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2026, Vol. 46 No. 02   Published date: 28 June 2026
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• Research on the Mechanism of Generating Multi-Scroll Chaos by Piecewise Affine Systems

  Wu Baolong

  2026, 46(02): 1-17. doi: 10.3969/j.issn.1006-8074.2026.02.001

  Abstract ( 109 )   PDF (806KB) ( 20 )   　　

  To elucidate the generation mechanism of multi-scroll chaotic attractors, this paper investigates a class of three-dimensional $n$-zone $(n≥2)$ piecewise affine systems. By analyzing the geometric structure of stable and unstable manifolds, sufficient conditions are established for the coexistence of $n−1$ heteroclinic cycles. Inspired by the Shilnikov-type theory , suitable cross-sections are constructed near the heteroclinic cycles, and the corresponding Poincaré map is derived. A rigorous analysis of the Poincaré map reveals the existence of a chaotic invariant set in the system. Numerical simulations are provided to support the theoretical results. The findings suggest that the multi-scroll chaotic structure may originate from the coexistence and interaction of multiple heteroclinic cycles.

• Gradient Estimates and Liouville-Type Theorems for a Class of Quasilinear Elliptic Equations on Complete Riemannian Manifolds

  Wang Youde, Yang Mixue, Zhang Liqin

  2026, 46(02): 18-39. doi: 10.3969/j.issn.1006-8074.2026.02.002

  Abstract ( 117 )   PDF (240KB) ( 33 )   　　

  This paper investigates gradient estimates for smooth solutions to a class of quasilinear elliptic equations of the form $\operatorname{div} \left((1 + |\nabla u|^2)^\theta \nabla u \right) + a|\nabla u|^r + h(u) = 0$ on a complete Riemannian manifold, where $a$ is a constant and $h(u)$ is a nonlinear term. By employing the Saloff-Coste type Sobolev inequality and the Nash-Moser iteration method, we derive, under the assumption of a lower bound on the Ricci curvature, a unified and concise local gradient estimate for solutions satisfying the structural conditions $-\frac{1}{2} < \theta \le 0$ and $r > 1 + 2\theta$. Moreover, an explicit upper bound for the gradient of entire solutions is obtained. As a direct consequence of these gradient estimates, we prove a Liouville-type theorem: if the manifold is noncompact and has nonnegative Ricci curvature, then any globally defined solution must be constant.

• Suitable Weak Solutions and Partial Regularity for Simplified Magneto-Viscoelastic Flows

  Liu Qiao, Zuo Zhongbao

  2026, 46(02): 40-68. doi: 10.3969/j.issn.1006-8074.2026.02.003

  Abstract ( 82 )   PDF (268KB) ( 25 )   　　

  In this paper, inspired by the celebrated work of Caffarelli et al. [2]on partial regularity for the classic incompressible Navier-Stokes system and by Du et al. [4] on suitable weak solutions for the co-rotational Beris-Edwards system, we establish the global existence of suitable weak solutions to a simplified magneto-viscoelastic flow system in three-dimensional space. This model couples the incompressible Navier-Stokes equation for the fluid velocity field, an evolution equation for the deformation tensor, and a gradient flow equation for the magnetization vector. Furthermore, we prove that the one-dimensional space-time Hausdorff measure of the potential singular set of such suitable weak solutions is zero.

• Convergence of Combinatorial p-th Ricci Flows in Euclidean Background Geometry

  Li Zhengkun, Song Kai

  2026, 46(02): 68-79. doi: 10.3969/j.issn.1006-8074.2026.02.004

  Abstract ( 104 )   PDF (183KB) ( 12 )   　　

  Lin and Zhang first introduced the combinatorial $p$-th Ricci flows and generalized Chow-Luo's classical results on combinatorial Ricci flows (when $p=2$) to any $p>1$. However, their generalization in Euclidean background geometry is incomplete: they only proved the convergence of discrete curvatures along the flow, but failed to establish the convergence of circle packing metrics. In this paper, by introducing a time-dependent normalization term $C(t)$, we overcome the core obstacle arising from the lack of compactness for solutions to combinatorial $p$-th Ricci flows. Our results improve those of Lin and Zhang and generalize Chow-Luo's classical results in Euclidean background geometry from $p=2$ to any $p>1$ in full generality.

• Direct and Inverse Approximation Theorems for Linear Combinations of Szász-Mirakjan-Durrmeyer Operators in Orlicz Spaces

  Han Lingxiong

  2026, 46(02): 80. doi: 10.3969/j.issn.1006-8074.2026.02.005

  Abstract ( 98 )   PDF (159KB) ( 15 )   　　

  This paper investigates approximation problems in Orlicz spaces generated by Young functions. Utilizing Jensen's inequality and the equivalence between the $K$-functional and the modulus of smoothness, we establish both direct and inverse theorems for approximation by linear combinations of Szász-Mirakjan-Durrmeyer operators in these spaces.

• Finite Groups with Few Cyclic Subgroups

  Kang Wangqiang, Wang Yu, Lu Jiakuan, Meng Wei

  2026, 46(02): 91-97. doi: 10.3969/j.issn.1006-8074.2026.02.006

  Abstract ( 119 )   PDF (181KB) ( 16 )   　　

  Let $G$ be a finite group and $c(G)$ denote the number of cyclic subgroups of $G$. In this paper, we discuss the

  structure of finite groups $G$ when $c(G)$ is small, and provide their isomorphic classifications.

• Reliability Analysis of Blockchain Incentive Mechanism Against Collusion Attacks Based on Markov Chain

  Cao Cong, Hou Muzhou, Shu Zhanjie, Lu Shoupeng

  2026, 46(02): 98-121. doi: 10.3969/j.issn.1006-8074.2026.02.007

  Abstract ( 114 )   PDF (2190KB) ( 22 )   　　

  This paper investigates the process by which multiple low-computing-power nodes in a blockchain network collude to exploit incentive mechanisms for excessive gains, comparing node profits under different attack strategies. First, aiming to maximize profits, we construct an optimal strategy combination for joint node attacks and establish a corresponding Markov chain model. We prove the irreducible and normal return properties of this Markov chain, thereby demonstrating the existence of its steady-state distribution. The steady-state distribution is derived by solving the transition rate matrix. Block final states are categorized into 15 types, and transition probabilities between states are analyzed to calculate the occurrence probabilities of each state. Based on this, a long-term node profit model is established. Furthermore, a real blockchain simulation environment is constructed to simulate attack strategies, validating the theoretical findings. Simultaneously, by controlling different variables in the strategy, the block production efficiency and overall profit differences under three scenarios--solo attacks, node defection, and full collusion--are compared and analyzed. The results indicate that some existing blockchain incentive mechanisms still have significant flaws. Nodes with low computing power can substantially increase their profits through collusive manipulation, thereby threatening the fairness and security of the system.