Search Results (594)

Generalized incommensurate fractional differential systems (GIFDSs) unify classical fractional frameworks via weight functions, capturing non-uniform multicomponent system dynamics. This paper fills a critical research gap by analyzing GIFDSs for both commensurate and incommensurate weight functions. For commensurate weights ($w_i\left(t\right)=w\left(t\right)$), classical IFDS equivalence is established via state transformation. Linear homogeneous mild solutions are derived using the incommensurate Mittag–Leffler function. Existence and uniqueness of nonlinear solutions are proved under continuity and Lipschitz assumptions. Hyers–Ulam stability is verified for linear non-homogeneous systems. For incommensurate weights (distinct $w_i\left(t\right))$, a novel framework is developed: by the integral bound lemma and Picard iteration, local existence (existence on $[a,t_1]$) is established, then it is extended to the full interval. The global uniqueness is obtained by Gronwall-type inequality via combined substitution. These results are applied to Hopfield Neural Networks, showing that one-layer HNNs with tanh or sigmoid activations admit unique mild solutions under GIFDS dynamics. Full article

This article focuses on the fixed-time synchronization problem for fractional-order fuzzy cellular neural networks (FOFCNNs) with information interactions and time-varying delays. To capture the complex dynamics of practical networks, nonlinear activation functions along with fuzzy AND and OR operators are incorporated into the master–slave systems. To achieve fixed-time synchronization despite these complexities, a novel adaptive multi-module controller is proposed. This controller integrates three functionally distinct components to accelerate the convergence rate, eliminate the effects of delays, and introduce negative feedback during communication, respectively. By employing fractional calculus tools, inequality techniques, and the proposed control law, sufficient criteria for the synchronization of the considered systems are rigorously established. Compared with existing synchronization works, this paper has significant advantages in model generality and controller design. Additionally, an explicit settling-time estimate is derived, which depends solely on control parameters and is independent of the initial conditions. Full article

This article proposes a novel time-space two-grid high-order compact difference scheme for the one-dimensional nonlinear Schrödinger equation subject to Dirichlet boundary conditions. In comparison with the fully nonlinear compact difference scheme, the proposed methodology combines a small-scale nonlinear fourth-order compact difference algorithm on a time-space coarse grid and a large-scale linearized correction compact difference algorithm on a fine grid. In contrast to the time two-grid compact difference method, the proposed scheme applies the two-grid technique in both the spatial and temporal domains, thereby further improving computational efficiency. Solutions from the coarse grid are projected onto the fine grid via a temporally linear and spatially cubic Lagrange interpolation operator. Unconditional stability and optimal convergence rates, which are fourth-order in space and second-order in time, are proven in both the discrete $L^2$ and $L^{\infty }$ norms, without any constraints on the grid ratio. In addition to the standard techniques of the energy method, a discrete Sobolev inequality and an a priori error estimate are employed to demonstrate stability and high-order convergence. Finally, the theoretical results are validated through numerical experiments, which confirm the robustness and reliability of the proposed approach. A single-soliton experiment demonstrates that, compared with the fully nonlinear compact difference scheme, the proposed method achieves a significant reduction in CPU time while maintaining a comparable level of accuracy. Additional experiments further illustrate the algorithm’s effectiveness in simulating two-soliton interactions and soliton birth. These findings establish the proposed scheme as a highly efficient alternative to conventional nonlinear approaches. Full article

This paper investigates the stability of nonlinear systems with truncated $\mathcal{V}$-fractional-order derivatives. Initially, based on the fundamental properties of $\mathcal{V}$-fractional calculus, the Bellman–Gronwall inequality for $\mathcal{V}$-fractional $\alpha$-differentiable functions is derived. Subsequently, several sufficient conditions for the stability of the considered systems are established via the Lyapunov direct method. For practical applications, multiple synchronization criteria for drive-response systems are further deduced by leveraging the aforementioned stability results. Finally, numerical examples are presented to verify the effectiveness and feasibility of the main theoretical findings. Full article

The rapid expansion of digitalization is reshaping factor mobility and income distribution between urban and rural areas, with important implications for inclusive and sustainable development. Using panel data for 277 prefecture-level cities in China from 2012 to 2022, this study examines how DE affects urban–rural income disparity from the perspectives of nonlinear effects, factor allocation, and spatial interdependence. Compared with existing studies based mainly on provincial data, this paper provides a more fine-grained analysis at the prefecture level and combines mediation, double-threshold, and spatial analysis within a unified framework. The results show that DE has a significant U-shaped effect on urban–rural income disparity, suggesting that digital development may initially narrow the gap but widen it after a certain stage. Urban–rural factor allocation acts as an important transmission channel, and its role exhibits a double-threshold characteristic. The effect of DE also varies across urban agglomeration types and stages of urbanization, with stronger impacts in more developed and urbanized regions. In addition, the direct effect of DE follows a U-shaped pattern, whereas its spatial spillover effect shows an inverted U-shape. These findings indicate that digitalization is not automatically equalizing and that its distributional consequences depend on factor allocation conditions, regional development stages, and spatial linkages. The study provides evidence for policies aimed at reducing urban–rural inequality and promoting more balanced and sustainable development. Full article

This study developed and tested a neighborhood-scale framework that integrates unmanned aerial vehicle (UAV)-based multispectral mapping and georeferenced socioeconomic data to identify inequities in urban green infrastructure and translate them into an operational prioritization tool for inclusive planning. Using object-based image analysis, impervious surfaces, low vegetation, and woody vegetation (trees and shrubs) were mapped across 33 Neighborhood Units in Temuco, Chile, and landscape metrics describing dominance, edge, isolation/connectivity, and diversity were derived. Socioeconomic conditions were summarized through Principal Component Analysis, and their relationships with vegetation metrics were evaluated using Generalized Additive Models. The results revealed strongly nonlinear and metric-specific associations, with the most robust relationships observed for woody-structure metrics, particularly total woody edge and built-environment isolation, whereas landscape diversity showed weaker but still significant dependence on resource-access gradients. To support inclusive planning, a dimensionless Green Infrastructure Prioritization Index (GIPI) was computed by combining standardized green deficit and standardized social vulnerability with equal weights. GIPI values ranged from 0.318 to 0.740 (median = 0.528), identifying 11 high-priority units characterized by higher social vulnerability and less favorable woody structure, including lower largest-patch dominance and greater isolation. Sensitivity analyses varying the deficit weight from 0.30 to 0.70 showed that 10 of the 11 high-priority units remained in the same class in at least 80% of weighting scenarios, indicating a stable priority set. Further classification of high-priority units according to dominant deficit type supported a staged intervention strategy, in which woody canopy is first increased in deficit nodes and subsequently reinforced through corridor-oriented greening to improve structural connectivity. These findings demonstrate the value of coupling fine-scale vegetation mapping with socioeconomic gradients to support more equitable urban green infrastructure planning. Full article

Extreme heat poses growing health risks in high-density cities, yet static assessments often fail to capture dynamic pedestrian exposure. This study quantifies the supply and demand disparity between urban shade provision and actual pedestrian demand in Fuzhou, China, during a specific extreme heat event. Integrating high-resolution mobile signaling data with dynamic urban shade simulations, we classified the road network into risk quadrants and analyzed behavioral drivers using XGBoost and SHAP algorithms. Results show a pronounced disparity: high-risk zones carry the highest pedestrian flows (a mean daily volume of 28.6 pedestrian trajectories per segment) but exhibit minimal shade coverage (3.14%), while comfort zones provide 5.5 times greater shading coverage for comparable activity levels. In contrast, surplus zones exhibit substantial shading capacity but limited pedestrian use, indicating inefficient spatial allocation of cooling resources. Further analysis shows that pedestrian accumulation in high-risk zones is primarily driven by functional necessity, whereas pedestrian flows in comfort zones are more sensitive to thermal conditions. These findings reveal structurally embedded thermal exposure risk and support a shift from static metrics toward dynamic urban planning to protect vulnerable pedestrian flows. Full article

In this paper, a moving horizon estimation (MHE)-based state estimation problem is studied for asynchronous multi-rate nonlinear systems. First, the asynchronous multi-rate system is transformed into a synchronous system at measurement sampling points through pseudo-measurement synchronization modeling. Secondly, a MHE strategy with a time-discounted quadratic objective is proposed. Under the detectability assumption, the exponential stability of the proposed MHE is established via the Lyapunov method, and the corresponding linear matrix inequality (LMI) constraints are derived. Moreover, to address the model mismatch after synchronization, a deep learning-based framework is proposed to approximate and learn the weighting parameters of the MHE. Then, barrier-function regularization is introduced to enforce the aforementioned LMI feasibility conditions, keeping the learned weights within the feasible region throughout training. Finally, the result is illustrated by a target tracking example. Full article

This paper investigates the spatial asymptotic behavior of solutions to a class of nonlinear parabolic equations defined on an exterior region in ${\mathbb{R}}^3$. By constructing a suitable weighted energy functional and employing a fractional-order differential inequality technique, we establish a sharp Phragmén–Lindelöf type alternative: the solution either ceases to exist at a finite radial distance or decays to zero as the radial variable $r\to \infty$ when the power $p>2$. In the decay case, we derive explicit polynomial type decay estimates. The analysis is conducted in unbounded exterior domains where traditional compactness arguments are not applicable, extending previous studies on semi-infinite cylinders to more complex geometric settings. Our results reveal distinct spatial behaviors compared to those observed in linear or differently nonlinear parabolic problems and can be seen as a version of Saint-Venant principle in exterior regions. Full article

Motivated by the recent progress in Finite-Time Fault Estimation (FTFE) and its application to very few classes of Nonlinear Dynamical Systems (NDSs), this paper aims to drive further advancements in the field. In this research direction, a review of the literature reveals that most studies integrate the Linear Matrix Inequality (LMI) approach with the Takagi–Sugeno fuzzy (TSF) models to approximate nonlinear dynamics. However, the Sum Of Squares (SOS) approach offers numerous advancements and improvements over the LMI approach for TSF models. As an initial effort, by applying the SOS approach, this paper proposes two design procedures to ensure the finite-time boundedness of the state and actuator estimation errors for a class of polynomial fuzzy (PF) models. The first result relies on a polynomial integral observer. The second result is derived using a polynomial proportional-integral observer. Simulation results are provided to compare the two design procedures. Full article

Accurately quantifying the inequality of plant organ size distributions, such as leaf area, is essential for understanding plant resource allocation strategies, and this is commonly achieved using Lorenz curves. Previous studies have shown that the performance equation (PE) and its generalized form (GPE) effectively describe Lorenz curves that are rotated 135° counterclockwise around the origin and shifted rightward by $\sqrt{2}$ units. However, few studies have compared the fitting performance of PE (and GPE) with other traditional equations generating Lorenz curves in modeling empirical leaf area distributions, and even fewer have considered the validity of linear approximation assumptions in these nonlinear models. To address this gap, we quantified the inequality of leaf area distributions in Semiarundinaria densiflora, a bamboo species for which the abundant and measurable leaves per culm provide an ideal system for examining the ecological strategies underlying leaf allocation patterns. Five nonlinear models were employed to fit the leaf area distribution: PE, GPE, the Sarabia equation (SarabiaE), the Sarabia–Castillo–Slottje equation (SCSE), and the Sitthiyot–Holasut equation (SHE). Model performance was assessed using root-mean-square error (RMSE) and Akaike information criterion (AIC), while nonlinearity curvature measures were applied to evaluate the close-to-linear behavior of parameter estimates. In addition, the Lorenz asymmetry coefficient (LAC) was used to quantify the asymmetry of the Lorenz curves. Our results showed a clear trade-off between predictive accuracy and linear approximation behavior. Among the five models, GPE achieved the best fit, with the lowest RMSE and AIC values, yet did not show good close-to-linear behavior. In contrast, SHE provided the poorest fit but demonstrated the strongest close-to-linear properties. LAC values indicated that relatively abundant, larger leaves disproportionately contributed to the inequality in leaf area distribution. These findings highlight an inherent trade-off in using Lorenz-based models to describe leaf area frequency distributions: predictive accuracy does not necessarily align with statistical validity. By integrating model fit, nonlinearity diagnostics, and asymmetry assessment, this study provides new perspectives and methodological tools for future investigations into inequality in plant organ size distributions and their ecological significance. Full article

This study examines a comprehensive class of coupled nonlinear $\varrho$-Hilfer fractional neutral impulsive integro-differential systems with mixed delays and non-local initial conditions. The primary contribution of this study is the creation of a unified analytical framework that encompasses coupled interactions, neutral-type dependencies, and impulsive disturbances, which have been studied separately by researchers. We utilize the Banach contraction principle and Krasnoselskii’s fixed-point theorem to provide suitable conditions for the existence and uniqueness of solutions within the product space of piecewise continuous weighted functions. In addition to existence, we examine Ulam–Hyers–Rassias (UHR) stability using a generalized Gronwall inequality, which guarantees the system’s robustness against functional perturbations. We also develop a controllability framework and a feedback control law that steer the system towards the desired terminal states. The theoretical results are supported by a numerical simulation using a complex kernel, implemented via a modified predictor-corrector algorithm, which validates the practical effectiveness of the proposed control and stability outcomes. Full article

This paper investigates the fixed-time (FXT) and predefined-time (PDT) synchronization of memristive neural networks (MNNs) subject to stochastic disturbances, reaction-diffusion terms, and time delays. First, a new PDT stability criterion is established for stochastic nonlinear systems, which permits a priori assignment of the settling time bound regardless of initial conditions, and offers a more concise form than prior results. Second, by leveraging Green’s formula, integral inequality, and stochastic analysis, some sufficient conditions are derived to guarantee FXT and PDT synchronization of introduced stochastic MNNs with reaction-diffusion terms. Finally, numerical simulations are given to validate the effectiveness of the proposed synchronization scheme. Full article

This paper systematically studies a class of weakly singular Wendroff-type integral inequalities with multiple variables and multiple nonlinear terms. We establish explicit bounds for the unknown functions by utilizing the method of characteristic functions and mathematical induction. These results generalize and improve existing inequalities found in the literature. Furthermore, we apply them to fractional partial differential equations to study the uniqueness, boundedness, and continuous dependence of solutions. Even in the presence of singularities, the proposed method proves effective. An application example is provided to illustrate the validity of the main results. Full article

This paper proposes a nonlinear robust H_∞ excitation controller based on an improved slime mould optimization algorithm (ISMA) to enhance the stability and anti-disturbance performance of synchronous generators (SGs) in power systems. First, a nonlinear dynamic model of the excitation system (ES) is established based on the electromechanical coupling mechanism of SGs, and it is transformed into an equivalent linear state-space form through feedback linearization. Subsequently, a controller design framework with linear matrix inequality (LMI) constraints satisfying H_∞ performance indicators is constructed, and ISMA is utilized to optimize the key design parameters, thereby balancing dynamic response and control robustness. Simulation results demonstrate that, compared with traditional excitation control strategies, the proposed method exhibits superior comprehensive performance in terms of transient response speed, steady-state regulation accuracy, and robust performance under parameter perturbations and disturbance conditions. The research results can provide a technical reference for achieving safe and stable operation of SGs in power grids. Full article