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2026, Vol. 46 No. 1   Published date: 28 March 2026
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• Randomized Numerical Schemes for Random Periodic Solutions of SDEs with Low Temporal Regularity

  Jiang Yingsong, Niu Yuanling

  2026, 46(1): 1. doi: 10.3969/j.issn.1006-8074.2026.01.001

  Abstract ( 378 )   PDF (276KB) ( 24 )   　　

  This paper investigates the numerical approximation of random periodic solutions for stochastic differential equations (SDEs)

  whose drift coefficient is only $\alpha$-H\"older continuous ($\alpha>0$) and diffusion coefficient only $(\frac12+\beta)$-H\"older continuous ($\beta>0$) in time. To overcome the limited convergence order of traditional methods caused by low temporal regularity, we propose two randomized schemes: the randomized Euler method (REM) and the randomized exponential integrator (REI).

  These schemes offer complementary advantages: REM is simple to implement and broadly applicable, while REI achieves higher accuracy with a convergence order independent of $\beta$. By incorporating uniformly distributed random variables to sample the drift at randomized intermediate points, the random schemes attain higher convergence orders under low regularity conditions. Theoretical analysis shows that the mean-square convergence order of REM is $\min\left( \frac{1}{2}+\alpha, \frac{1}{2}+\beta, 1 \right)$, while REI achieves the order $\min\left( \frac{1}{2} + \alpha, 1 \right)$. Furthermore, we establish the existence and uniqueness of random periodic solutions for both numerical schemes and demonstrate their mean-square convergence to the exact random periodic solution of the SDE at the aforementioned orders. Numerical experiments are conducted to validate the theoretical findings.

  
  

• A Mixed Primal-Dual Dynamical System with Hessian-driven Damping and Its Convergence Analysis

  Zhang Xiqiao, Liu Lingling, Ding Kewei

  2026, 46(1): 18. doi: 10.3969/j.issn.1006-8074.2026.01.002

  Abstract ( 345 )   PDF (714KB) ( 18 )   　　

  This paper proposes a mixed primal-dual dynamical system with constant damping and Hessian-driven damping for solving linearly constrained optimization problems. The system consists of a second-order ordinary differential equation (ODE) with Hessian-driven damping for the primal variable and a first-order ordinary differential equation for the dual variable. By constructing an appropriate Lyapunov function, we analyze the convergence properties of the primal-dual gap, the feasibility measure and the objective function value, and establish exponential convergence rates under suitable scaling coefficients. Based on a time discretization of the continuous-time system, we derive an inertial primal-dual algorithm and validate the theoretical findings through numerical experiments, demonstrating the effectiveness and robustness of the proposed method. 

• Asymptotic Stability of Near-constant Steady States of the Chemotaxis Models in Symmetric Planar Domains

  Wang Hongze

  2026, 46(1): 37. doi: 10.3969/j.issn.1006-8074.2026.01.003

  Abstract ( 357 )   PDF (230KB) ( 23 )   　　

  The stability of non-constant steady states in chemotaxis systems remains a challenging problem. In this paper, we employ local bifurcation theory to establish the existence of near-constant steady states in bounded planar domains. Building on the local bifurcation structure, together with asymptotic expansions of solutions and inequalities for Neumann eigenvalues, we investigate the asymptotic stability of these bifurcating steady states for the minimal model. In particular, we provide a detailed stability analysis for near-rectangular and near-disk planar domains with two axes of symmetry.

• Laplacian Spectrum of the Nil-clean Graph of the Ring $\mathbb{Z}_{n}$

  Su Huadong, He Qing

  2026, 46(1): 58. doi: 10.3969/j.issn.1006-8074.2026.01.004

  Abstract ( 381 )   PDF (187KB) ( 20 )   　　

  In this paper, we determine the Laplacian spectrum of the nil-clean graph $G_{NC}(\mathbb{Z}_{n})$ for the ring $\mathbb{Z}_n$, obtain a necessary and sufficient condition for the Laplacian spectral radius of $G_{NC}(\mathbb{Z}_{n})$ to be equal to $n$, and furthermore, address the problem of the coincidence of the algebraic connectivity and the vertex connectivity of $G_{NC}(\mathbb{Z}_{n})$.

• Ground State Solutions for the Schrödinger-Poisson-Slater Equation with Double Critical Exponents

  Shao Yumi, Lei Chunyu

  2026, 46(1): 72. doi: 10.3969/j.issn.1006-8074.2026.01.005

  Abstract ( 409 )   PDF (174KB) ( 13 )   　　

  This paper is concerned with the Schr\"odinger-Poisson-Slater (SPS) equations incorporating both Coulomb-Sobolev and Sobolev critical exponents in $\mathbb{R}^3$. These double critical exponents, specifically 3 and 6, lead to the non-compactness of embeddings in the relevant function space. By utilizing the Pohozaev identity and the Br\'{e}zis-Lieb lemma, we show the existence of ground state solutions.

• Estimation of the Number of Solitary Periodic Wave Solutions for High-Order Singularly Perturbed Generalized KdV Equations

  Kou Guiyan, Xie Jiaqi, Yuan Xiaoping

  2026, 46(1): 81. doi: 10.3969/j.issn.1006-8074.2026.01.006

  Abstract ( 372 )   PDF (332KB) ( 16 )   　　

  This paper investigates the number of solitary periodic wave solutions for a class of perturbed generalized KdV (pgKdV) equations with two arbitrarily high‑order nonlinear terms. By applying the traveling wave transformation, the original partial differential equation is reduced to a planar ordinary differential system. Using geometric singular perturbation theory, the existence and counting of solitary periodic wave solutions are transformed into the problem of zero distribution of an Abel integral. For this Abel integral, the Chebyshev system criterion is employed to prove that any linear combination of its generating elements has at most one zero in the energy interval. Consequently, the original system admits at most one solitary periodic wave solution.

• Research on Total Variation Regularization Inversion Methods for Two-Dimensional Wave Equations

  Yu Fan, Feng Guofeng

  2026, 46(1): 95. doi: 10.3969/j.issn.1006-8074.2026.01.007

  Abstract ( 376 )   PDF (2212KB) ( 12 )   　　

  This paper investigates the propagation patterns and characteristics of wave equations in inverse problems. Using the two-dimensional wave equation as the core theoretical framework, we systematically develop and numerically validate an efficient numerical method for solving the forward problem and a robust reconstruction method for the inverse problem. First, the finite difference method is employed to discretize the two-dimensional wave equation in time and space, revealing the dynamic response characteristics of wave field propagation under specific parameter configurations. Subsequently, regularization is applied to the nonlinear wave equation, transforming the inverse problem into a well-posed minimization problem. By integrating total variation regularization with the finite volume method, iterative inversion algorithms are constructed based on nonlinear optimization techniques, including the steepest descent method, Newton's method, and the conjugate gradient method. Finally, numerical simulations of the parameters in the two-dimensional wave equation are conducted using a three-layer medium model. The results demonstrate the feasibility and computational efficiency of the proposed algorithm.

• Finite-Time Control and Extensions of Lyapunov Stability Criteria

  Liu Yuanshan, Xia Yude, Jiang Wenxiang, Lin Yawen, Xu Zhaomeng

  2026, 46(1): 112. doi: 10.3969/j.issn.1006-8074.2026.01.008

  Abstract ( 348 )   PDF (273KB) ( 16 )   　　

  While traditional asymptotic control methods are limited in convergence rate and precision, finite-time control theory provides a superior alternative. However, its practical application is constrained by the dependence of the settling time on the system's initial conditions. To overcome this limitation, fixed-time stability theory was developed, which establishes a deterministic upper bound for the settling time independent of initial states. Further advancing this concept, predefined-time (or prescribed-time) stability theory allows this upper bound to be directly assigned as a tunable parameter by the designer. This theoretical evolution has opened new avenues for achieving fast and accurate control of complex nonlinear systems. This paper systematically reviews this important theoretical progression. It clarifies the core definitions of various finite-time stability concepts and offers a detailed comparative analysis of the corresponding Lyapunov-based stability criteria, highlighting their differences in settling time estimation. Finally, the characteristics of these criteria are summarized, and future research directions are outlined to provide a clear theoretical framework and guidance for the field.