Discrete Nonlinear Model Research Articles

Invariant Sets, Global Dynamics, and the Neimark–Sacker Bifurcation in the Evolutionary Ricker Model

In this paper, we study the local, global, and bifurcation properties of a planar nonlinear asymmetric discrete model of Ricker type that is derived from a Darwinian evolution strategy based on evolutionary game theory. We make a change of variables to both reduce the number of parameters as well as bring symmetry to the isoclines of the mapping. With this new model, we demonstrate the existence of a forward invariant and globally attracting set where all the dynamics occur. In this set, the model possesses two symmetric fixed points: the origin, which is always a saddle fixed point, and an interior fixed point that may be globally asymptotically stable. Moreover, we observe the presence of a supercritical Neimark–Sacker bifurcation, a phenomenon that is not present in the original non-evolutionary model.

Comparative assessment and policy analysis of forecasting quarterly renewable energy demand: Fresh evidence from an innovative seasonal approach with superior matching algorithms

In the pursuit of sustainable development, accurate renewable energy demand forecasting holds great significance for climate change mitigation and promoting sustainability. However, renewable energy forecasting has been consistently challenged by seasonality and nonlinearity. Identifying the periodic and nonlinear characteristics concealed within renewable energy sources accurately is still an unexplored problem. Consequently, an innovative nonlinear discrete seasonal grey model is proposed for renewable energy forecasting, which incorporates seasonal dummy variables and a power exponent term for handling the seasonality and nonlinear patterns in time series. Furthermore, an intelligent algorithm matching framework is proposed to augment the flexibility of the newly developed model. For practical purposes, the new methodology is contrasted against a range of benchmarks encompassing statistical, machine-learning, and traditional grey models in forecasting the quarterly total renewable energy consumption in the United States. The proposed model exhibits over 27% improvement rates over its counterparts, achieving the most superior predictive accuracies of 1.45%, 39.27, and 0.79 in MAPEP, RMSEP, and MASEP metrics, respectively. Furthermore, the probability density and sample size analyses are conducted to validate the robustness of the new model, confirming its adaptability and stability towards algorithm randomness and historical information volume. Consequently, the novel model is employed to forecast the short-to-long-terms renewable energy consumption in the U.S., showcasing an upward trend and seasonal fluctuations of the consumption for the forthcoming 24 quarters from 2023Q4 to 2029Q3. These insights can offer valuable implications to the stakeholders such as energy suppliers, utility managers, and policy advocates, highlighting actionable strategies for optimizing renewable energy consumption forecasting and aiding sustainable development initiatives.

A novel intervention effect-based quadratic time-varying nonlinear discrete grey model for forecasting carbon emissions intensity

In the context of severe global warming, accurately exploring the trend of carbon emissions intensity (CEI) changes is of great significance for mitigating climate change issues. The implementation of China's Carbon Emissions Trading Scheme (ETS) in 2013 is a policy intervention aimed at influencing CEI. The impact of intervention events makes forecasting a complex problem, which poses significant challenges to the construction of forecasting models. We first develop a quadratic time-varying nonlinear discrete grey model (QDNDGM(1,1)) to assess the intervention effect of the ETS policy. Then, a novel intervention effect-based quadratic time-varying nonlinear discrete grey model (IE-QDNDGM(1,1)) is developed to conduct the prediction under intervention effect, including an intervention term. The Whale Optimization Algorithm (WOA) is used to calculate a nonlinear parameter. We assess the intervention effect of the ETS policy in China and find that it can indeed reduce CEI. We verify the IE-QDNDGM(1,1) model’s superiority by comparing its predictive performance with that of three grey models, one statistical technique, and one artificial intelligence model. The comparative study shows the proposed model’s excellent fitting and prediction performance. An ablation experiment is conducted to validate the design of the IE-QDNDGM(1,1) model. Policy implications of the ETS intervention effect are discussed.

Learning-based double layer control method of yaw stability simulation research for rear wheel independent drive electric tractor plowing operation

Complex field environments and variable soil conditions make tractor plowing operations susceptible to maneuvering instability, especially challenging in yaw stability. A learning-based double layer control (LDLC) method of yaw stability for rear wheel independent drive electric tractor (ET) plowing operation is proposed in this paper in order to solve this problem. Firstly, a sliding window self-learning traction resistance predictor is proposed based on Gaussian process regression (GPR) theory to predict the future traction resistance of plowing operation. Then, a discrete nonlinear model predictive control (NMPC) supervisor yaw stability controller is designed based on the dynamics model of the plowing unit and the tire-soil model to solve for the additional yaw torque required to maintain the yaw stability of the tractor. Finally, considering the additional yaw moment and driver demand torque, the subordinate controller composed of a particle swarm optimization algorithm optimally allocates the driving torque of the two driving wheels. Hardware-in-the-loop test results show that the maximum mean tracking errors of yaw rate and sideslip angle are 0.002 rad/s and 0.01 rad for the LDLC controlled ET during straight line and continuous steering under normal plowing operation, the maximum mean tracking errors of yaw rate and sideslip angle are 0.007 rad/s and 0.012 rad for straight line and continuous steering plowing operations under harsh plowing operation. The tracking accuracy is significantly higher than traditional cascade PID and equal distribution control methods. The proposed LDLC method effectively ensures the yaw stability of the ET during straight line and continuous steering under normal and harsh plowing operation conditions.

Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions

Abstract We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence.

Application of terminal region enlargement approach for discrete time quasi infinite horizon nonlinear model predictive control

SummaryEnsuring nominal asymptotic stability of the nonlinear model predictive control (NMPC) controller is not trivial. Stabilizing ingredients such as terminal penalty term and Terminal Region (TR) are crucial in establishing the asymptotic stability. Approaches available in the literature provide limited degrees of freedom for the characterization of the TR for the discrete time quasi infinite horizon NMPC formulation. Current work presents alternate approaches namely arbitrary controller based approach and linear quadratic regulator (LQR) based approach, which provide larger degrees of freedom for enlarging the TR. Both the approaches are scalable to system of any dimension. Approach from the literature provides a scalar whereas proposed approaches provide two additive matrices as tuning parameters for shaping of the TR. Proposed approaches involve solving modified Lyapunov equations to compute terminal penalty term, followed by explicit characterization of the TR. Efficacy of the proposed approaches is demonstrated using benchmark two state system. TR obtained using the arbitrary controller based approach and LQR based approach are approximately 10.4723 and 9.5055 times larger by area measure when compared to the largest TR obtained using the approach from the literature. As a result, there is significant reduction in the prediction horizon length while retaining the feasibility of the controller.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings.

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Computational study of a homogenized nonlinear generalization of Timoshenko beam proposed by Turco et al.

<p>Mechanical metamaterials are most often assemblies of stocky beam elements connected through rigid connections, hinges, or flexural joints. The description of these materials through classical beam theories is challenging because of the wide variety of complex phenomena observed in the severe deformation regime mechanical metamaterials must undergo and because most classical beam theories can only be applied to elements with sufficiently high slenderness. In the spirit of Hencky, Turco et al. (2020) has recently formulated an intrinsically discrete nonlinear elastic model suitable for the design of mechanical metamaterials. The objective of this contribution was to present a numerical study of the nonlinear generalization of the Timoshenko beam that results from the asymptotic homogenization of the discrete model introduced by Turco et al. The present numerical study took into account several loading cases and elucidated the sensitivity of the homogenized continuum with respect to axial, bending, and shear stiffness parameters, as well as to load imperfections, in terms of mechanical behavior, including buckling onset and post-critical behavior. It was found that the predictions obtained with the homogenized model in the large deformation regime matched excellently with those of the discrete model proposed by Turco et al.</p>

A novel grey multivariate convolution model based on the improved marine predators algorithm for predicting fossil CO2 emissions in China

The effective prediction of fossil carbon dioxide (CO2) emissions provides a foundation for environmental protection authorities to make relevant management and decisions. CO2 emissions are influenced by various factors and exhibit a nonlinear and complex trend. First, the weighted conformable fractional accumulation (WCFA) method is proposed. WCFA is a generalized form of the existing weighted accumulation and conformable fractional order accumulation. A nonlinear discrete grey Bernoulli multivariate convolution model is introduced to model the sequences processed by WCFA. Then, an improved marine predators algorithm (IMPA) is obtained by improving the marine predators algorithm by the tent map and cosine control parameter. The performance of IMPA is verified by algorithm comparison experiments and Friedman test. The feasibility and predictive effectiveness of the proposed model is validated through three case studies. In prediction of fossil CO2 emissions in China, the fitting and prediction errors of the proposed model reach 1.1773% and 0.5402%, respectively. Finally, the total fossil CO2 emissions in China from 2022 to 2026 are predicted and analyzed. The results indicate a gradual decrease in fossil CO2 emissions over the next five years. Some reasonable suggestions are made based on the prediction results and sources of fossil CO2 emissions.

Fuzzy cellular automaton with slow-to-start property

We propose a nonlinear discrete model which incorporates the effects of slow-to-start property into rule 184 fuzzy CA as a mathematical traffic flow model. The exact solutions corresponding to free-flow and congested states are given for the new model. The exact average fluxes of both states are also given. By studying the stability of the uniform solution corresponding to the congested state, the critical point at which congestion does not occur is obtained analytically. This may be the first analytical result on stability of congested states of the models with slow-to-start property. The stability of some typical free-flow states are also discussed analytically.

Creep Instability Mechanism and Control Technology of Soft Coal Roadways Based on Fracture Evolution Law

To address the challenging issues of large deformation, control difficulties, and susceptibility to failure in the support structure of soft coal roadways, this study utilizes the CVISC block creep model in UDEC software. The model incorporates Coulomb slip without cohesive contact to simulate the characteristics of soft coal, such as its loose, fragile, and small-block nature. Additionally, a soft coal nonlinear discrete element creep model is developed to investigate the creep characteristics of soft coal under triaxial compression, with the aim of revealing the underlying creep destabilization mechanism in soft coal tunnels. Based on the research findings, a primary, strong active support technology is proposed. This approach involves the use of high-preload, high-strength anchor rods and anchor cables, as well as the implementation of steel mesh and plastic woven mesh to enhance surface protection. The study highlights that: (1) The shear cracks inside the coal body of the soft coal specimen transform into tensile cracks under external force, leading to an increase in the number of tensile cracks. This is an important symbol of accelerated creep in soft coal. Improvement in peripheral pressure helps inhibit the generation of tensile cracks inside the specimen. (2) The rapid development of creep and inter-particle tensile fissures within the coal body particles themselves, along with the change in stress state after the excavation of the roadway, are the main reasons for the overall creep damage of the roadway. (3) The support force in the early stage of shed cable support is small, which cannot inhibit the accelerated development of tensile fissures. This leads to continuous deformation of the roadway, resulting in the failure of the support structure in the later stage and aggravated roadway damage. (4) The new support technology helps control surface deformation by enhancing the strength of the roadway protection surface. This suppresses the development speed and number of tensile fissures during roadway deformation, improves the starting strength of the roadway for accelerated creep, and enables effective control of the overall deformation of the soft coal roadway. Thus, the effectiveness of roadway support is remarkable.

Forecasting renewable energy generation with a novel flexible nonlinear multivariable discrete grey prediction model

Forecasting renewable energy generation with a novel flexible nonlinear multivariable discrete grey prediction model

Synchronization of Fractional Partial Difference Equations via Linear Methods

Discrete fractional models with reaction-diffusion have gained significance in the scientific field in recent years, not only due to the need for numerical simulation but also due to the stated biological processes. In this paper, we investigate the problem of synchronization-control in a fractional discrete nonlinear bacterial culture reaction-diffusion model using the Caputo h-difference operator and a second-order central difference scheme and an L1 finite difference scheme after deriving the discrete fractional version of the well-known Degn–Harrison system and Lengyel–Epstein system. Using appropriate techniques and the direct Lyapunov method, the conditions for full synchronization are determined.Furthermore, this research shows that the L1 finite difference scheme and the second-order central difference scheme may successfully retain the properties of the related continuous system. The conclusions are proven throughout the paper using two major biological models, and numerical simulations are carried out to demonstrate the practical use of the recommended technique.

Construction of abundant solitons in a coupled nonlinear pendulum lattice through two discrete distinct techniques

In this paper, we deal with two distinct discrete techniques named the discrete Tanh method and the differential-difference Jacobi elliptic functions sub-ODE method to extract abundant soliton solutions for a coupled nonlinear pendulum chains. According to the two straightforward schemes’, we construct various solutions likewise, kink, anti-kink, bright, dark, singular-bright solitary wave, traveling compacton, traveling gray soliton-like, periodic soliton-like have not reported on the most recently studied mechanical model. We extract the sought solutions with less approximation (introduction of a discrete wave transformation) and display some graphical representations showing well known, novel shapes discovered in a coupled nonlinear pendulum lattice. More often, the tediousness computations of discrete techniques lead some researchers to changing a nonlinear differential-difference equation into a nonlinear ordinary differential equation throughout the semi-discrete approximation before introducing a wave transformation. Nonetheless, the semi-discrete approximation involve a Taylor bounded expansion’s such as the threshold order depends on the ability and skills of researchers to get solutions through available methods and those that they can perform. Hence, while an algorithm can be lead us to avoiding approximation (eventual source of errors), tediousness analytical computations, we perform by using it, in addition of software availability. And therefore, we establish as is the case in the present work various soliton-like solutions by means of two direct discrete techniques. The effect of some physical parameters are displayed on the shaped through 2D and 3D graphical simulations. The used methods (tanh, sn, cn, dn Jacobi elliptic functions approaches) previously applied on other physical models (discrete electrical transmission lattices) are presented here in manner to be replicable for exploring more soliton solutions in different nonlinear discrete physical models.

Folding a flat rectangular plate of uniform-thickness panels using Miura-ori

In the realm of space technology, aerospace solar arrays are conventionally employed as efficient means of energy harvesting. The solar arrays need to be packaged in a payload capsule before launch and unfold as a standard rectangle with uniform thickness and a flat surface in orbit. In this paper, we introduce a novel methodology for folding a standard rectangular plate with uniform thickness and a flat surface based on Miura origami. Importantly, these plates can be unfolded with a single degree of freedom after a special design. The package direction is defined by analyzing the distinct characteristics of wave creases and straight creases. To achieve motion equivalence, different lengths of tape spring hinges are used along the package direction, while door hinges are employed at the wave crease. In order to address the motion interference problem during folding and maintain structural stability, constraints are derived, and a discrete nonlinear optimization model is established. The particle swarm optimization algorithm is used and improved to enhance convergence performance and computational efficiency. According to the calculation results, the corresponding folding plates were made, and the folding process and results are consistent with zero-thickness origami. These findings suggest that the proposed thick-panel origami method not only achieves efficient optimization of the geometric design but also preserves the essential characteristics of traditional zero-thickness origami. To aid designers, some key design recommendations based on the parameter analysis are provided for enhancing the deployment ratio.

On the intrinsic pinning and shape of charge-density waves in 1D Peierls systems

Within the standard perturbative approach of Peierls, a charge-density wave is usually assumed to have a cosine shape of weak amplitude. In nonlinear physics, we know that waves can be deformed. What are the effects of the nonlinearities of the electron-lattice models in the physical properties of Peierls systems? We study in details a nonlinear discrete model, introduced by Brazovskii, Dzyaloshinskii and Krichever. First, we recall its exact analytical solution at integrable points. It is a cnoidal wave, with a continuous envelope, which may slide over the lattice potential at no energy cost, following Fröhlich’s argument. Second, we show numerically that integrability-breaking terms modify some important physical properties. The envelope function may become discontinuous: electrons form stronger chemical bonds which are local dimers or oligomers. We show that an Aubry transition from the sliding phase to an insulating pinned phase occurs when the model is no longer integrable.

Voltage monitoring based on ANN-aided nonlinear stability analysis for DC microgrids

Due to the low inertia of the DC microgrid, the DC bus voltage is prone to drop or oscillate under disturbance. It is also challenging to supervise the stability of a DC microgrid since it is a highly nonlinear dynamic system with high dimensionality and randomness. To tackle this problem, this paper proposes a new method using ANN-aided nonlinear dynamic stability analysis for monitoring the DC bus voltage, which is combined with two steps. The first step is to establish six corresponding nonlinear accurate discrete iterative models of six switching modes of the PV-battery-load-based DC microgrid system, based on the Poincaré map theory, in order to judge the stability quantitatively with a promoted stability margin index. The second step is to use artificial neural networks (ANNs) to forecast the operating mode of the system when random changes occur in environmental circumstances and load power; this will aid the first step in being efficient and adaptable while determining stability cases. And the employed ANNs are trained with the datasets, including the circuit data, ambient temperature, irradiance, and load power, which are generated by MATLAB/Simulink simulation. Theoretical and simulation analyses are carried out under different operating conditions to validate the proposed method’s efficacy in judging the DC microgrid’s destabilizing oscillation and stable running.

Генетическая дивергенция в системе смежных популяций при плотностно-зависимом лимитировании гаметопродукции

The paper studies the mechanisms leading to the emergence of genetic divergence (stable genetic differences) between two populations coupled by migration. We considered the classical system of panmictic populations with Mendelian rules of inheritance and monolocus selection directed against heterozygotes. In order to limit the growth of populations, we propose to assume that gamete production and total fertility (birth) decreases with population growth due to limited resources. We have proposed a non-linear discrete time model that describes the concentration dynamics of one of the alleles and each population abundance. To calculate the coordinates of all fixed points corresponding to different types of the limiting genetic structure and the abundance ratio, we have proposed a method for calculating their coordinates. It is shown that with a density-dependent birth limitation in the model, a set of fixed points corresponding to a homogeneous and heterogeneous distribution is possible. At a homogeneous distribution, both populations are monomorphic, with individuals belonging to only one genotype. At this, the limiting values of population abundance do not always coincide. With a non-homogeneous distribution, adjacent populations are polymorphic with individuals of different genotypes, but they differ significantly in the frequencies of alternative alleles and asymptotic population abundance. Bifurcations of the fixed points birth corresponding to heterogeneous distribution and genetic divergence are described. It is found that the movement towards one of the possible limiting genetic structures is accompanied by a change in the reproductive capabilities of populations.It is shown that a reduced fitness of heterozygotes in monomorphic populations results in a higher birth rate as compared to polymorphic populations obviously containing individuals with different reproductive capabilities. As a result, monomorphic and polymorphic populations correspond to different limiting abundances and cycles with different periods and oscillation phases after the loss of stability, even if the populations are completely identical.

Construction of continuous and discrete non-linear prognostic models of the control system for the process of grinding ore materials

The article analyzes the developed non-linear predictive model of the control system in continuous time of the state space, including states, control and controlled values. The transformation of the continuous form of a function into a discrete one is carried out according to the trapezoid rule. Based on the discrete model, the optimality criterion for the objective function is formulated. The efficiency of the grinding process is considered based on the weight components of the model, since it entirely depends on the accuracy of estimating the particle size of the product and the performance of the plant. To control the volume of water in the sump, a dynamic inversion controller is provided, which makes it possible to reduce control problems to a solution in a closed form and guarantee asymptotic stability of the error dynamics.

On the Asymptotic Stability of Discrete Crocodilians Model

The sex ratio of crocodiles is strongly biased towards females, often as high as 10 females to 1 male. In crocodilians, the temperature of egg incubation is the environmental factor determining sex. If the temperature is low, around 30˚C, the hatchlings are all females. Higher temperature, around 34˚C, hatch all males. This study was made to consider the asymptotic stability of a positive equilibrium point in a nonlinear discrete model of the basic nesting population model, which is described in three-region depending on the temperature of egg incubation. This model is based on key life-historical data and Murray’s research. To study above, we have applied the classical linearization method and P. Cull’s method and moreover, we employ non-standard discretization methods for later our Equations (6)-(8) and (15).