Nonlinear Part Research Articles

ON THE EXISTENCE OF PERIODIC SOLUTIONS OF A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH QUASI-HOMOGENEOUS NON-LINEARITY

In the paper was investigated the a priori estimate and the existence of periodic solutions of a fixed period for a system of second-order ordinary differential equations with the main quasi-homogeneous non-linearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have non-zero bounded solutions. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions under any perturbation from a given class is formulated and proven. The results obtained differ from earlier results in that the set of zeros of the main non-linear part is not taken into account.

A novel surface deformation prediction method based on AWC-LSTM model

Severe surface deformation can damage the ecological environment, trigger geological disasters, and threaten human life and property. Reliable surface deformation prediction is conducive to reducing potential risks and mitigating disaster losses. Currently, machine learning-based surface deformation prediction models have shown significant improvements in prediction performance. However, most prediction models do not sufficiently consider the characteristics of surface deformation, exhibit subjectivity in parameter settings, and inadequately capture local features in time series data. We introduce the AWC-LSTM model to predict surface deformation. Initially, leveraging the strengths of the autoregressive integrated moving average (ARIMA) model in handling linear signals, the obtained surface deformation information is decomposed to linear and nonlinear parts, and the linear part is predicted. Secondly, by incorporating convolutional neural network (CNN) layers into the long short term memory (LSTM) model, the ability to learn local features is enhanced and the whale optimization algorithm (WOA) is introduced to determine the optimal hyperparameters of the model, thereby predicting nonlinear deformation. The proposed AWC-LSTM model was validated using the Shilawusu coal mine and Beijing as case studies. The outcomes indicate that the deformation predictions for the Shilawusu coal mine and Beijing exhibit a high degree of consistency with the monitored data, with root mean square errors (RMSE) not exceeding 3 mm. This underscores the model’s reliability and applicability across different areas. Comparisons with existing prediction models indicate that the AWC-LSTM model achieves higher predictive accuracy, with an average improvement in accuracy ranging from 28.38 % to 80.59 % over other models.

Stability and [formula omitted]-gain analysis of switched positive systems with unstable subsystems and sector nonlinearities

In this paper, we are concerned with exponential stability and L1-gain performance of a type of switched nonlinear positive systems (SNPSs) with time-varying delay made up of partially unstable subsystems. What sets it apart from other articles is that nonlinear parts of this article are confined in a certain region. Besides, the adequacy criteria are obtained in a linear programming format via designing the novel nonlinear Lyapunov-Krasovskii (L-K) functionals and a set of special switching sequences under average dwell time (ADT) switching rule. Eventually, with the help of comparison principle, the theoretical results can be further expanded to time-varying switched systems. As practical applications, we apply the theoretical achievements to stability of switched neural networks to confirm the feasibility of the conclusions.

Adaptive sliding mode-based feedback linearization control for floating offshore wind turbine in region II

ABSTRACT Wind turbine systems are highly nonlinear and time-variable. Under external interference, the normal and stable operation of the system is seriously affected. In addition, the inertia of the wind turbine causes a serious lag in speed tracking. These effects are even more severe for floating offshore wind turbines (FOWT). To solve these problems, this paper proposes a new optimal torque control form, and further improves and optimizes it. Firstly, the feedback linearization is used to eliminate the nonlinear part of the system and the time-varying parameters, and a new torque control form is obtained. Then, the adaptive sliding mode is used to further optimize the torque controller to enhance the robustness of the system. Finally, adaptive sliding mode-based feedback linearized optimal torque control (ASMFLOTC) was obtained. ASMFLOTC was applied to the FOWT system to verify the effectiveness of its maximum power point tracking (MPPT) control. The results show that ASMFLOTC can better track the reference speed, effectively reduce the relative error, and improve the utilization rate of wind energy. And from the results, the platform motion of the proposed controller is not significantly different from that of other controllers. The proposed controller does not exacerbate the platform motion while increasing the output power. This shows its feasibility.

LIMO-GCN: a linear model-integrated graph convolutional network for predicting Alzheimer disease genes.

Alzheimer's disease (AD) is a complex disease with its genetic etiology not fully understood. Gene network-based methods have been proven promising in predicting AD genes. However, existing approaches are limited in their ability to model the nonlinear relationship between networks and disease genes, because (i) any data can be theoretically decomposed into the sum of a linear part and a nonlinear part, (ii) the linear part can be best modeled by a linear model since a nonlinear model is biased and can be easily overfit, and (iii) existing methods do not separate the linear part from the nonlinear part when building the disease gene prediction model. To address the limitation, we propose linear model-integrated graph convolutional network (LIMO-GCN), a generic disease gene prediction method that models the data linearity and nonlinearity by integrating a linear model with GCN. The reason to use GCN is that it is by design naturally suitable to dealing with network data, and the reason to integrate a linear model is that the linearity in the data can be best modeled by a linear model. The weighted sum of the prediction of the two components is used as the final prediction of LIMO-GCN. Then, we apply LIMO-GCN to the prediction of AD genes. LIMO-GCN outperforms the state-of-the-art approaches including GCN, network-wide association studies, and random walk. Furthermore, we show that the top-ranked genes are significantly associated with AD based on molecular evidence from heterogeneous genomic data. Our results indicate that LIMO-GCN provides a novel method for prioritizing AD genes.

A nonlinear manifold model order reduction approach for thermo‐mechanically coupled plasticity

AbstractThis study employs a nonlinear manifold mode order reduction (MOR) approach to address the high computational cost inherent in multi‐physical nonlinear models, which are discretized by the finite element method (FEM), with a particular focus on thermo‐mechanically coupled plasticity under finite strain conditions. Utilizing nodal snapshots obtained from a full‐order simulation, a projection matrix is constructed through singular value decomposition (SVD), facilitating the construction of a reduced system within a lower‐dimensional space. This reduction process entails the multiplication of the discretized residual vector and stiffness matrix by a linear projection matrix, complemented by the incorporation of a nonlinear projection component to ensure accurate reconstruction within the nonlinear part. Subsequently, a comprehensive three‐dimensional benchmark test is conducted to assess the accuracy and efficacy of the reduced‐order model.

A Novel Method for Designing Electronic Logic Circuits

This scientific paper presents a novel method for designing single and multi-output electronic logic circuits. The produced circuit consists of two parts. The first part consists of (n-1) XOR gates, where "n" is the number of input signals to the logic circuits. This part is called the linear part of the circuit. The second part is designed by any method used in designing logic circuits. This part called the non-linear part of the circuit. The suggested method can be programmed by any high level programming language to be used on any digital computer especially with the circuits which have a large number of input signals.

Discussion of S. Ponsioen, S, Jain and G. Haller: “Model reduction to spectral submanifolds and forced-response calculation in high-dimensional mechanical systems”, Journal of Sound and Vibration 488, 2020, pages 1–23

The authors of the discussed article considered two examples (1) a finite element model of a linear beam with a cubic spring attachment having 10 to 10,000 degrees of freedom, and (2) a finite element model of a geometrically nonlinear Timoshenko beam having 21 degrees of freedom. The authors concluded that Harmonic Balance (HB) becomes intractable for those examples. In this discussion, we falsify this conclusion. We demonstrate that very well-tractable computation effort is obtained when properly using the HB tool NLvib that was also employed in the discussed article. In particular, for (1), the well-known exact condensation to the nonlinear part of the equation system is carried out. For (2), suitable parameters are set for the path continuation and the Newton-type solver. With this, the computation takes from split seconds to about two minutes instead of the many hours reported in the discussed article.

Exact solitary wave solutions for a coupled gKdV–Schrödinger system by a new ODE reduction method

AbstractA new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with ‐power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with . No solutions for were known previously in the literature. For , four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized Hénon–Heiles form.

A Linear Quadratic Regulation Controller Based on Radial Basis Function Network Approximation

This paper proposes a linear quadratic regulation (LQR) tracking control method based on a radial basis function (RBF) that successfully compensates for the shortcomings of the LQR method. The LQR method depends on the linearity of a model. Specifically, an RBF neural network is used to approximate and compensate for the nonlinear part of a controlled object in the PID type-I, type-II and type-III control loops to improve the performance of the system. Through the simulation of different industrial systems, such as underdamped, overdamped and critically damped systems, the method significantly improves the dynamic response performance indices, such as the rise time and settling time, of the system.

Parametrical synthesis of various radio devices with the set quantity of identical cascades of type «the nonlinear part – the mixed two-port network»

Background. Presence of possibility of analytical definition of a part of parametres of various radio devices, optimum by criterion of maintenance of preset values of modules and phases of transfer functions on necessary quantity of frequencies, considerably reduces time of numerical optimisation of other part of parametres by criterion of formation demanded frequency response and phase response in a strip of frequencies. Till now such problems dared concerning radio devices only with one cascade of type «a nonlinear part - the coordination the device» or «the coordination the device – a nonlinear part». In quality «the coordination devices were used the jet, resistive, complex or mixed two-port networks. The problem of multicascade radio devices with jet two-port networks is solved also. Change of basis for the coordination two-port networks and a place of inclusion of a nonlinear part leads to change of area of a physical realizability. Aim. Working out of algorithms of parametrical synthesis of radio devices with any quantity of identical and unequal cascades of type «a nonlinear part ‑the coordination the mixed two-port network» by criterion of maintenance of the set frequency characteristics. Nonlinear parts are presented in the form of a nonlinear element and parallel either consecutive on a current or pressure of a feedback. Methods. The theory of two-port networks, matrix algebra, a decomposition method, a method of synthesis of actuation microwave devices, numerical methods of optimisation. Results. In interests of achievement of the specified purpose systems of the algebraic equations are generated and solved. Models of optimum two-port networks in the form of mathematical expressions for definition of interrelations between elements of their classical matrix of transfer and for search of dependences of mixed of two-poles from frequency are received. It is shown, that at certain parities between quantity of cascades and values of resistance of a source of a signal and loading of the one-cascade radio device frequency characteristics of one-cascade and multicascade radio devices appear identical or similar. Such schemes are named by equivalent. Conclusion. The comparative analysis of theoretical results (frequency response and phase response radio devices, value of parametres), received by mathematical modelling in system MathCad, and the experimental results received by схемотехнического of modelling in systems OrCad and MicroCap, shows their satisfactory coincidence.

Identification of a Non‐Commensurate Fractional‐Order Nonlinear System Based on the Separation Scheme

ABSTRACTThis article is aimed to study the parameter estimation problems of a non‐commensurate fractional‐order system with saturation and dead‐zone nonlinearity. In order to reduce the structural complexity of the system, the model separation scheme is used to decompose the fractional‐order nonlinear system into two subsystems, one includes the parameters of the linear part and the other includes the parameters of the nonlinear part. Then, we derive an auxiliary model separable gradient‐based iterative algorithm with the help of the model separation scheme. In addition, to improve the utilization of the real time information, an auxiliary model separable multi‐innovation gradient‐based iterative algorithm is presented based on the sliding measurement window. Finally, the feasibility of the presented algorithms is validated by numerical simulations.

The Effects of Loading Angles on the Failure of Cross-Ply Notched Bio-Basalt Composites

A cross-ply basalt V-notched butterfly specimens were subjected to pure tension, combined tension-shear, and shear stress-strain state using a modified Arcan test fixture with loading angles from 0° to 90° with 15° increment. Multiaxial stress and strain states were studied using the principal stress and strain ratio, from which the principal angle was determined and used to represent principal states in the gauge section. The quasi-elastic to non-linear transition stresses were determined for each loading angle. In biaxial stress-strain states and pure shear, the deformation and consequently the shear-induced damage start to accumulate significantly. Also, in biaxial stress-strain states between 15°-75°, the shift from tension-shear to pure shear was observed after the transition to the non-linear part of the stress-strain curve. The digital image correlation (DIC) images and microscopic evaluation show that a large extent of damage is the consequence of the shear deformation after the rotation of 0° clamped fibres, while the 90° fibres maintained their original straight form. In off-axis tests, the principal strain axis rotates towards the weakest material axis even at small off-axis angles. This causes a transition from a tension-shear biaxial state in the linear loading part to shear in the non-linear part, leading to irreversible damage beyond the transition point.

Active Disturbance Rejection Control via Neural Networks for a Lower-Limb Exoskeleton

This article presents the design of a control algorithm based on Artificial Neural Networks (ANNs) applied to a lower-limb exoskeleton, which is aimed to carry out walking trajectories during lower-limb rehabilitation. The interaction between the patient and the exoskeleton leads to model uncertainties and external disturbances that are always present. For this reason, the proposed control considers that the non-linear part of the model is unknown and is perturbed by external disturbances, which are estimated by an active disturbance rejection control via Artificial Neural Networks. To validate the proposed approach, a numerical simulation and an experimental implementation of the ANN-Controller are developed.

Wind Speed Modeling under Quasi‐Linear Autoregressive Neural Network Model for Prediction of Production Power

Accurate wind speed modeling is beneficial for the design of wind energy conversion systems. Models of wind speed are used to assess the adequacy and dependability of a power supply. However, precise wind speed modeling is challenging due to the sporadic availability of wind speed. In this note, we propose a wind speed model with an autoregressive (AR) structure. A hybrid model is developed under linear and nonlinear parts based on a quasi‐linear autoregressive exogenous neural network (Q‐ARX‐NN). The model's structure is composed of a regression vector and its coefficients. The coefficients are divided into linear and nonlinear coefficients. A set of linear coefficients is identified under the algorithm of least square error (LSE), and a set of nonlinear coefficients is modeled by using a neural network to refine the residual error of the nonlinear part. Some artificial neural network (ANN) models can be set as nonlinear part sub‐models to sharpen the model's accuracy. The proposed model is tested for wind speed modeling to estimate wind energy production. Various nonlinear parts of the sub‐model are tested, such as neural networks, radial basis function networks, and ANN networks. Moreover, we evaluate the effects of the order of the model by varying hidden and output nodes, which can be summarized as the number of coefficients of the regression vector. Using specific wind turbine performance data, prediction models estimate production power. © 2024 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Developing a Hybrid ARIMA-XGBOOST Model for Analysing Mobile Money Transaction Data in Kenya

This study formulated a hybrid ARIMA-XGBOOST model using the dataset from the Central Bank of Kenya and the objective was to formulate a Hybrid ARIMA-XGBOOST models to capture the patterns and dynamics in Mobile Money Transactions in Kenya. The study was motivated by the gaps that existed from the previous studies which lacked the ability to model both linear and non-linear trends in the data across time period. Since 2007 when Mpesa was invented, the rate of Mobile Money transactions has rose steadily and this required a complex model that will provide a clear trend. To accurately model the dynamics of Mobile Money Transactions in Kenya, this study presented a complex Hybrid model. The model was formulated by combining Autoregressive Integrated Moving Averages (ARIMA) and Extreme Gradient Boosting (XGBOOST). The ARIMA captures the linear trends of the data while the XGBOOST models the non-linear part and trained to model the ARIMA residuals. The model parameters were evaluated using Mean Absolute Error, Root Mean Square Error among others and confirmed to be accurate and reliable. Based on the findings from the ADF coefficient, the stationary condition was met (P-value=0.01), and therefore we proceeded to develop the ARIMA models. Initial diagnoses included model identification and examination of autocorrelation to determine the ARIMA configurations, whereas the Box-Jenkins test confirmed the models' adequacy (P-value=2.220e-16). Based on Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), and Mean Percentage Error (MPE), which are all below 1 percent, indicates that the performance of the models had high prediction accuracy.

Application of the Cubic Exponential B-spline Collocation Operator Splitting Method for Numerical solutions of Burgers Equation

In this study, the cubic exponential B-spline collocation method has been proposed for the numerical solutions of the Burgers equation with the operator splitting. To apply the operator splitting method, the Burgers' equation has decomposed into two sub-equations based on the time term: the linear part (diffusion) and the nonlinear part (convection). Subsequently, for each sub-equation, Crank-Nicolson finite difference schemes in the temporal direction and cubic exponential B-spline functions and their derivatives, have applied at the x_m nodal points in the spatial direction. The algebraic equation systems obtained have been solved numerically using the Lie-Trotter and Strang splitting schemes to get the solutions of the main equation. Some advantages of the splitting methods include preserving the physical characteristics of the solution, yielding more convergent results over long time intervals, enabling simpler algorithms, and facilitating the storage of solution vectors on computer. To assess the accuracy of the computed numerical results the L_2 and L_∞ error norms have been used. Additionally, the obtained results have been compared with some studies in the literature. The stability analysis of the applied method has been investigated using the von Neumann Fourier series method.

Adaptive sliding mode control of petrochemical flare combustion process based on radial basis function network

Steam-assisted combustion elevated flares are currently the most widely used type of petrochemical flares. Due to the complex and variable composition of the waste gas they handle, the combustion environment is severely affected by meteorological conditions. Key process parameters such as intake composition, flow rate, and real-time data of post-combustion residues are difficult to measure or exhibit lag in data availability. As a result, the control methods for these flares are limited, leading to poor control effectiveness. To address this issue, this paper proposes an adaptive sliding mode control method based on the Radial Basis Function (RBF) network. Firstly, the operational characteristics of the petrochemical flare combustion process are analyzed, and a control model for the combustion process is established based on carbon dioxide detection. Secondly, an RBF neural network-based unknown function approximator is designed to identify the nonlinear part of the actual operating system. Finally, by combining the control model of the petrochemical flare combustion and designing the RBF sliding mode controller with its adaptive control law, fast and stable control of the flare combustion state is achieved. Simulation results demonstrate that the designed control strategy can achieve tracking control of the petrochemical flare combustion state, and the adaptive law also accomplishes system identification.

Alternative split-step method for solving linearly coupled nonlinear Schrödinger equations

In this paper we introduce an alternative method for solving linearly coupled nonlinear Schrödinger equations by using a split-step approach. This methodology involves approximating the nonlinear part of the evolution operator, allowing it to be solved exactly, which significantly enhances computational efficiency. The dispersive component is addressed using a spectral method, ensuring accuracy in the treatment of linear terms. As a reference, we compare our results with those obtained using the Runge-Kutta method implemented using a pseudo-spectral technique. Our findings indicate that the proposed split-step method achieves precision comparable to that of the Runge-Kutta method while nearly doubling computational efficiency. Numerical simulations include the evolution of a single soliton in each field and a collision between two solitons, demonstrating the robustness and effectiveness of our approach.

Numerical Solutions of Some Weak Singular Nonlinear Integral Equations of The First Type Using The Spectral Collocation Method

In this paper, a collocation-spectral approximation is proposed for weakly nonlinear and neutral singular Volterra integral-differential equations with rough solutions. We used some appropriate transformations to transform the equation of the equation into a new equation, so that the solution of the new equation has a better order (smoothness) and Jacobi's orthogonal polynomial theory can be easily used. Under appropriate assumptions on the nonlinear part, we were able to perform an acceptable error analysis on the soft and the weighted soft. To obtain a numerical approximation, some numerical examples (linear and non-linear) with uneven solutions are considered and numerical results are also presented. Also, a comparison between the proposed method and some existing numerical methods is also provided.