The rapid integration of inverter-based renewable resources poses significant challenges to power systems' stability and resilience. This paper presents a data-driven variable-order fractional control framework that enhances grids' resilience through adaptive memory management. The proposed controller employs a hybrid Caputo-Hadamard structure, in which the fractional order $\alpha (t)$ adapts in real time occording to wide-area frequency measurements. The Caputo component captures short-memory transient dynamics associated with power electronic responses, while the Hadamard component represents long-memory logarithmic effects arising from variability in the load and renewable generation. Rigorous stability analysis establishes Mittag-Leffler stability under bounded order variation and Ulam-Hyers practical stability, ensuring robustness against modeling uncertainties and numerical discretization errors. Numerical validation using realistic US power system data from Pennsylvania-New Jersey-Maryland (PJM) Interconnection, California Independent System Operator (CAISO), National Renewable Energy Laboratory (NREL), and Frequency Monitoring Network (FNET/GridEye) demonstrates consistently improved performance compared with integer-order and fixed-order fractional controllers, including up to $67\mathrm{\%}$ reduction in voltage overshoot and $72\mathrm{\%}$ reduction in the duration of rate of change of frequency violations under compound disturbance scenarios. The proposed framework provides a mathematically rigorous and practically viable approach for adaptive control in renewable-rich power systems, aligning with ongoing grid modernization efforts that seek to balance fast transient response with long-term stability in the system.

The research developed a resilient time-varying formation control strategy with prescribed-time convergence to a bounded residual set for non-strict-feedback second-order MASs to maintain accurate tracking under these conditions. Neural networks function to predict unknown nonlinear dynamics, while a state observer based on neural networks uses partial leader information to reconstruct unmeasured states. The effects of FDI attacks and communication uncertainties were addressed through matrix equalities/inequalities that solve Laplacian asymmetry problems. The proposed method achieves semi-global practical finite-time stability because it maintains all closed-loop signals within their bounded limits while tracking errors stay within their defined performance limits. The simulation results showed that formation errors achieve the prescribed bounds in finite time while maintaining stability and reliable coordination under adversarial and uncertain conditions, which demonstrates the method's robustness and scalability.

In this article, we study the stochastic complex coupled Kuralay model, which possesses some applications in various fields, including physics, biology, and engineering, to obtain new solitary wave solutions. The explicit analytical solutions are obtained by using the Sardar subequation method, which helps to illuminate the dynamics of oscillators under random (noisy) effects. The integration of the Wiener process along a given method is a precise approximation of the stochastic behavior of the system. The proposed strategy enables the derivation of several exact solitary wave solutions under stochastic conditions, including bright, dark, and singular wave profiles. More significantly, the obtained solutions are also represented by 3D surface and contour plots that clearly show how solitary waves change and evolve when noise is introduced. Other stochastic models in physics and engineering can use the proposed approach to understand the workings of complex systems.

This paper investigates a class of fractional differential equations (FDEs) that involve the generalized Katugampola fractional derivative (FD) subject to nonlocal boundary conditions. By transforming the considered boundary value problems (BVPs) into equivalent integral equations, we establish several results concerning the existence and uniqueness of solutions. The analysis is carried out using classical fixed point (FP) techniques, including the Banach contraction principle(BC), as well as Schaefer's FP theorems under appropriate assumptions. In addition, we examine the Lyapunov stability of nontrivial solutions and derive sufficient conditions to ensure asymptotic stability. The obtained results extend and complement the existing contributions in the literature on fractional BVPs with nonlocal conditions. Finally, illustrative examples are provided to demonstrate the applicability of the theoretical findings.