Fractals

This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis functions to compute space derivatives while Caputo derivative scheme utilizes for time-fractional integration to semi-discretize the model equations. To assess the accuracy, maximum error norm is used. In order to validate the proposed method, some non-rectangular domains are also considered.

The neutral delay model has many applications in biology, ecology, man-made neural networks, mechanics and engineering. The main purpose of this paper is to establish new oscillation criteria for the two-dimensional neutral delay dynamical system

This paper is devoted to the study of the behaviors of the solution to Fujita-type equations on finite graphs. Under certain conditions given by absorption term of the equations, we prove respectively local existence and blow-up results of solutions to Fujita-type equations on finite graphs. Our results contain some previous results as special cases. Finally, we provide some numerical experiments to illustrate the applicability of the obtained results.

Taking into account the quarantine for an infectious disease, a susceptible-exposed-infected-quarantined-recovery-susceptible (SEIQRS) epidemic model with time delay on the finite scale-free network is given. The basic reproduction number $R_0$, which is dependent not only on all kinds of transfer rates, but also on the topology of the network, is derived. By constructing the Lyapunov function, it is asserted that the disease-free equilibrium of system is locally asymptotically stable if $R_0<1$, moreover, disease-free equilibrium of system is globally asymptotically stable when $R_0<1$. In addition, the influence of network nodes on the spread of diseases is discussed. Finally, the theoretical results are verified by corresponding numerical simulation.

Fractal analysis is a totally new area of research based on local fractional calculus. It has interesting applications in various fields such as a complex graph, computer graphics, the music industry, picture compression and many more fields. In this paper, we present new variants of Hadamard–Mercer-type inequalities on fractal sets ${ℝ}^{\alpha }$ ($0<\alpha \le 1)$ by employing generalized convex function. We establish two new lemmas involving local fractional integrals. By using these lemmas, we obtain several results related to generalized Hadamard–Mercer-type integral inequalities for local differentiable generalized convex functions on real linear fractal space. Finally, we give applications for probability density functions and compute new generalized means.

This study explores some significant consequences of discrete $\widehat{\hslash }$-proportional fractional sums $(D\widehat{\hslash }PFs)$ having an exponential function as a nonlocal kernel. Certain novel weighted versions comprising a group of positive mappings via $(D\widehat{\hslash }PFs)$ are given. A variety of refinements can be derived by taking into account the extraction of the new estimates and the nabla $\widehat{\hslash }$-fractional sums. The suggested technique is a revolutionary formulation of conventional operators that may be used to design efficient mechanism descriptions in short time spans by provoking certain noteworthy properties of chaos theory. Moreover, novel generalizations of the discrete $\hslash$-fractional sum can be generated by the specific value of the proportionality index. Derived outcomes and investigation confirm that the proposed plan will offer gains in many modeling and chaotic framework applications.

The purpose of this paper is to determine the assets of Small and Medium-sized Enterprises (SMEs), reasonably value this type of enterprise, and effectively use available data to measure their financing risks. Based on the previous research, the nonlinear expectation and the stochastic differential equation are adopted to combine the enterprise’s financing risk with the psychological factors of the consumer market to construct different risk measurement models according to the current technology-based enterprise’s pricing and financing problems. Besides, the nonparametric estimation method is applied to the nonlinear expectation, which improves the application of the model in finance. The effectiveness of the model in technology-based SMEs pricing and financing risk measurement evaluation is verified using the instance data of technology-based SMEs. Results demonstrate that the traditional pricing models for technology-based SMEs need to ensure that every investor is rational. Although the price of the enterprise cannot be effectively predicted, the nonlinear expectation and stochastic differential equation can reflect the changes in the value of the enterprises and the impacts of investors’ psychological factors on the financing effect of the enterprises. Different risk measurement models constructed can effectively price and measure technology-based SMEs. The non-parametric estimation method can effectively improve the accuracy of the model prediction. The results can provide some research ideas and practical value for the market valuation and financing risk assessment of technology-based SMEs.

The aim of this paper is to improve the accurate analysis of the sports training project, research its video data profoundly, and discover deficiencies and summarize experience from the existing videos, thereby promoting China’s sports industry. According to previous works, the existing sports training target detection system has the problem of nonlinear video and low target detection accuracy, which can easily cause target loss or tracking failure. Therefore, the Kalman Filter (KF) is studied in-depth and applied to detect the targets in sports videos. Combined with the Multi-Innovation (MI) theory, an MI-Extended KF (EKF) algorithm model based on nonlinear dynamic technology is proposed. This model can effectively solve the filtering accuracy problem under the strong nonlinear system. Finally, the performance analysis through different datasets has verified the effectiveness of the model proposed further. Results demonstrate that the model proposed can effectively improve the filtering accuracy of nonlinear systems. As the angle module changes, the accuracy of sports recognition also changes. The final accuracy of the model proposed can reach above 96%. The simulation results and convergence are better than other algorithms, which also proves the effectiveness of the model proposed. The results can provide a theoretical basis for the research related to the detection of training sports targets in sports videos.

The aim of this paper is to discuss the accidental events’ impacts on the complicated supply chain systems and reveal the dynamic nonlinear system’s role in enterprises’ logistics financial management. First, the logistics distribution network and nonlinear system are analyzed and introduced, revealing the complexity of enterprises’ supply chain systems. Second, considering the situation of enterprises’ financial management, four major entities, the supplier, the distributor, the retailer, and the manufacturer, are introduced to construct a dynamic nonlinear model for the supply chain system. Finally, the model before and after accidental events are simulated and analyzed using the AnyLogic software. Results demonstrate that in normal states, each node’s inventory in the system changes periodically, while the distributor’s state is the most stable. A notable decrease in market demand has the largest impact on the distributor. Once the market demand is recovered, the decrease in the distributor’s inventory is the most apparent, and the inventories of the supplier, the retailer, and the manufacturer also decrease. The supplier has the highest cost at the beginning of system operation, and the distributor’s cost is the highest after the operation becomes stable. Once an accidental event caused a sharp decrease in market demand, the costs of all major entities in the system show an increasing trend. Once the market demand is recovered, the costs of different node entities will increase instead. As for the profits, the distributor has the highest profit initially, which is significantly influenced by the accidental events. The results provide a reference for revealing the changing laws of complicated supply chain systems and promoting the development of enterprises’ financial management.

To improve the integrity of the supply chain and get the optimal joint scheduling of the complete supply chain, in this exploration, first, the supply chain related theory is elaborated. Then, the dynamic model of product recovery in enterprise joint scheduling is established. Finally, based on the differential game theory (based on the continuity of nonlinear differential equation function) and Hamilton function, the optimal scheduling strategies with manufacturer and retailer as the main recycling body in the complete supply chain are calculated respectively. The results show that the product recovery rate with manufacturer as the main recycling body is higher than that with retailer as the main recycling body. There is no transfer price in the recovery based on the manufacturer. However, in the recovery based on the retailer, if the transfer price increases, the product recovery rate of the system will also increase; if the transfer price increases, the manufacturer’s instant profit will appropriately increase; if the transfer price value is high $(v=1.5)$, the manufacturer’s instant profit is higher. If the transfer payment is low $(v=1)$, the instant profit of manufacturer with manufacturer as the main recycling body will be first lower and then higher than that with retailer as the main recycling body. When the system is stable, the instant profit with the manufacturer as the main recycling body is higher than that with retailer as the main recycling body; the total profit of the manufacturer with the retailer as the recycling body is first lower and then higher than that with the manufacturer as the main recycling body. Whether the transfer payment is high or low, the retailer’s profit level is higher when the manufacturer is the main recycling body. When the retailer is the main recycling body, the retailer’s profit level is inversely proportional to the transfer payment. Finally, in the centralized supply chain environment, enterprises can maximize their total profits and promote the good development.

Based on the improved BP neural network, this paper establishes an adaptive online controlling model and adopts the model to optimize the controlling accuracies in discrete nonlinear dynamic systems and inverted pendulum systems. To avoid the local minimum problem of the BP neural network’s objective function in the training process, this paper proposes a neural network training method based on the quasi-Newton method (BFGS) optimization algorithm. Compared with other control methods, the neural network-based inverted pendulum control method proposed in this paper has higher control accuracy. Through the control simulation of its power system uses a discrete control method and the control of the inverted pendulum model system, this paper verifies the validity and good control has significantly improved the control method.

To explore the application of improved variational iterative method in solving fractional differential equations and the effect of DNA coding algorithm based on logistic chaos mapping in image encryption, combining Chebyshev polynomial with variational iterative method, a new algorithm for solving fractional differential equations is proposed. In order to solve the initial value problem, the simulation software is used to compare the approximate solution after the change of values of $\alpha$ and $k$ in the solution process to investigate the influence of different parameters on the accuracy of fractional order system. Subsequently, the logistic chaos sequence is used to generate discrete binary sequences, and an image encryption algorithm based on DNA coding is proposed. In order to verify the encryption performance of the proposed algorithm, Matlab simulation software is used to simulate and verify the image encryption processing. The results show that when $\alpha$ approaches 1, the solution is closer to the exact solution than that when $\alpha$ approaches 0.5, 0.7, and 0.9; compared with 1, 3, 5, and 7, the constructed algorithm has the highest accuracy when $k$ value is 9. The simulation results of the fuzzy partial fractional order system show that the chaotic sequence is sensitive to the initial value, and the logistic mapping sequence is suitable for the information encryption of the secure communication system. The simulation results of the image encryption algorithm based on DNA coding show that the algorithm can effectively encrypt the image, and the image histogram after encryption is approximately a horizontal line; slight changes in the initial value will affect the decryption effect of the image; the correlation analysis results show that there is no correlation between adjacent pixels in the image after encryption, which indicates that combining Chebyshev polynomials with variational iteration method can reduce the computational burden and improve the computational accuracy, and the image encryption algorithm proposed in this study can improve the reliability and security of image encryption.

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

The purposes of this paper are to optimize the financial management methods of small- and medium-sized enterprises (SEMs) and deeply explore asset-liability ratio’s relevant theories. A medium-sized manufacturing enterprise’s financial conditions and the influencing factors of enterprise’s asset-liability ratio are researched by a dynamic model with two-way effects and the enterprises’ dynamic nonlinear simulation model. Meanwhile, capital injection’s impacts on asset-liability ratio are explored. The results show that different industries’ enterprise asset-liability ratios vary. The real estate enterprises’ asset-liability ratio is 61.35%, the highest, and the cultural communication industry is the lowest, 35.62%. Meanwhile, the asset-liability ratios of energy industry, manufacturing industry, and the wholesale and retail industry are also relatively high. The larger the enterprise’s scale, the higher its asset-liability ratio, long-term liability ratio, current liability ratio, and loan financing. The proposed simulation software performs satisfactorily in its application. The simulation system’s analysis shows that the target enterprise’s asset-liability ratio is gradually increasing, and its return on equity and capital turnover rate is decreasing, reaching the critical bankruptcy point in the 36th month. With a 20 million CNY capital injection, its asset-liability ratio decreases significantly, and return on equity increases steadily. In the 49th month, it resumes normal operations. The results provide a useful reference for subsequent research on enterprises’ asset-liability ratios and SMEs’ financial management.

The purposes are to solve the option pricing problem in financial derivatives and provide a practical basis for reducing the risk rate of the financial market. First, the classification of fractal theory and option pricing is outlined, the connotation of stochastic volatility is analyzed, and a stochastic volatility model is established. According to the Fourier transform and nonlinear differential equation, the pricing method in the volatility swap process and the pricing method of volatility derivatives are analyzed. Finally, the option pricing parameters are set, and the Origin 75 software is employed to visually analyze the mathematical results. The forward prices under stochastic volatility are compared monthly and weekly, and the resulted values are all close to 235. There is a gap between the forward prices of each month and each quarter, and the difference between the forward prices of each quarter is large, close to 50. The average relative percentage errors (ARPEs) within and out of the sample calculated by the stochastic volatility model are close to 16%, and the error gaps between different monetary intervals are also large. With the increase in strike price, the change rate of options under the random volatility model is faster than that of options under VIX index. When the strike price increases to 0.6, the two options are equal. The fitting trend of the stochastic volatility model for implied volatility at different maturity dates first decreases and then tends to a fixed value. The implied volatility of the 60-day-expiring option has changed significantly, showing a trend of increasing first and then tending to a fixed value (0.8). In contrast, the implied volatility of the 90-day-expiring option and the 120-day-expiring option almost have no change. When $w$ (fractional operator) takes 1 and $Y$ takes 1.6 and 1.3, the solution to the stochastic volatility model is consistent with that to the classic CGMY model. When $w$ takes 1, the stochastic volatility model has the same solution as the classic CGMY and BS models. As the fractional order of time increases, the trend of asset price changes becomes smaller.

In this paper, a financing decision-making method is proposed based on the nonlinear differential equations (NDEs) to promote the healthy development of high-technological (high-tech) small and medium enterprises (SMEs). An assessment index system for financing risks is established, and the financing decision-making method is proposed based on the backward stochastic differential equation (BSDE). Then, the financing risk is evaluated with analytic hierarchy process (AHP) combined with fuzzy evaluation based on the financing status of Peilin Company. Finally, the corresponding financing decision is proposed based on the BSDE. It is found that the comprehensive score of financing risk of Peilin Company is 63.6124, indicating that the risk is high. The financing decision obtained shows that the current equity fund financing should be 396.24million to achieve the future goal. It is proved that the proposed financing risk evaluation method can assess the financing risks accurately, and the financing decision-making method based on NDE can obtain more appropriate financing decisions through calculation. This study can provide a crucial reference for management and control of financing risk for enterprises.

Based on the analysis of the self-similarity of network traffic, a network anomaly detection technology is proposed by combining with the fuzzy logic so as to explore the fractal characteristics of network traffic. The concepts of network traffic and network security are introduced. Then, a network traffic model of network traffic is proposed based on the fractal theory and wavelet analysis. Finally, a distributed denial of service (DDoS) that attacks the monitoring and intensity judgment method is put forward based on the fuzzy logic theory. The results show that the autocorrelation function of the multifractal wavelet model constructed based on the local Hurst exponent (LHE) can reach a mean square error (MSE) of $4.762\times 10^{-4}$, which proves that the network traffic model proposed can reduce the impact of the non-stationary characteristics of the network traffic on the modeling accuracy. The network security detection method proposed can monitor the DDoS attacks and can accurately judge the attack intensity in real time. The research in this study provides an important reference for the scientific operation of the network.

To improve the product quality and production efficiency in industrial production, the industrial production process is explored. First, for the production state in industrial production, the radio frequency identification technology (RFID) is used to monitor the state of the nodes in the production process and collect data. Based on the problem of blind area in industrial production, an adaptive controller design based on uncertain nonlinear dynamic systems is introduced; second, based on this, for the problem that product quality is difficult to guarantee in the process flow, the improved multi-layer feedforward network (back propagation, BP) algorithm is used to predict it. Finally, the control diagnosis algorithm based on multi-order least squares support vector machine (LS-SVM) is used to analyze the abnormal cause of the product. The results show that the use of the RFID can monitor the production process in real time and obtain real-time product quality information data. The proposed adaptive controller based on uncertain nonlinear dynamic system has good applicability in solving the problem of blind area. The improved BP neural network algorithm improves the prediction accuracy in the process of product quality prediction and increases the convergence speed of the algorithm. The control diagnosis algorithm based on multi-order least squares that support vector machine (SVM) can classify and learn the historical data of products, as well as perform multi-order search and analysis on the causes of abnormal quality. It helps relevant personnel to carry out process improvement and control, thereby achieving the purpose of improving product quality. Through this investigation, it is found that the method proposed can monitor the state of industrial production in real time and help relevant personnel to control and predict product quality. It can be applied in actual processes.

The purposes of this paper are to improve the scientific processing level of risk management in the financial field, enrich the application range of mathematical models in financial calculations, and comprehensively discuss the theories and concepts of mathematical finance and stochastic differential equations. More importantly, the common option pricing issues in financial risk management have been researched using the forward–backward stochastic differential equation. The fully discrete and uncoupled forward–backward stochastic differential equation is employed to analyze the spread option and the better-of option, the complicated multi-asset options. Results demonstrate that the fully discrete and uncoupled forward–backward stochastic differential equations can effectively price the spread option and the better-of option. Simulation by the MATLAB software suggests that the value of spread option pricing is 0.0264, and the value of the better-of option pricing is 0.0251. The above results can provide scientific and useful references for the subsequent application research on forward–backward stochastic differential equations in the financial field; simultaneously, they also have important practical significance for researching on and developing the financial risk management.

The purpose of this paper is to control and judge the big data of students’ learning and living conditions in college education. College students’ book-borrowing data are mined deeply from three aspects, the multi-source preprocessing of students’ borrowing data from university and college libraries, the quantification of students’ book borrowing, and academic performance prediction by learning and book borrowing. The data mining technology analyzes and processes students’ primary information, score information, and book-borrowing information. Students’ book borrowing is modeled and analyzed using the backpropagation neural network (BPNN) algorithm, and the constructed BPNN book-borrowing model’s loss function is optimized based on the partial differential equations. The library access control data and book-borrowing data are used for statistics of the learning behavior frequency. Data such as students’ stay duration in the learning area and attendance rate are input into the analysis model for experiments, and the average absolute error, the mean square error, and the determination coefficient evaluate the prediction results. The results show that as students’ booking borrowing frequency decreases, their scores decrease, and students who often borrow books have strong learning motivations. In Experiment 4, when $R^2$ reaches its maximum value, 0.594, the predicted scores by the students’ book-borrowing model have a high correlation with students’ actual scores, indicating that the BPNN algorithm has the best prediction results. The results show that the indicator of students’ book borrowing has significantly improved the model’s prediction performance, and the borrowed book number and book-borrowing frequency are significant in the prediction model construction.

In order to improve the intelligent tracking efficiency of switched nonlinear dynamic system, first, the optimized backpropagation neural network (BPNN) is proposed and the switched nonlinear dynamic system is constructed. Then, the optimized BPNN is extended and the adaptive control is studied. Finally, the extended neural network system related to the error is formed, and then the simulation analysis is carried out. The results show that under different reference signal conditions, the output peak values of the system are 1.7 and 1.5 at 2s, respectively, while the response peak value of the reference signal is 1. However, after 4s, the peak values of the two signals are very close; under different reference signals, the scaling factor decreases with the increase of time. After 6s, the value of scaling factor is 0. Under the condition of reference signal 1, the estimated values of parameters decrease with the increase of time, and tend to 0 after 6s. However, under the condition of reference signal 2, the estimated values of parameters in the first 6s increase with the increase of time; the peak values of the state variable $x_2$ and the adaptive parameters in the system are in a stable state, while the peak values of the system controller and the system tracking error gradually decrease with the increase of time, and finally tend to a stable value. It suggests that the technique proposed in this work can guarantee the signal in switched nonlinear dynamic system to be bounded.

The purpose is to optimize the optical fiber Fabry–Perot (F-P) sensor’s performances to provide a realistic basis for the optical fiber’s development. First, the classification of the optical fiber F-P sensors is summarized. The optical fiber F-P sensors’ fundamental principles are analyzed, focusing on the phase demodulation method that converts the optical fiber F-P sensing system’s input light source into a broadband light source. The broadband light source’s lightwave changes are analyzed, and the Fourier transform method and the fringe counting method are proposed to optimize the phase demodulation method’s function and analyze the light source’s nonlinear dynamic characteristics. According to the proposed demodulation method, the optical fiber F-P sensor with a cavity length change of $[180,190]$$\mu$m is simulated, and the cavity length selected for the simulation spectrum is the change range’s center value, 185um. The wavelength division multiplexer (WDM) decomposes the two optical signals, and an equation directly calculates the wavelength value orthogonal to its phase. The results show that the light intensity’s initial value of the actual spectrum is 0.04, and that of the standard function is 0.06. The light intensities for the maximum peak values are the same, 0.12, with the wavelength in $1.455\times 10^{-6}$–$1.47\times 10^{-6}$m. When the wavelength exceeds $1.47\times 10^{-6}$m before the next peak, the light intensity difference between them is large. The two orthogonal optical signals’ peaks change between the cavity length 0.18nm and 0.19nm after the wavelength selected, and the two signals’ peak values are orthogonal with the cavity length changes. The two signals’ initial and end peak values are different. The curve obtained by the two signals division calculation shows a tangent trend with the highest peak value of 87 and the lowest of $-58$. The peak curve changes obtained by the two signals’ arctangent calculation are in an orthogonal state, with the maximum peak value of 1.3 and the minimum of $-1.3$. The reflected light peak value changes of the optical fiber F-P sensor are in an orthogonal state, with the highest value of $-12$ and the lowest $-31$, and the adjacent peak changes in the wavelengths of 1500–1580 are the same. The curve’s fitting linearity is 1, and the linearity in the dynamic demodulation is 0.9776. The sampling frequency in the phase demodulation is $1\times 10^3$Hz, and the sampling frequency in the dynamic demodulation is $1\times 10^5$Hz, showing that upgrading the interface can increase the dynamic sampling frequency continuously to improve the device performance.

Using computer vision technology to obtain and analyze biomechanical information is an important research direction in recent years. However, the linear model in the computer vision system cannot accurately describe the geometric relationship of the camera imaging, so it is difficult to realize human posture recognition in high-precision mechanics information. Therefore, how to improve the recognition accuracy is very important. In this paper, we apply nonlinear differential equations to stereo computer vision (SCV) information systems. And based on the median theorem, a nonlinear posture recognition and error compensation algorithm based on BP neural network is proposed to reduce the recognition error. The test set uses the Leeds Motion Pose (LSP) dataset to verify the performance of the algorithm. Experimental results show that the compensated median filter of BP neural network can eliminate glitches in attitude data. Superimposing the output attitude error compensation value with the attitude estimation value can greatly reduce the root-mean-square error of the attitude angle. The result of gesture recognition is closer to reality. Compared with traditional algorithms, the cyclomatic complexity of the proposed BP neural network algorithm has a much lower growth rate in high-order calculations, which indicates that the proposed BP neural network algorithm is more concise and scalable.

The purposes of this paper are to analyze the financial management and influencing factors of small and medium-sized enterprises (SMEs) and explore the nonlinear relationship in enterprises’ financial market structure. The small and medium-sized enterprises in Eastern and Western China are taken as the research objects. First, the relationships between asset–liability constraints and financial management are discussed, analyzed, and explained. The development of enterprises’ financial management under the asset–liability constraints system is emphasized. Second, a self-adaptive nonlinear dynamic system is proposed based on dynamic surface control and the multi-directional and uncertain control of the financial market structure. Finally, a dynamic nonlinear panel estimation model for enterprises is constructed based on the nonlinear system. The simulation and empirical analysis results confirm that the proposed nonlinear system is useful in dynamic, uncertain control problems. The statistical results of the three primary indicators, economic variables, financial indicators, and control variables, reveal the significant regional differences of different financial market structure indicators. Model estimates based on the three key sub-indicators, the leverage ratio, return on assets, and debt interest rate, reveal the differences in financing and leverage ratio of small and medium-sized enterprises located in the eastern and western regions. SMEs in Eastern China are taken as examples; the direct financing rate and insurance proportion are negatively correlated with the leverage ratio and debt interest rate. In contrast, they are positively correlated with asset returns. In conclusion, there are noticeable differences in the financial market structure between different regions.

To estimate the parameters of the nonlinear output error system (nonlinear system), a variational Bayesian estimation method (VB method) is proposed based on the probabilistic graphical model (PGM). First, the related theories are introduced in this study such as the PGM and nonlinear systems. Then, the parameter estimation model of the nonlinear system is established. Finally, a VB method is proposed based on the PGM to estimate the parameters of the nonlinear system, which is tested and verified by numerical simulation experiments. It is found that the parameter estimation model of nonlinear system and the proposed method can estimate the parameters of relevant nonlinear system better, and the minimum error between the parameter estimated value and the actual value is only about 0.0001. It proves the feasibility of the VB method based on PGM in the identification of nonlinear systems. The results of this study provide an important reference for the control and identification of nonlinear systems.

This paper aims to study the processing and repairing methods of blurred images, promote the development of partial differential equations in the field of image processing, and expand the application of stochastic fuzzy differential equations in the field of image processing and repair. This study starts with a typical blurred image repair method. First, a comparative analysis of several common blurred image repair methods including Wiener filtering restoration, inverse filtering restoration, and Lucy–Richardson (L-R) filtering restoration is performed. Second, based on the linear partial differential equation learning model (LPDE), the concept of fuzzy integral is introduced, and an improved stochastic fuzzy partial differential equation learning model (SFCPDE) is proposed. The effect of the learning model before and after improvement on blurred color image processing is compared and analyzed. Finally, based on the total variation (TV) blurred image repair algorithm, an improved TV blurred image repair algorithm is proposed. The comparison and analysis of the repair effects of several blurred image repair algorithms are performed. The results show that there are obvious differences in the repairing methods with or without noise. Inverse filtering works best when there is no noise. L-R filtering has the disadvantage of amplifying noise. Compared with LDPE, the training speed of SFCPDE is significantly improved, and the training error is less than LDPE. The SFCPDE learning model performs better in the processing of blurred color images. After 10 iterations, the improved TV algorithm is significantly better than the TV algorithm and the CDD algorithm in repairing blurred images. The PSNR value of the TV algorithm and the curvature-driven diffusion (CDD) algorithm after 10 iterations corresponds to about 60% of the PSNR value of the improved algorithm. The algorithms and models of stochastic fuzzy partial differential equations proposed in this paper have great application potential in the processing and repair of multiple blurred images.

For the problem of parametric design of anisotropic curves and cambers in the practical application field of environmental landscape design, in this exploration, the modeling of design object is carried out through computer parametric technology. It is based on the information collection and data extraction of design object and the formulation of logical relationship between parametric variable factors related to design. For nonlinear systems, a necessary condition to judge parameters is proposed. This criterion is a natural extension of the PBH criterion, which makes it easy to identify the parameters of the pseudo-linear system in the generalized case, and avoids a lot of complex operations; at the same time, a parametric design method of dynamic environment landscape is proposed systematically. In the first order, second order, high order and generalized nonlinear model, the complete parametric optimization method of environmental landscape design is established. The $f_{min}$ search function in MATLAB optimization toolbox is used to solve the generalized nonlinear system. The performance of the algorithm is tested by the examples of pseudo velocity, pseudo velocity uniformity and parameter optimization of the parametric curve called by the software environment. The results show that system dynamic process to simulate the dynamic environmental changes in landscape design by using the optimized parameter method is faster than that by using the unoptimized method. Meanwhile, the optimization method in this exploration can reduce the amplitude of change when external factors interfere, which shows that the optimized parametric design model has better transient performance; the curve generated by calling the drawing function provided in this exploration is smoother when the number of drawing points is the same.

In order to study the application of fuzzy partial fractional-order equation based on infinite impulse response (IIR) digital filter in noise big data monitoring, in this study, first, the noise level is calculated through IIR filter to obtain the sound signal sequence after C weight and A weight; the weight equivalent continuous sound level of C, A, and Z is calculated. Then, the vibration condition of fuzzy partial fractional-order equation is studied; the vibration acceleration signal is converted into the vibration velocity and displacement signal; the corresponding effective values are calculated according to the root mean square (RMS), and the noise monitoring platform is designed from hardware and software aspects. Finally, the calibrated sound level value is compared with the standard signal value generated by the digital signal generator; the sequence of sound signal after weight of the original signal, the actual acceleration signal, velocity signal, and displacement signal are calculated; the acceleration signal, velocity signal, and theoretical signal values are compared at different frequencies. The results show that the error between the calculated and calibrated sound level and the standard signal value is very small, and the results of weight counting of C and A are good and meet the requirement of calculation; the error between the measured value of actual acceleration, velocity, and displacement signal and the theoretical value is small, which can realize the accurate acquisition of vibration signal; the error between the actual measured value and the theoretical value of acceleration and velocity at different frequencies is small, about 1% and 10%, respectively. Therefore, the fuzzy partial fractional-order equation based on IIR digital filter can effectively monitor the big noise data.

To further explore the specific application of nonlinear system in the construction of landscape architecture and promote the parametric development of landscape design, in this exploration, based on the digital elevation model (DEM) and nonlinear algorithm theory, the road selection scheme using cost distance algorithm and path distance algorithm is explored. The results suggest that there are obvious differences between cost distance algorithm and path distance algorithm in the consideration of parameters in use. The path distance algorithm needs to further consider the influence of surface grid and vertical system parameters. Environmental factors and different classification methods have great influence on road selection. When the limit value is greater than $\pm 30^{\circ }$, there is no obvious change in road selection. However, when SLOPE values are $15^{\circ }$, $30^{\circ }$ and $45^{\circ }$, respectively, the difference in road selection is small, and the application effect is good. This exploration can provide the sufficient reference for the follow-up study of landscape road selection.

To solve the problem of insufficient combination of neural mechanism and psychological mechanism in the study of human brain cognitive memory activity, in this exploration, the memory forgetting model in psychology is analyzed and constructed by fractional order model of linear differential equation. The study of memory in cognitive psychology is combined with the neurophysiological mechanism of memory. At the same time, for the stability of long-term memory, a long-term memory neural network model (LTMNNS) is proposed. The validity and accuracy of the model are tested by the recognition of memory eigenvectors. Numerical test and simulation results show that when the difference between any initial values of the system is not more than $\delta$, the corresponding system solution difference is not more than $𝜀$ in the interval $[0,T]$. Reasoning correctness of fractional order computation model of memory neural network is verified. The LTMNNS algorithm in this exploration guarantees a high number of completed tasks, and the average number of failed tasks is less than MemNNs, DNC and LSTM algorithm. Compared with other models, the proposed LTMNNS has better generalization ability and higher stability.

The nonlinear dynamic characteristics of human brain cognition and music stimulation are introduced based on the negative emotions of college students to explore the influence of music stimulation on human brain cognition activities and analyze the nonlinear dynamic characteristics of electroencephalogram (EEG) signals. First, the correlation between EEG signals and human brain cognition, as well as the nonlinear dynamics of cognition, are explained. The average correlation dimension and the maximum value of the Lyapunov exponent are selected to characterize the nonlinear dynamic characteristics of EEG signals. Second, the power spectrum and EEG coherence are considered, and an emotion assessment algorithm is proposed. Finally, music stimulation is introduced, and its role in adjusting college students’ negative emotions is discussed. Results demonstrate significant differences in the average correlation dimension and the maximum value of the Lyapunov component of the participants in different cognition states under normal physiological conditions. The cognition function state shows chaotic behaviors, explaining the nonlinear dynamic characteristics of EEG signals. After music stimulation, a significant increase in EEG relative power is mostly located in the frontal and temporal regions of the brain. The EEG coherence in the same brain region shows growth changes. After the intervention of music stimulation, the emotion assessment scores of the self-rating anxiety scale, the positive and negative affect schedule, and the self-rating depression scale are reduced, and the entire changing process presents statistical significance. In conclusion, music stimulation intervention can affect human brain cognition activity, playing a positive role in emotion regulation.

The purpose of this paper is to improve the English teaching levels of colleges and universities. First, the primary attributes of dynamic nonlinear system theory in second language acquisition are analyzed. Second, the influencing factors of English grammar application ability in theoretical dimension and measurement methods are proposed. Six first-year students majoring in English in a university were randomly selected as research objects, who were followed up for a semester, during which three identical oral English tasks were finished (in September, October, and November of 2020). The fluency, accuracy, and complexity of students’ utterances are statistically analyzed using SPSS 26.0. Results demonstrate that the fluency of students’ utterances has increased from 110 syllables per minute on average in September to 129 syllables per minute on average in November. The proportion of error-free clauses has increased from 0.5 to 0.55, and the number of clauses in the T unit has increased from 1.42 to 1.51. Students ${\mathrm{\text{P}}}_1$, ${\mathrm{\text{P}}}_2$, ${\mathrm{\text{P}}}_3$, ${\mathrm{\text{P}}}_5$, and ${\mathrm{\text{P}}}_6$ have a gradually increasing accuracy of utterance, while student ${\mathrm{\text{P}}}_4$ shows a gradually decreasing utterance accuracy. Students ${\mathrm{\text{P}}}_1$ and ${\mathrm{\text{P}}}_5$ have a decreasing utterance accuracy, while students ${\mathrm{\text{P}}}_2$ and ${\mathrm{\text{P}}}_6$ present an increasing utterance accuracy. Besides, the utterance accuracy of students ${\mathrm{\text{P}}}_3$ and ${\mathrm{\text{P}}}_4$ first decreases and then increases. The performances in utterance complexity of students ${\mathrm{\text{P}}}_2$, ${\mathrm{\text{P}}}_3$, ${\mathrm{\text{P}}}_4$, ${\mathrm{\text{P}}}_5$, and ${\mathrm{\text{P}}}_6$ have been increasing gradually, while the performance of student ${\mathrm{\text{P}}}_1$ has been decreasing gradually. The normal distribution of each student’s test scores is different, as well. Hence, any minor changes will affect the overall oral English levels of students.

In the process of task scheduling, due to the large search space of resources, it takes a long time to allocate appropriate resources for tasks, which results in the increase of execution time and execution cost of algorithms. For this reason, this paper proposes a cloud computing task fuzzy scheduling strategy based on hybrid search algorithm and differential evolution, which incorporates the classification based on normal distribution and a variety of mutation strategies on the basis of standard differential evolution algorithm. In mutation strategy, the individual difference vector assigns priority to each task and arranges tasks according to resource allocation rules. It improves the slow convergence speed and easy to fall into local optimum of standard difference algorithm, and can effectively solve the task scheduling problem of cloud computing. A cloud computing task scheduling algorithm with time and cost constraints is designed and tested in a simulation environment. The experimental results show that the algorithm can not only shorten the task processing time and reduce the execution cost, but also fully satisfy the users’ actual quality of service requirements by adjusting the weights of time and cost factors.

To further explore the research of financial time series prediction and broaden the application scope of Gaussian distribution probability density equation, based on the nonlinear differential equation of Gaussian distribution probability density, the semi-supervised Gaussian process model is taken as the research object to discuss the application of semi-supervised Gaussian process model in the stock market. Shanghai 180 Index, Shanghai (Securities) Composite Index and the yield of three stocks are studied in detail. The specific results are as follows. The prediction accuracy of Shanghai 180 Index is 83%, and the prediction accuracy of Shanghai (Securities) Composite Index is 78%. The stock yield series curves of the three stocks have the characteristics of peak and thick tail, which do not obey the normal distribution. Among the three models of semi-supervised Gaussian process model, initial Gaussian process model and SVM algorithm model, the prediction accuracy of semi-supervised Gaussian process model is the best, and the prediction accuracy of the yield of three stocks is 82.45%, 85.03% and 84.53%, respectively. The research results can fully prove that the semi-supervised Gaussian process model has good application effect in stock time series prediction. The research content can provide scientific and sufficient reference for the follow-up research of financial time series, and also has important significance for the research of Gaussian process model.

The single-degree-of-freedom and three-degree-of-freedom viscous damping systems are simulated based on the Morlet wavelet function transformation in an effort to study the wavelet transform and further promote the optimization of nonlinear system modal identification. In the meantime, the modal animation display technology is studied using the Visual Basic (VB) 6.0 software and Open GL three-dimensional graphics library. The research object is a thin plate member with five degrees of freedom. The research results are as follows. In the single-degree-of-freedom viscous damping system, the identification frequency is 11.545rad/s, and the damping ratio is 2.87%. The simulation result has a small gap with the set damping ratio, and the identification in the system is accurate and reliable. In the three-degree-of-freedom damping system, the recognition accuracy of the first-order wavelet coefficient model is higher. Besides, the recognition accuracy of the natural frequency in the second-order is better, and the damping ratio error value is 11.08%. In the third-order, the natural frequency and the damping ratio have a large error from the theory; the error values are 24.53% and 32.11%, respectively. In the meantime, VB 6.0 software and Open GL software can effectively identify the actual shape of the research object, showing a good application effect. The above results can provide scientific and effective reference materials for subsequent research on nonlinear system modal identification.

Due to the complexity of the problem in the actual group management decision, the ambiguity of the information of all parties and the uncertainty of the decision environment, it is often difficult for decision-makers to describe the decision information by quantitative methods, but to give uncertain linguistic variables. Based on the analysis of previous research results, this paper proposes a group management decision-making aggregation model based on relative entropy and norm operator. First, the model uses the operation of $T$-norm and new $T$-norm operator to make the fuzzy set satisfy the ranking rate and contradiction rate. Then, based on the reliability of relative entropy, the attribute weights of different expert decisions are obtained, and the optimal assembly model of the corresponding group decision is established, respectively. The optimal solution and decision algorithm are given for each optimization model. The last example of the algorithm proves the validity and feasibility of the model, and can effectively avoid the distortion and distortion in the process of group management decision preference aggregation.

The purpose is to apply the multifractal analysis method to the research of human physiology in the process of sports. The mass index spectrum is used for multifractal analysis of heart rate variability signal based on the heart rate variability signal analysis theory and fractal theory. Finally, 10 healthy college students are selected as the experimental subjects. RAC-3003 portable electronic measuring instrument is used to collect heart rate signals in different exercise stages. Finally, the data are analyzed by $R/S$ and Lo-$R/S$ analysis methods. The results show that the lowest value of ln$(R/S)$ is 3.1 and the highest value is 7.0 in different stages in the morning, and the lowest value is 3.3 and the highest value is 7.5 in different stages in the afternoon. The average value of random signal ln$(R/S)$ gradually increases from 2.6 to 3.7; whether in the morning or in the afternoon, the average Hurst exponent during exercise is lower than that before and after exercise, and the average Hurst exponent after exercise is slightly higher than that before exercise; the long-range correlation index of heart rate variability signal in each exercise stage first increases and then decreases, and the changes of short-range correlation index and long-range correlation index are opposite; the average of fitting intercept of $R/S$ curve is lower than that of Lo-$R/S$ curve in the first and third stages; the fractal coefficient of the original data in the first and third stages of exercise is significantly higher than that in the second stage, which indicates that the overall fractal degree of heart rate variability signal before and after exercise is higher.

In view of the recognition of human motion function in physical education, a multidimensional evaluation index of motion function based on fatigue, muscle strength and muscle contraction, and relaxation ability is proposed, and the mapping rules among fatigue characteristic value, muscle strength, fractal dimension characteristic change, and motion function state are studied. Based on the characteristics of surface electromyography (sEMG) signal and its generating principle, the collected signal is preprocessed by median filtering to obtain the filtered signal. The effectiveness of the filtering method is proved by comparing the signals before and after filtering. The current muscle strength calculation method based on sEMG signal is analyzed and studied. The muscle strength calculation method of Hill three-element model is used to establish the muscle model through the muscle activity and force–length curve calculated by sEMG signals. The muscle strength is obtained by muscle model, and the feasibility of the experimental method is verified by comparing the calculated actual muscle strength with the theoretical muscle strength. In the experiment, the distribution of different subjects in the multidimensional evaluation index space is studied. Moreover, the mapping relationship between sEMG signal characteristics and human motion function is discussed comprehensively based on theory and data.

In order to further explore the important impact of inbound tourism on economic development, six cities such as X city are selected as the research objects. Based on the panel threshold regression model and relevant data, the dynamic nonlinear relationship between inbound tourism and the economic development of the tertiary industry is explored through mathematical empirical analysis. The specific research results are as follows. The inbound tourism of X city develops rapidly and is in the stage of accelerating development. Meanwhile, the development of inbound tourism in X city has a certain pulling effect on the development of related industries and the tertiary industry. Among them, the elasticity coefficients of inbound tourism in X city to transportation industry, accommodation and catering industry, wholesale and retail industry, finance and insurance industry, and real estate industry are 1.137, 1.104, 1.134, 1.105, and 1.062, respectively; the elasticity coefficient to the income of the tertiary industry is 0.436. Moreover, the analysis of the relationship between inbound tourism and the development of the tertiary industry in six cities shows that there is a dynamic nonlinear relationship between them, and the research results are in line with the expected hypothesis. The research results can provide scientific and effective reference for the follow-up study of inbound tourism and economics.

The purposes are to accurately perceive the network security situations and predict the development trend and effectively defend against network attacks during the operation of the Internet. The Long Short-Term Memory (LSTM) network is adopted as the subject of the network security situation awareness and prediction model. Moreover, it is optimized by the Genetic Algorithm (GA) to improve its global search capability. Then, a Fractal Neural Network (FNN) is constructed in combination with fractal theory, which is utilized in network security situation awareness to avoid the exploding or vanishing gradient problems. The KDD CUP 99 standard dataset is applied to verify the performance of the proposed GA-LSTM FNN; results demonstrate that its accuracy of network security situation awareness can reach 90.22%. The experimental results confirm that using the fractal difference function as the activation function can deliver the gradient variation in a balanced and stable manner. Besides, it can improve the feasibility and effectiveness of the neural network structure for network security situation awareness and prediction. The FNN studied is of practical significance for assessing the current network security situation and predicting its evolution trend, providing a reference for protecting the operation of the Internet from network attacks.

Based on back propagation neural network (BPNN), we design a dynamic nonlinear intelligent control system for signal processing and analysis in this work. Performances of the traditional PID and BPNN control systems is compared in a case of heat power engineering system design. In the design, the training times, regulation time of the BPNN control system are evaluated when $k=-1.7$ and $k=-2.0$, respectively. Furthermore, the regulation time, oscillation time and amplitude of the traditional PID and BPNN control systems are compared in the dead zone. Results of the comparison show that the BPNN based dynamic nonlinear intelligent control system demonstrates superior performances over the traditional PID control system. The research results prove the advantage of BPNN in the design of nonlinear control systems and provide a scientific and effective reference for the follow-up research works.

To further expand the application range of fractal theory and promote the application of sensors in the era of intelligent interconnection, we design a wireless sensor network (WSN) using statistical fractal measurements in simulation. Fractal features and network complexity of the WSN are analyzed. Besides, the related influencing factors of the WSN transmission success rate are explored. When the box size is smaller than 100, the relationship between the box size and the number of boxes tends to be a straight line, the relationship between them is more stable when the number of nodes increases gradually. Moreover, the degree distribution of nodes presents more obvious symmetry with the increase of network nodes. The simulation results of the network transmission success rate show that under certain other conditions, with the increase of node hops, the network transmission success rate can be promoted significantly. When the transmission failure probability $f_1$ is 0.1 and hop count is 6, the transmission success rate can reach more than 90%. This exploration can provide a scientific and effective reference for the follow-up research of the fractal theory.

The purpose is to optimize the training methods of athletes and improve their competitive ability. First, the corresponding research model is established by analyzing the influencing factors of the athlete training system and combining with the dynamic nonlinear theory. Next, based on the deterministic learning theory, the dynamic nonlinear state identification model of athletes is established to obtain the state information of athletes in the process of competitive sports training and competition. Then, based on the established nonlinear state model, the nonlinear staged athlete training mode is designed. The experimental results show that the designed gait recognition algorithm based on deterministic learning can effectively collect the nonlinear signals generated by the movement process; the accuracy of gait recognition is about 90%, and it is less affected by the training and test datasets. The nonlinear staged training mode can effectively improve the overall index of maximum strength and strength endurance of tennis players, and effectively improve the training effect of players. This exploration provides a reference for the study of nonlinear competitive ability training mode of sports athletes.

The purpose of this paper is to improve the financial management efficiency of large enterprises and enhance the overall operation vitality of enterprises. First, the connotation and characteristics of enterprise resource planning (ERP) are analyzed, and the financial ERP system is established. Then, the relevant dynamic models of nonlinear systems are classified and their characteristics are analyzed. Moreover, the system model of enterprise financial risk management is constructed based on the key success factors of project implementation risk and control flow chart of project life cycle. Finally, based on MATLAB software, $Z$ large enterprise is taken as an example to evaluate the implementation effect of analytical hierarchy process (AHP) algorithm and back propagation neural network (BPNN) algorithm in ERP system. The results reveal that compared with 2019, the capital concentration in 2020 increases by 8%, the operating cost decreases by 23.6%, and the expense reimbursement process time decreases from 60–80 days to about 6 days. The expected output and assessment result of AHP are 6.912 and 6.823, respectively, and the error between them is 0.0196. The expected output and assessment result of BPNN are 6.798 and 6.675, respectively, and the error between them is 0.0121. The error value of BPNN in ERP implementation effect assessment is less than that of AHP, which indicates that the assessment effect of BPNN is better than that of AHP.

The dynamic nonlinear system with saturation constraints and unknown information is studied to design a better control system. The controller of the general n-order affine nonlinear system is constructed utilizing the backstepping method and the improved radial basis function (RBF) that can approximate the unknown parameters. Then, the appropriate Lyapunov function is selected. The mature stability judgment theorem can prove that the designed system is semi-globally uniform and ultimately bounded. Finally, the controller is added to the system for the experiment on MATLAB. The experimental results show that the controller is reliable and the system output can approach the ideal value through the limited range of control signal. During the simulation, the reference value can be quickly tracked by the system output, and the tracking error is small only within 0.04. The proposed method can provide a reference for the design of control systems in practical engineering.

The purpose is to study the handling stability and energy saving mode of energy saving electric vehicles. First, the differences between electric vehicles and traditional fuel vehicles are analyzed. Moreover, based on active safety technology, the control stability, handling stability and driving safety of electric vehicles are improved. Then, based on the vehicle dynamics theory and the proposed assumptions, the modular electric vehicle dynamics model is established, including the wheel model, tire model and driver model, which can effectively control the error of the vehicle model and reduce the difficulty and complexity of vehicle modeling. Next, based on the model predictive control theory, the vehicle steady-state control strategy is established, which can predict and control the state of the electric vehicle in the compound condition and improve the control stability of the vehicle. Finally, the vehicle dynamic model and steady-state control system are simulated and tested. The experimental results show that the simulation results of the established dynamic model of the electric vehicle in angle step condition and single line shifting condition are consistent with the results of professional software, which shows that the established dynamic model can accurately describe the dynamic state of electric vehicle and provides the basis for the research of vehicle stability control strategy. Experimental results of the control strategy under the condition of double line shifting show that the designed control strategy can accurately track the state of the system and constrain the deviation in a small range, so as to ensure the driving stability of the electric vehicle. This exploration provides a reference for the study of dynamic analysis and handling stability control strategy of electric vehicles.

The purpose is to apply deep learning to facial expression recognition and improve the efficiency of facial expression recognition in real scenes: first, convolutional neural network (CNN) theoretical model is proposed. On this basis, a new CNN model is proposed, which has six learning layers, including three convolutional layers and three fully connected layers. Then, based on the shortcomings of the model, weight decay and dropout optimization methods are proposed. Moreover, the attention module is designed based on the theory of attention mechanism. After each convolutional layer, a hybrid attention module (channel domain attention and spatial domain attention) is added. Finally, the frame error rate (FER) 2013 dataset and Cohn–Kanade (CK)+ dataset are used to test and compare the performance of the model. The results show that after 50Epoch, the accuracy rate of the original model fluctuates greatly in the process of convergence; however, after attention mechanism is added, the fluctuation of the accuracy of the model is obviously reduced. In the CK+ dataset, the accuracy of each model is maintained at about 95%, while the accuracy of FER2013 dataset is about 71%. After attention mechanism is added, the recognition rate of dynamic nonlinear expression is higher than that of basic model. The recognition rate of the original model and the optimized model with attention mechanism for the dynamic nonlinear expression with occlusion decreases in varying degrees, but the reduction of the recognition rate of the latter is alleviated.

With the development of industrial technology, acceleration sensors have a wide range of applications. The application environment has strict requirements on acceleration sensor, which needs it to get accurate input and output response in complexity. According to the Hammerstein model, this paper studies the dynamic nonlinear relationship between the output voltage and the input acceleration of the acceleration sensor that can be divided into static nonlinear component and dynamic linear component. We combine the least square method with adaptive neural network to calculate the parameters of static nonlinear and dynamic linear components. The least square method is used to improve the training performance of the network and avoid the network falling into the local minimum of the traditional neural network. Experimental results show that compared with other methods, the proposed method has the advantages of less training steps and strong approximation ability, and the algorithm is less affected by external noise. This method can realize nonlinear system identification of acceleration sensor and provide reliable basis for compensation.

In rail transit systems, improving transportation efficiency has become a research hotspot. In recent years, a new method of train control system based on virtual coupling has attracted the attention of many scholars. Virtual coupled train is a typical nonlinear system. How to control the nonlinear system with high precision is the key to realize the safe and efficient operation of virtual coupled train. Therefore, this paper proposes a virtual coupling scheme based on local leader–follower method. On this basis, the controller based on artificial potential field with variable parameter mixed is applied to carry out the collaborative and anti-collision control for the virtual coupled train system with nonlinear dynamics. Finally, a section of the Beijing–Shanghai high-speed railway is taken as the background to verify the effectiveness of artificial potential field with variable parameter mixed controller. The results show that the variable parameter artificial potential field controller with changing the weight can reduce the average error of the stop by 0.4304m compared with the traditional controller.

The fractal Iterated Function System (IFS) algorithm is studied to simulate swinging plants’ animation effect under natural conditions. First, the collage method and the IFS attractor parameter morphing method render the fractal plant. Second, the IFS attractor parameter morphing is added on this basis. Third, the swinging plant’s animation morphing is achieved by combining the physics and adjusting the morphing parameters. Finally, the effects of leaves transforming into trees and swinging in the wind are simulated. The simulation results verify the effectiveness of the proposed algorithm. Compared with traditional methods, it reduces the time required to render images and overcomes interactive control limitations. The ability of attractors with parameter variables to control images is very stable. Continuous adjustment of parameters can effectively solve image distortion caused by matrix decomposition during fractal morphing. The proposed algorithm provides some references for applying fractal morphing technology in animation design and computer graphics.

With the acceleration of industrialization and urbanization, most lakes and reservoirs have been in eutrophication state. Eutrophication of water body will produce a series of environmental problems, among which cyanobacteria bloom is one of the most studied and seriously polluted problems. It is of great significance to effectively control the occurrence of cyanobacteria blooms by predicting and simulating the outbreak process of cyanobacteria blooms and accurately forecasting the relevant governance departments. However, there are two problems in the existing analysis of algal blooms: on the one hand, it is difficult to consider the impact of other factors on cyanobacteria blooms by taking chlorophyll concentration as the main influencing factor, and it is also unable to determine the relationship between various factors. On the other hand, only based on the field monitoring data research, lack of comprehensive analysis of the whole water area. The remote sensing image can reflect the change of the whole water area, but the traditional analysis method is difficult to deal with the massive remote sensing data effectively. In this study, eutrophication level was used as characterization index of cyanobacteria bloom, and the remote sensing image and its inversion map were taken as the main research data, and a new method of cyanobacteria bloom prediction based on four-dimensional (4D) fractal CNN was proposed. The prediction model uses 4D fractal CNN to extract the features of multi factor remote sensing images, capture the temporal and spatial characteristics and the interaction between multiple factors, and predict the eutrophication level of water body. In this study, a total of 216 remote sensing images of Taihu Lake Basin were selected from 29 groups with fine weather from 2009 to 2010 obtained by MODIS satellite. The simulation results show that the method proposed in this paper has excellent prediction performance, and the accuracy rate of 85.71% is better than that of common 3D CNN and 4D CNN models.

The purpose is to further explore the application effect of the neural network algorithm in defense audit and improve the user information security performance. Based on the relevant theoretical basis of neural network in machine learning, the back propagation neural network (BPNN) algorithm model is constructed and optimized. Moreover, by comparing with the classification and prediction effect of the decision tree method, the application effect of BPNN is further clarified. Through statistical analysis, a total of six risk users are screened out. The test data are classified into non-risk user group and risk user group to study the prediction of classification. The specific results are as follows. The prediction accuracy of non-risk group is 99% by using the BPNN algorithm and that is improved to 99.5% by using the optimized BPNN; for risk group, the prediction accuracy of BPNN is only 50% and that of optimized BPNN is 83.3%. Meanwhile, the prediction error rate of the BPNN algorithm is significantly lower than that of the decision tree algorithm, which further verifies the good application effect of the BPNN algorithm. This study can provide scientific and effective reference for the follow-up research of defense audit.

The purpose is to apply the fractal statistical model to analyze tourist flow feature of scenic area, and promote the effective performance of scenic area. Zhangjiajie scenic area is taken as an example. First, its natural resources characteristics, ecological environment and problems in the development of scenic area are analyzed in detail; moreover, fractal theory and fractal dimension calculation method are elaborated; finally, wavelet transform is proposed to analyze the nonlinear transformation characteristics of tourist flow. The results show that the changes of the high-frequency layer $H_1$, $H_2$, $H_3$ and $H_4$ all have cyclical fluctuations in the change of tourist flow. The change range of tourist flow in spring and autumn is relatively large, and the tourist flow in winter is low. Inbound tourism tourist flow has a certain seasonal regularity, with obvious characteristics of low and peak seasons. Wulingyuan scenic spot and Tianmen Mountain scenic spot of Zhangjiajie have three non-scale ranges, which are related to the scale and change of tourist flow. The concentration of inbound tourist flow in the peak season still exists in both scenic spots, and it is more and more serious. Tourist flow of inbound tourists from April to May and from September to October is large, accounting for more than half of the annual tourist flow, while most of the other periods are in the off-season.

In order to further explore the intelligent research of the pressure sensor in the pipeline, based on the existing pipeline detection methods of the Internet of Things (IoT), the signal processing method and correction research in the wireless sensor network of the IoT are explored. A multi-pressure sensor detection method of negative pressure wave is proposed. Scientific and intelligent means are used to explore the pipeline remote intelligent detection. The specific research results are as follows. In the simulation study, when the working condition of pump station 1# changes, the pressure wave first reaches A₁ and B₂ sensors, and finally reaches C₁ and C₂ sensors. If there is leakage in the pipeline, the simulation results show that when the pressure of pump station rises to 5MPa, the pressure of B₁ and B₂ test points decreases to about 4.9MPa, and the pressure of A₁, A₂, C₁ and C₂ test points is about 3MPa; when the pressure of the pump station drops to 3MPa, the pressure of all test points decreases. Generally speaking, the pressure of the front-end pressure sensor of the pump station is basically unchanged while the pressure of the back end increases when the pressure increases; when the pressure decreases, the pressure of the front-end pressure sensor of the pump station is basically unchanged while the pressure of all the back-end pressure sensor decreases. After correction, the signal data is obviously smooth and the error is obviously reduced. This study can provide scientific and effective reference for the pressure monitoring of IoT sensor network.

The purposes are to recognize the facial expression information and establish a fast and accurate facial expression recognition algorithm. First, the noise caused by face image shooting is eliminated by analyzing the primary methods of face recognition and preprocessing the images. Then, the Nonlinear Support Vector Machine (NSVM) is combined with the decision tree to obtain the Nonlinear Support Vector Machine Decision Tree (NSMMD) algorithm. By introducing the binary tree, NSVM has the ability of multi-classification and can extract the feature information of the face image to obtain the characteristic face. Then, the algorithm flow is introduced. Second, the characteristic face data are recognized and classified based on Generalized Regression Neural Network (GRNN), and the expression data contained in the face images are obtained to recognize the expressions of the face image. Finally, the designed face recognition algorithm and facial expression recognition algorithm are tested. The results show that the designed NSMMD algorithm has a weak dependence on feature dimension, and the face recognition rate for different datasets is above 80.1%. The GRNN algorithm will be affected by smooth parameters, its optimal parameter value is 0.1, and its recognition rate of expressions on various databases is above 92%. Therefore, the designed face recognition algorithm and facial expression recognition algorithm can provide sound recognition effects, which can reference the study of facial image feature recognition.

The identification of nonlinear system is studied to establish an accurate model of coordination system. The original dynamic fuzzy neural network (DFNN) is first used to identify the nonlinear system for finding the existing problems. Aiming at the problems found, two improvements are made. For the problem of too many pre-set parameters in the original algorithm, the fuzzy completeness $𝜀-$ is introduced to allocate the parameters, and the width of the membership function is modified. The simulation results reveal that the fuzzy neural network (FNN) model not improved produces seven fuzzy rules, and the root mean square error (RMSE) of training is 0.0261, while the improved FNN model produces six fuzzy rules, and the RMSE of training is 0.0161. The improved network has more advantages in performance. The model proposed provides some references for the application of FNN in dynamic nonlinear systems.

With the acceleration development of urbanization, the infrastructure construction plays an increasingly crucial role in the expansion process of cities. However, the existing decision-making methods of urban infrastructure investment still stay at the static level, resulting in a huge waste of resources, which does not meet the needs of dynamic urban development and management. Based on geometric Brownian motion, option function and the theory of real option, this paper establishes a dynamic differential optimal investment opportunity decision model applied to urban infrastructure construction. An example was then conducted to test the operability and application of the proposed model. The results confirm that the proposed approach contributes to improving the scientificity of urban infrastructure planning and construction, and realizing the maximization of infrastructure’s social value. Meanwhile, this paper contributes to promoting the development of city infrastructure investment decision-making methods from static to dynamic, and from partial to overall.

With the rapid development of tourism economics, personalized and diversified demand of tourists and the advancement of modern information technologies, the investment of scenic spots faces multitudinous uncertainties and emergencies and can be considered as a nonlinear dynamic system. Given this situation, the existing investment decision-making model (e.g. net present value, NPV) is difficult to estimate the realistic value of scenic spots, and new theory and decision-making model are needed in the area. Based on real option theory and option function, the paper establishes a nonlinear dynamic investment decision model of scenic spots with uncertainties and emergencies, and then, the model is verified with an example. The results show that with the impacts of emergencies, when the investment value is considered, the option value of the investment project in the scenic spot can be maximized, thus maximizing the total value of the investment project. The proposed nonlinear dynamic investment decision model proves that the favourable impact of emergencies on the investment of scenic spots increases the investment opportunity value of scenic projects, thus increasing the investment value of scenic projects, emphasizes more on the dynamic nature of investment and the favourable conditions in the face of crisis, and helps enterprises investing in scenic spots make full use of its uncertain environment and the favourable impact of emergencies to choose the optimal investment opportunity. Therefore, this study has both theoretical and practical significance.

Tourism destination system can be considered as a complex nonlinear dynamic system. It’s difficult to accurately evaluate the operational orderliness with existing evaluation methods due to the complexity and dynamics of system. This paper attempts to introduce the concept of entropy in thermodynamics into organizational management, hoping to establish a nonlinear dynamic method for assessing destination system orderliness from management lens. Combining information entropy formula and the characteristics of tourism destination, this paper analyzed the entropy flow and negative entropy flow in the destination business ecosystem. Then an entropy evaluation index was constructed and the weight of each specific primary indicator and secondary indicator were calculated with the combination of analytic hierarchy process (AHP) and principal component analysis (PCA). Finally, taking JIUZHAI Valley as an example, this paper tested and modified the nonlinear dynamic evaluation approach. The nonlinear dynamic evaluation approach not only emphasize the overall operation state and process of destination system, but also take into account the nonlinear relationships amongst the internal elements of destination, contributing to overcoming the defects and deficiencies of traditional evaluation methods.

Discrete fractional calculus (${𝒟}{ℱ}{𝒞}$) is significant for neural networks, complex dynamic systems and frequency response analysis approaches. In contrast with the continuous-time frameworks, fewer outcomes are accessible for discrete fractional operators. This study investigates some major consequences of two sorts of inequalities by considering discrete Atangana–Baleanu $({𝒜}{ℬ})$-fractional operator having $\hslash$-discrete generalized Mittag-Leffler kernels in the sense of Riemann type (${𝒜}{ℬ}{ℛ}$). Certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete ${𝒜}{ℬ}$-fractional operators having $\hslash$-discrete generalized Mittag-Leffler kernels are given. Moreover, several other generalizations can be generated for nabla $\hslash$-fractional sums. The proposing discretization is a novel form of the existing operators that can be provoked by some intriguing features of chaotic systems to design efficient dynamics description in short time domains. Furthermore, by combining two mechanisms, numerous new special cases are introduced.