condensates. Nonlinear Dyn. 2023;111(4):3653–64.10.1007/s11071-022-07986-3Search in Google Scholar

[8] Wang HF, Zhang JJ, Zhang JJ. Classical-field description of Bose-Einstein condensation of parallel light in a nonlinear optical cavity. Chin Phys B. 2021;30(11):110301.10.1088/1674-1056/abf4f9Search in Google Scholar

[9] Zhu KQ, Yu ZF, Gao JM, Zhang AX, Xu HP, Xue JK. Periodically modulated interaction effect on transport of Bose-Einstein condensates in lattice with local defects. Chin Phys B. 2019;28(1):266–72.10.1088/1674-1056/28/1/010307Search in Google Scholar

[10] Znojil M. Bose-Einstein condensation processes with nontrivial geometric multiplicities realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks. Quantum Rep. 2021;3(3):517–33.10.3390/quantum3030034Search in Google Scholar

[11] Wen XL, Xiong QH. Bose-Einstein condensation of exciton polariton in perovskite semiconductors. Front Optoelectron. 2020;13(3):193–5.10.1007/s12200-020-1086-zSearch in Google Scholar PubMed PubMed Central

[12] Liu Q, Torres PJ, Xing M. Modulated amplitude waves with non-trivial phase of multi-component Bose-Einstein condensates in optical lattices. IMA J Appl Math. 2019;84(1):145–70.10.1093/imamat/hxy053Search in Google Scholar

[13] Aveline DC, Williams JR, Elliott ER, Dutenhoffer C, Thompson RJ. Observation of Bose-Einstein condensates in an Earth-orbiting research lab. Nat. 2020;582(7811):193–7.10.1038/s41586-020-2346-1Search in Google Scholar PubMed

[14] Ghasemian E, Tavassoly MK. Chaotic dynamics of an atomic Bose-Einstein condensate in a frequency-modulated cavity QED. J Opt Soc Am B. 2021;38(11):3385–94.10.1364/JOSAB.437963Search in Google Scholar

[15] Sadgrove M, Horikoshi M, Sekimura T, Nakagawa K. Coherent control of ballistic energy growth for a kicked Bose-Einstein condensate. Eur Phys J D. 2021;45(2):229–34.10.1140/epjd/e2007-00277-3Search in Google Scholar

[16] Ruban VP. Systems of vortices in a Binary Core-Shell Bose-Einstein condensate. JETP Lett. 2022;116(5):329–34.10.1134/S0021364022601579Search in Google Scholar

[17] Niu ZX, Tai YH, Shi JS, Zhang W. Bose-Einstein condensates in an eightfold symmetric optical lattice. Chin Phys B. 2020;29(5):56103.10.1088/1674-1056/ab8211Search in Google Scholar

[18] Zhao Q, Zhao J. Optical lattice effects on Shannon information entropy in rotating Bose-Einstein condensates. J Low Temp Phys. 2019;194(1-4):302–11.10.1007/s10909-018-2099-5Search in Google Scholar

[19] Grillakis M, Machedon M. Uniform in N estimates for a Bosonic system of Hartree-Fock-Bogoliubov type. Commun Part Differ Eq. 2019;44(12):1431–65.10.1080/03605302.2019.1645696Search in Google Scholar

[20] Qu X, Tong H, Wang C, Wang S. Neutron matter properties from relativistic Brueckner-Hartree-Fock theory in the full Dirac space. Sci China Phys. 2023;66(4):8.10.1007/s11433-022-2048-3Search in Google Scholar

[21] Oztas Z. Spin orbit coupled Bose Einstein condensate in a two-dimensional bichromatic optical lattice. Phys Lett A. 2019;383(6):504–8.10.1016/j.physleta.2018.11.022Search in Google Scholar

[22] Rasooli Berardehi Z, Zhang C, Taheri M, Roohi M, Khooban MH. A fuzzy control strategy to synchronize fractional-order nonlinear systems including input saturation. Int J Intell Syst. 2023;2023:1550256.10.1155/2023/1550256Search in Google Scholar

[23] Roohi M, Mirzajani S, Basse-O’Connor A. A no-chatter single-input finite-time PID sliding mode control technique for stabilization of a class of 4D chaotic fractional-order laser systems. Math. 2023;11(21):4463.10.3390/math11214463Search in Google Scholar

[24] Roohi M, Zhang C, Taheri M, Basse-O’Connor A. Synchronization of fractional-order delayed neural networks using dynamic-free adaptive sliding mode control. Fractal. 2023;7(9):682.10.3390/fractalfract7090682Search in Google Scholar

[25] Cancès E, Kemlin G, Levitt A. Convergence analysis of direct minimization and self-consistent iterations. Siam J Matrix Anal A. 2021;42(1):243–74.10.1137/20M1332864Search in Google Scholar

[26] Drossel B. What condensed matter physics and statistical physics teach us about the limits of unitary time evolution. Quantum Stud-Math Fd. 2020;7:217–31.10.1007/s40509-019-00208-3Search in Google Scholar

[27] Carlen EA, Fellner K, Gallagher I, Emmanuel P. Classical and quantum mechanical models of many-particle systems. Oberwolfach Rep. 2021;17(4):1857–902.10.4171/OWR/2020/38Search in Google Scholar

[28] Zhang X, Pan W, Scattolini R, Yu S, Xu X. Robust tube-based model predictive control with Koopman operators. Automatica. 2022;137:110114.10.1016/j.automatica.2021.110114Search in Google Scholar

[29] Shi Y, Lan Q, Lan X, Wu J, Yang T, Wang B. Robust optimization design of a flying wing using adjoint and uncertainty-based aerodynamic optimization approach. Struct Multidiscip Optim. 2023;66(5):110.10.1007/s00158-023-03559-zSearch in Google Scholar

[30] Zheng B, Lin D, Qi S, Hu Y, Jin Y, Chen Q, et al. Turbulent skin-friction drag reduction by annular dielectric barrier discharge plasma actuator. Phys Fluids. 2023;35(12):125129.10.1063/5.0172381Search in Google Scholar

[31] Goerigk L, Casanova-Paéz M. The trip to the density functional theory zoo continues: making a case for time-dependent double hybrids for excited-state problems. Aust J Chem. 2020;74(1):3–15.10.1071/CH20093Search in Google Scholar

[32] Waziri TA, Yakasai BM. Assessment of some proposed replacement models involving moderate fix-up. J Comput Cogn Eng. 2023;2(1):28–37.10.47852/bonviewJCCE2202150Search in Google Scholar

[33] Maihulla AS, Yusuf I, Bala SI. Reliability and performance analysis of a series-parallel system using Gumbel-Hougaard family copula. J Comput Cogn Eng. 2022;1(2):74–82.10.1504/IJMOR.2023.128637Search in Google Scholar
Received: 2023-10-10
Revised: 2024-01-09
Accepted: 2024-01-22
Published Online: 2024-03-01

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  33. Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
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  36. In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
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  44. Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
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  46. Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
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  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
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  60. Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
  61. Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
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  63. Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
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  66. Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot
  67. Thermal analysis of extended surfaces using deep neural networks
  68. Steady-state thermodynamic process in multilayered heterogeneous cylinder
  69. Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
  70. Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
  71. Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
  72. The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
  73. Research on optimization of combustor liner structure based on arc-shaped slot hole
  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
  77. Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
  83. Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
  84. Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
  100. Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
  101. Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
  116. Special Issue on Nanomaterial utilization and structural optimization - Part II
  117. Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
  118. Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
  119. Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
  120. Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
  121. Special Issue on Nanofluids: Synthesis, Characterization, and Applications
  122. Exact solutions of a class of generalized nanofluidic models
  123. Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
  126. Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
  127. Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
  129. Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
  130. Comparative study of couple stress fluid flow using OHAM and NIM
  131. Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
  134. Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
  136. Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
  143. Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
  147. Special Issue on Advanced Energy Materials - Part III
  148. Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
https://doi.org/10.1515/phys-2023-0196
Keywords for this article
Hartree–Fock mean field theory; BEC system; chaotic control; constant shift method; Lyapunov index; control response system
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BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
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  17. Chaotic control problem of BEC system based on Hartree–Fock mean field theory
  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
  22. Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
  23. Model of conversion of flow from confined to unconfined aquifers with stochastic approach
  24. Study of fractional variable-order lymphatic filariasis infection model
  25. Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
  26. Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
  27. Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
  28. Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
  29. Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
  30. Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
  31. Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
  32. A two-stage framework for predicting the remaining useful life of bearings
  33. Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
  34. Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
  35. Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
  36. In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
  37. Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
  38. Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
  39. A comparative analysis of shielding effectiveness in glass and concrete containers
  40. Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
  41. Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
  42. Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
  43. Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
  44. Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
  45. Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
  46. Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
  48. Fractional view analytical analysis of generalized regularized long wave equation
  49. Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
  50. Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
  51. Estimation of monotone α-quantile of past lifetime function with application
  52. Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
  55. Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
  56. An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
  57. CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
  58. Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
  59. A short report on a probability-based interpretation of quantum mechanics
  60. Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
  61. Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
  62. Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
  63. Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
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  68. Steady-state thermodynamic process in multilayered heterogeneous cylinder
  69. Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
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  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
  77. Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
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  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
  83. Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
  84. Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
  100. Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
  101. Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
  116. Special Issue on Nanomaterial utilization and structural optimization - Part II
  117. Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
  118. Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
  119. Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
  120. Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
  121. Special Issue on Nanofluids: Synthesis, Characterization, and Applications
  122. Exact solutions of a class of generalized nanofluidic models
  123. Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
  126. Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
  127. Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
  129. Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
  130. Comparative study of couple stress fluid flow using OHAM and NIM
  131. Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
  134. Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
  136. Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
  143. Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
  147. Special Issue on Advanced Energy Materials - Part III
  148. Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
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