The restricted three body problem (RTBP) describes the motion of a restricted mass (also known as infinitesimal mass), which is moving under the effect of gravitational field of two massive bodies (called as primaries), without disturbing the actual motion of these primaries about their common centre of mass [25]. The RTBP is a demonstrative model of many problems in celestial mechanics, and it is used as a base model to describe the dynamics of small bodies (asteroid, comet, satellite, spacecraft, etc.) in solar as well as in extra-solar planetary systems [3, 11].

In the study of RTBP, the perturbing forces such as radiation pressure, Poynting-Robertson drag, solar wind drag, oblateness of massive bodies, etc. have an effective influence on celestial bodies. A number of researchers have contributed to the study of motion of infinitesimal mass under the frame of RTBP, Chermnykh-like problem etc. through different approaches and techniques in the presence of different perturbations in the form of oblateness, radiation pressure force, disc's effect, drag forces etc. [4, 6, 12, 13, 24, 29]. AbdulRaheem and Singh [1] have studied the combined effect of Coriolis force, centrifugal force, radiation pressure force and oblateness of the primaries on the equilibrium points and found that only Coriolis force acts as a stabilising factor, whereas centrifugal force, radiation pressure force and oblateness act as destabilising factors that cause a contraction in the stability range of the triangular equilibrium points. A few authors [15, 23, 28] have discussed the effects of radiation pressure, triaxial rigid bodies, oblateness of massive bodies and presence of a disc during the analysis of equilibrium points and their stability. They have found a deviation in the positions of equilibrium points and stability region because of these perturbations.

In the solar system dynamics, the analysis of periodic orbits produces many important ideas to understand the general properties of different dynamical systems. Companys et al. [8] have determined the existence of quasi-periodic orbits near the equilibrium points of the Earth-Moon system. The quasi-periodic orbits play an essential role in the designing of new space missions due to which fuel consumption in the spacecraft reduces. Motion of asteroids in the surroundings of triangular equilibrium points _{4} and _{5} in the Sun-Jupiter system can be more understandable by studying the tadpole and horseshoe orbits about the _{4} and _{5} points, respectively [9, 16]. Rabe [22] and Taylor [26] have determined approximate initial conditions to compute a series of periodic orbits around the _{4} and _{5} points of the Sun-Jupiter system and they have found stable horseshoe-shaped periodic orbits.

In order to understand and to analyse the evolution of periodic orbits, Winter [27] studied the evolution of periodic and quasi-periodic orbits and their stability property in the phase space of the Earth-Moon system using a numerical technique known as Poincaré surfaces of section (PSS) and found that these are sensitive dependence on the initial conditions. Many authors [10, 14, 21] have analysed the nature and stability of the periodic and quasi-periodic orbits with the help of PSS under the frame of perturbed restricted three-body problem and have noticed the effects of perturbing parameters. Abouelmagd et al. [5] have investigated the analytical derivations to study periodic solutions for the two-body problem perturbed by the first zonal harmonic parameter by using different methods. Further, Abouelmagd and Ansari [3] have analysed the cases, which agree that Jacobian and energy conversation are constants, in the Sun-perturbed Earth-Moon system and have illustrated the equilibrium points and their stability and PSS. Again, Pathak et al. [17, 19] have studied interior higher order resonant orbits in the perturbed photo-gravitational restricted three-body problem and have found that there exist periodic orbits for seventh- and ninth-order resonance, which are passing around the Earth. They also analysed the effect of radiation pressure and oblateness on these orbits.

In this paper, we are interested to determine the tadpole and horseshoe orbits and to observe the dynamical behaviour of orbits using PSS technique under the frame of RTBP with oblate primaries. The paper is organised as follows: Section 2 presents the equations of motion of the infinitesimal mass in the presence of oblateness. Sections 3 and 4 are devoted to the determination of equilibrium points and their linear stability test under the effect of oblateness, respectively. Tadpole and horseshoe orbits in the neighbourhood of equilibrium points are discussed in Section 5. Computation of PSS and the analysis of the effect of perturbing parameters on the PSS are presented in Section 6. Finally, the paper is concluded in Section 7. Numerical and semi-analytic computations are performed by using MATHEMATICA software (Version 12.0). Accuracy and precision goals of the values are achieved by considering or taking the values up to seven decimal places.

Let _{1} and _{2} be the masses of bigger and smaller oblate primaries, respectively. Assume that the primaries rotate about their common centre of mass in the _{1} + _{2}) = 1. Since the primaries are considered as oblate spheroid therefore, the angular speed (mean motion) _{2i}, _{1} and _{2} of the infinitesimal mass from the bigger and smaller primaries are defined as

Now, multiplying Eq. (1) by

On integration, we obtain

In this section, we shall discuss the existence of those points at which the velocity of the infinitesimal mass become zero. These points can be obtained by solving Ω_{x} = 0, Ω_{y} = 0, simultaneously i.e.

_{1},

_{2},

_{3}

Collinear equilibrium points are those points, which lie on the line joining the primaries. Therefore, for collinear equilibrium points _{1} = |_{2} = |

In the case of collinear equilibrium points, Eq. (6) is unimportant due to

For simplicity, collinear axis is divided into three parts as _{1}, _{2} and _{3}, respectively.

In case of _{1} point (i.e for region −

Similarly, in case of _{2}: _{3}: _{1}, _{2} and _{3}, respectively. We have computed the collinear equilibrium points _{1}, _{2} and _{3} by varying the oblateness coefficient _{21} and _{22} in Jupiter-Europa system as shown in Table 1. From Table 1 it is observed that on increment in the oblateness coefficient _{21} of the Jupiter shows more variation in the position of _{1}, but a small effect is seen both in case of _{2} and _{3} points. Due to the oblateness coefficient _{22} of Europa, we found a considerable change in the position of _{1}, _{2} and _{3} points. An increment in the value of _{22} corresponds to a shift in the position of _{1} and _{2} points towards Europa, but _{3} slightly move away from the Jupiter. Further, points _{1} and _{3} move towards Jupiter on increment in the oblateness _{22} of Europa while _{2} moves away from the Europa.

Position of _{1}, _{2}, _{3}, _{4} and _{5} in Jupiter-Europa system at

_{21} | _{22} | _{1} | _{2} | _{3} | _{4,5} : ( |
---|---|---|---|---|---|

0 | 0 | 0.9798121 | 1.0204124 | −1.0000104 | 0.4999749, 0.8660254 |

0.0001 | 0 | 0.9798138 | 1.0204108 | −1.0000104 | 0.5000249, 0.8659965 |

0.0002 | 0 | 0.9798155 | 1.0204091 | −1.0000105 | 0.5000749, 0.8659677 |

0.0003 | 0 | 0.9798172 | 1.0204074 | −1.0000105 | 0.5001248, 0.8659388 |

0 | 0.0035 | 0.9646568 | 1.0350740 | −0.9982666 | 0.4982325, 0.8650171 |

0 | 0.0070 | 0.9597175 | 1.0394326 | −0.9965348 | 0.4965052, 0.8640129 |

0 | 0.0105 | 0.9563164 | 1.0422269 | −0.9948149 | 0.4947928, 0.8630128 |

0.000285 | 0.007198 | 0.9595035^{*} | 1.0396141^{*} | −0.9964397^{*} | 0.4965504^{*}, 0.8638773^{*} |

represent the values are at actual oblateness parameters

_{4},

_{5}

Since, for triangular equilibrium points _{1} = _{2} = 1 which implies
_{1} = 1 + _{1} and _{2} = 1 + _{2}, where _{1}, _{2} are very small real numbers. Substituting the value of _{1} and _{2} in Eqs (5) and (6) then, solving for _{1} and _{2} and their product, we get

The expression for _{1} and _{2} are obtained by solving the following system of equations

The system of Eqs (13) and (14) are achieved by putting the value of _{1} and _{2}, where

Thus, substituting the value of _{1} and _{2} in Eqs (11) and (12) we get triangular equilibrium points _{4}; _{5}. In Table 1, apart from collinear equilibrium points, we have also presented the positions of triangular equilibrium points _{4} and _{5} at different values of oblateness coefficient _{21} and _{22} of the primaries. From Table 1, it is observed that the effect of the oblateness coefficient _{21} of the Jupiter causes large variation in the positions of _{4,5}. However the oblateness coefficient _{22} of Europa influence more in the positions of _{4} and _{5} points compared to that of _{22}. A combined effect of the approximate value of oblateness coefficients _{21}, _{22} on equilibrium points _{1},_{2}, _{3}, _{4} and _{5} have also been obtained as shown in bottom row of Table 1.

Let the co-ordinate of the equilibrium point be denoted by (_{0},_{0}). Suppose _{0},_{0}), then position of infinitesimal mass at any future time is given as _{0} + _{0} + _{x} and Ω_{y} about the equilibrium point (_{0}, _{0}) up to linear order terms in

Therefore, the equations of motion (1) and (2) in the neighbourhood of (_{0},_{0}) can be written by using Taylor series expansion in the linear order as
_{0}, _{0}) and the expressions of Ω_{xx}, Ω_{xy}, Ω_{yy} are given by

Now, we investigate the motion of infinitesimal body about the equilibrium point (_{0}, _{0}) in ^{λ t}^{λ t}

The above equations have a non-trivial solution for _{21} and _{22}. For the linear stability of equilibrium points, the nature of roots play a key role, which depends on the sign of discriminant
_{j}_{j}

For the collinear equilibrium points

Thus, solutions of Eqs (15) and (16) can be written as

The exponential factors ^{at}^{−at} in Eqs (21) and (22) results in unbounded motion in the _{1}, _{2} and _{3} remain unstable for all 0

In case of triangular equilibrium points,

In absence of the oblateness, i.e. when _{21} = 0, _{22} = 0, the Eq. (23) gives the critical mass ratio _{c}_{21} = 0.000285, _{22} = 0.007189), the critical value of mass ratio becomes _{c}

^{2} − 4

Let

_{c}^{2} − 4

This case implies

_{c}^{2} − 4

In this case, we found that the roots are of the form

Further, we found that all characteristic roots at _{21} = 0.000285, _{22} = 0.007198 are purely imaginary for mass ratio 0 _{4,5} are stable there. Also, we have observed a variation in the nature of characteristic roots with respect to oblateness _{21}, _{22} as given in Figure 3 and noticed that the oblateness of Europa affects more than that of oblateness of Jupiter on the stability region.

Suppose, the coefficients _{j}_{j}_{j}_{j}_{j}_{j}_{j}_{j}

Therefore, after simplification solutions (19) and (20) reduces to

In particular at _{4} (0.4965504, 0.8638773), _{21} = 0.000285 and _{22} = 0.007198, the perturbed solution ^{−4}, ^{−4} and
_{1,2} _{3,4} _{4,5} are stable. The semi analytical solution

In the RTBP, there exist different types of motion of an infinitesimal mass in the vicinity of the _{4,5}, valid only for small magnitude of displacement from the equilibrium point. Rabe [22] has computed the approximate initial conditions to analyse the periodic orbits close to _{4,5}. He has found two set of initial conditions for horseshoe orbits and for long tadpole-shaped orbits around the _{4,5}. Also, Taylor [26] briefly has described the smooth horseshoe orbits around _{4,5} for different initial conditions in the Sun-Jupiter system without perturbation. We are interested to compute the tadpole and horseshoe periodic orbits in the Jupiter-Europa system to analyse the effect of oblateness on these orbits. To determine the tadpole and horseshoe orbits in Jupiter-Europa system, we have followed the method described in [16, 26].

We assume that the infinitesimal mass _{4} or _{5}, which is the closest approach for _{4} or _{5}. A good approximation for periodic orbit about _{4,5} is that the magnitude of velocity _{4} be _{4} and _{5} in the Jupiter-Europa system in presence and absence of oblateness of Jupiter and Europa. For the initial conditions corresponding to the value of _{4} (also, similar tadpole orbits about _{5} are obtained which are not presented here) at _{21} = _{22} = 0. We have analysed the effect of oblateness on tadpole and horseshoe orbits as plotted in Figures 5(b,c,d) and 6(b,c,d). We have found that tadpole orbits gets elongated gradually due to oblateness of Jupiter (_{21} = 0.0285), but a slight increment in the oblateness of Europa (_{22} = 0.0007189) results in more elongation of the tadpole orbits (Figures 5b, 5c, 5d) with small deviation in amplitude of loops. On the other hand, in Figure 6(a–d), the horseshoe orbits about _{4} and _{5} at

The position coordinate (

We have computed the PSS at different values of mass ratio _{j}_{21}, _{22} by the method described in [7,16,27]. The starting value of _{j}_{j}_{j}_{j}_{j}_{21} and _{22} varies, the structural region and islands changes in character and one particular structure may become more dominant as shown in Figures 9 and 10, but reduction in the chaotic zones is more due to oblateness of smaller primary _{22} than that of bigger primary _{21}. In presence of oblateness, it is found that the curves or chain of islands representing quasi-periodic orbits predominates the chaotic regions.

We have considered the Jupiter-Europa system with oblateness of Jupiter and Europa and have found the equations of motion of infinitesimal mass. We have determined the equilibrium points and examined its linear stability under the influence of oblateness of both the primaries. It is found that due to oblateness of Jupiter and Europa, the stability range of the triangular equilibrium point is deviated from 0 < _{c}_{c}_{1,2,3,4,5} have shown deviations from their respective positions in classical cases (Table 1). Again, we have reckoned tadpole and horseshoe orbits in the Jupiter-Europa system under the influence of oblateness for different values of _{4} at = 0.0000251 and _{21} = _{22} = 0. The effect of oblateness on tadpole and horseshoe orbits are analyzed (Figures 5b, 5c, 5d and 6b, 6c, 6d) and it is observed that tadpole orbits gets elongate gradually due to oblateness of Jupiter (_{21} = 0.0285), but a slight increment in the oblateness of Europa (_{22} = 0.0007189) results in more elongation of the tadpole orbits (Figures 5b, 5c, 5d) with small deviation in amplitude of loops. On the other hand, from the horseshoe orbits about _{4} and _{5} at _{j}_{21} and _{22} (Figures 7–10). It is noticed that the oblateness of the primaries minimize the chaotic regions and induce the formation of a chain of islands that corresponds to quasi-periodic orbits. Also, we have found that the oblateness parameters _{21} and _{22} affect the structure and nature of orbits. The PSS in our model can be degraded to that of PSS without the oblateness of primaries [7]. Finally, it is concluded that shape and size of the primaries play a significant role in the study of motion of the infinitesimal mass. These results may help to the study of more generalized problem of three bodies in space.

#### Position of L1, L2, L3, L4 and L5 in Jupiter-Europa system at μ = 0.0000251

_{21} |
_{22} |
_{1} |
_{2} |
_{3} |
_{4,5} : ( |
---|---|---|---|---|---|

0 | 0 | 0.9798121 | 1.0204124 | −1.0000104 | 0.4999749, 0.8660254 |

0.0001 | 0 | 0.9798138 | 1.0204108 | −1.0000104 | 0.5000249, 0.8659965 |

0.0002 | 0 | 0.9798155 | 1.0204091 | −1.0000105 | 0.5000749, 0.8659677 |

0.0003 | 0 | 0.9798172 | 1.0204074 | −1.0000105 | 0.5001248, 0.8659388 |

0 | 0.0035 | 0.9646568 | 1.0350740 | −0.9982666 | 0.4982325, 0.8650171 |

0 | 0.0070 | 0.9597175 | 1.0394326 | −0.9965348 | 0.4965052, 0.8640129 |

0 | 0.0105 | 0.9563164 | 1.0422269 | −0.9948149 | 0.4947928, 0.8630128 |

0.000285 | 0.007198 | 0.9595035^{*} |
1.0396141^{*} |
−0.9964397^{*} |
0.4965504^{*}^{*} |

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Detection Enterprise Financial Risk Early Warning System Based on Structural Equation Model A Study on the Application of Quantile Regression Equation in Forecasting Financial Value at Risk in Financial Markets Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment Information transmission simulation of Internet of things communication nodes under collision free probability equation Image denoising model based on improved fractional calculus mathematical equation Random Fourier Approximation of the Kernel Function in Programmable Networks The Complexity of Virtual Reality Technology in the Simulation and Modeling of Civil Mathematical Models University Library Lending System Model Based on Fractional Differential Equations Calculation and Performance Evaluation of Text Similarity Based on Strong Classification Features Intelligent Matching System of Clauses in International Investment Arbitration Cases Based on Big Data Statistical 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Dominate Color Method Financial Risk Prediction and Analysis Based on Nonlinear Differential Equations Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching Multiple Effects Analysis of Hangzhou Issuing Digital Consumer Coupons Based on Simultaneous Equations of CDM Model Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations A Hybrid Computational Intelligence Method of Newton's Method and Genetic Algorithm for Solving Compatible Nonlinear Equations Pressure Image Recognition of Lying Positions Based on Multi-feature value Regularized Extreme Learning Algorithm English Intelligent Question Answering System Based on elliptic fitting equation Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Application Research of Mathematica Software in Calculus Teaching Computer Vision Communication Technology in Mathematical Modeling Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for English-Chinese Cross-Language Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontal-torsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation Construction of an enterprise financial risk management system based on F-score model Research on personalized Japanese language teaching model based on big data technology Research on education management system based on machine learning and multidimensional data modeling A multidimensional interpretation of Marxist globalization theory based on decentralized internet Analysis of the improvement path of English listening and speaking ability based on big data technology Multi-featured multi-scale combination of high-resolution remote sensing images for building extraction Study on the absorption technology of distribution network with high-proportion distributed power supply Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Electronic properties of diamond semiconductor materials: based on response surface model Application of BOPPPS teaching model in online and offline blended teaching of college Public Basic Mathematics Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching Deep learning techniques in modern women’s smart clothing design Grouting reinforcement strategy for tunnel sand layer based on BP neural network Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology A study on the aesthetic characteristics of ethnic music based on radial basis neural network model Research on university laboratory network security based on Cloud Computing The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Analysis of the application methods of film and television media and images in the era of big data cloud Títul Mobile visual media design based on digital media technology and augmented reality Air data computer simulation and design credibility assessment considering ADC calculation model Spatial planning layout design of national land in the context of rural revitalization based on MOLA model Analysis of news communication strategies in the era of full media based on big data mining On the improvement, innovation and inheritance of stage makeup styling in opera under the background of big data Influence of 3D characters image transfer on animation drawing based on detail feature preservation Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Evaluation Mechanism of Teaching Staff’s Building in Local Vocational Colleges Based on Big Data Technology Títul Sports consumption behavior and consumption demand of urban residents in the context of “Internet+” Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based 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mechanical device Public Management Model Based on Mathematical Equation of Statistics of Probability Theory 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System An analysis of time commitment for college students to online English writing and peer assessment learning Optimization path of corporate ESG performance and compensation performance management under carbon neutral target Research on network security management monitoring platform based on Internet of things technology New anergy tide control strategy based on Eviews econometric model Study of frictional wear properties of materials for mechanical seals Optimization of vocal singing training method based on intelligent big data technology Fractional Differential Equations in Electronic Information Models Exploring the construction of an all-round cost management system for investment control in university engineering construction The construction of blockchain platform for engineering museums based on vectorized image processing technology BIM Engineering Management Oriented to Curve Equation Model Relationship Between Physical Activity Sedentary Behavior and Body Mass Index in Children and Adolescents Based on FIT Modelo Green supply chain innovation management strategy based on the combination of low carbon economy and e-commerce with big data technology Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Grid data asset relationship and intelligent classification integrating knowledge graph and Internet of Things Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's 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and character intelligence analysis based on semantic ontology and Poisson equation Analysis of digital humanistic knowledge production based on natural language processing Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering The Construction of Judgment Model for Cultivating Intelligent Accounting Professionals in Universities Based on Logarithmic Periodic Power Law Model Action of Aut( G ) on the set of maximal subgroups ofp -groupsDigital Regeneration and Database Construction of Hunan Embroidery Needlework under the Perspective of Artificial Intelligence Research on the practice of innovation and entrepreneurship education in universities under the environment of big data Application of digital technology in environmental art design Research on loyalty prediction of e-commerce customer based on data mining Research on the evaluation system of tourism management classroom teaching based on multimedia big data analysis The application of ARCS motivation model in college oral English teaching Research on Innovation of Translation Teaching and Translation Strategies for College Students in Multimedia Background A multi-channel convolutional neural network based on attention mechanism fusion for facial expression recognition Exploring the development of cultural and creative tourism under the perspective of ecological theory of tourism destination management Modeling Analysis of User Consumption Data of Internet Culture Industry in the Context of Emerging Technologies Reference: An algorithm for recognizing the main melody of orchestral music based on artificial intelligence of music melody contour Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and 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game review IoT-based interaction design of smart home products for elderly families Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Model Construction of Influencing Factors of Physical Education Core Quality Cultivation Based on Multiple Linear Regression A multidimensional interpretation analysis of Marxism globalization theory based on distributed Internet Application of Nonlinear Differential Equation in Electric Automation Control System Research on the operation and management mechanism of cross-border e-commerce shared practice training base Strategy for financing mode optimization in international trade supply chain based on deep learning model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Strategies for improving Chinese language proficiency based on artificial intelligence technology Early warning and detection of geological disasters based on intelligent genetic algorithm The success rate model of college students’ new entrepreneurship based on nonlinear differential equations Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Research on the application of marketing strategy of national fitness exercise and dance events in the construction of sports culture based on big data technology Exploring the Application of Modern Fine Arts Artistic Modeling and Color in Modern Art Design Based on Information Fusion Technology Decisions of competing supply chain with altruistic retailer under risk aversion Strategies for constructing mathematical models of nonlinear systems based on multiple linear regression models Research on China’s tourism marketing model in the data perspective: based on ecological perspective Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations Research on human resource management performance evaluation based on BP algorithm The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Badminton players’ trajectory under numerical calculation method Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis The Mental Health Education Management of Higher Vocational Students Based on Fractional Differential Equations Application of regression function model based on panel data in financial risk management of bank resource allocation University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation Research on Resonance Properties of Semantic Wave Fractal Fractals Based on Quantitative Analysis of English Corpus Research on urban landscape big data information processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on urban energy Internet planning based on fuzzy comprehensive method Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the intelligent path of college students’ network ideological and political education based on big data mining technology Research on the application of PLC technology in electrical automation engineering Application of matrix multiplication in signal sensor image perception Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering AtanK-A New SVM Kernel for Classification The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Rhino application in ceramic packaging design teaching Study of a linear-physical-programming-based approach for web service selection under uncertain service quality Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory Virtual Museum Scene Design Based on VRAR Realistic Interaction under PMC Artificial Intelligence Model A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Research on innovative cultivation mode of English and American literature translation talents based on internet environment Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Analysis of quantitative management of online intelligent monitoring of tailing ponds based on the perspective of safety prevention and control Optimization of college physical education curriculum based on G-DINA Optimization of local ethnic music teaching transmission path based on logistic regression model RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning Design and evaluation of intelligent teaching system on basic movements in PE Research on identifying psychological health problems of college students by logistic regression model based on data mining Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Retraction: Wei, Wei and Lu, Liu. “The influence of X fuzzy mathematics method in basketball tactics scoring” Applied Mathematics and Nonlinear Sciences, vol.7, no.1, 2022, pp.677-684 Retraction Retraction: Retraction of: Mundewadi, R. A. and S, Kumbinarasaiah. “Numerical Solution of Abel's Integral Equations using Hermite Wavelet” Applied Mathematics and Nonlinear Sciences, vol.4, no.2, 2019, pp.395-406