Bursting, a common electrical behavior observed in neuronal membranes, has been recognized as a predominant pattern in various biological systems. This paper explores the firing pattern of a single-compartment pre-B & ouml;tzinger complex (PBC) model, with a focus on the influence of stimulation current similar to the persistent sodium current 'NP and calcium-activated nonspecific cationic current 'CAN. Through multi-time scale analysis, we investigated how different stimulation currents interact. Our findings suggest that both the somatic and calcium subsystems can induce various degrees of firing pattern transitions under the influence of stimulation current. Notably, Hopf/Hopf bursting via Fold/Hopf hysteresis loop, which was not found in previous studies, was discovered by fast and slow decomposition techniques. Furthermore, a two-parameter bifurcation analysis was conducted to elucidate the transition dynamics resulting from stimulation current. This study provides valuable theoretical insights into the generation mechanism of respiratory rhythm.

This paper presents a comparative study of the Homotopy Analysis Method (HAM) and Incremental Harmonic Balance Method (IHBM) for solving nonlinear periodic responses in dynamical systems. Both methods are evaluated in terms of accuracy, computational efficiency, convergence behavior, and applicability to strongly nonlinear regimes. Numerical results on Duffing and van der Pol oscillators show that IHBM achieves high accuracy and rapid convergence for steady-state solutions, making it well-suited for engineering simulations when adequate harmonic content and good initial guesses are provided. In contrast, HAM offers analytical flexibility and controllable convergence through auxiliary parameters, enabling solution construction even without prior knowledge of periodicity. However, higher-order approximations in HAM incur significant computational cost. The results highlight the complementary strengths of the two methods: IHBM excels in efficiency for regular periodic responses, whereas HAM provides greater analytical insight into complex nonlinear dynamics. These findings offer practical guidance for selecting appropriate semi-analytical tools based on problem characteristics.

This study systematically elucidates the intrinsic mechanisms by which a nonlinear energy sink (NES) suppresses multimodal coupled vibrations in cable structures. First, a spatial model of the cable-NES system and a simplified model considering only the first three in-plane modes were established. Using the Galerkin method, the strongly nonlinear governing equations of the system were derived, and high-precision analytical solutions were obtained via the homotopy analysis method, with their validity verified by Runge–Kutta numerical integration. Comparative analysis of the two models demonstrates that neglecting out-of-plane modes significantly underestimates the system's vibrational energy, while incorporating out-of-plane degrees of freedom is essential for accurately characterizing the targeted energy transfer path. Furthermore, the vibration suppression performance of the NES was quantified using amplitude-frequency response curves, and the effects of the NES damping ratio, nonlinear stiffness, and installation location on the suppression bandwidth were systematically investigated. The findings reveal that out-of-plane modes play a significant role in coupled vibrations; the NES achieves targeted energy transfer by activating nonlinear energy channels, thereby effectively suppressing multimodal coupled vibrations. Parametric analysis shows that optimizing the NES damping characteristics, nonlinear stiffness, and placement can substantially broaden the suppression bandwidth, enabling sustained and broadband vibration attenuation. This research provides theoretical foundations and design references for the nonlinear vibration control design of engineering cable structures.

Nonlinear Energy Sink (NES), as an innovative passive vibration control technology, has attracted attention for broadband vibration suppression owing to its high efficiency. This paper investigates the parametric resonance and bifurcation behavior of a spatial cable coupled with an NES under parametric excitation. An accurate nonlinear dynamic model of the cable-NES system is first formulated, capturing the essential nonlinear coupling mechanisms and energy transfer pathways. Theoretical analysis reveals complex nonlinear dynamics, including bifurcations induced by system parameters. The dynamic responses under primary parametric resonance are systematically examined using the Homotopy Analysis Method (HAM), which highlights the significant influence of subtle NES parameter variations on the cables vibration. Furthermore, the fourth subharmonic resonance of the cable under parametric excitation is explored. Results demonstrate that optimal placement of the NES at the cable end yields superior vibration mitigation compared to mid-span attachment. Moreover, appropriate tuning of NES damping enables the cable response to settle into a stable steady state. This study elucidates the internal mechanism of parametric resonance in the cable-NES system, providing a theoretical basis and practical guidance for vibration control in bridge engineering.

Experimental results reveal that the dynamic response of a stay cable under forced excitation exhibits significant nonlinear characteristics, manifested by the presence of higher harmonics in addition to the excitation frequency component. To investigate the underlying mechanism, this study establishes multi-mode uncoupled and coupled systems for the first three symmetric inplane modes based on the governing equations of modal motion. The equations are discretized into ordinary differential equations using the Galerkin method, and the homotopy analysis method (HAM) is employed for solution. Response curves are constructed using the NewtonRaphson method combined with the pseudo-arclength algorithm. A systematic investigation is conducted into the effects of key parameters, including excitation amplitude, damping ratio, and excitation under different modal conditions. The results indicate that higher harmonics originate from strong nonlinear internal resonance mechanisms. When the natural frequencies satisfy omega 3 approximate to 2 omega 1, a 2:1 internal resonance pathway facilitates efficient energy transfer from the excited higherorder mode to the lower-order mode. Even when external excitation is applied only to higherorder modes such as the third or fifth, a 3:1 internal resonance near Omega approximate to omega 3 can excite double resonance peaks in all three modes, thereby breaking the one-to-one correspondence between excitation and response observed in linear systems. Under high-amplitude excitation, parameter variations further induce typical nonlinear phenomena such as response curve bifurcation, jumping, and phase drift. This study reveals the nonlinear vibration mechanism of stay cables under the combined action of internal resonance and external excitation resonance, providing a theoretical basis for the nonlinear vibration control of cable structures.

Modal analysis is a widely applied method to study the vibration phenomenon of continuum structures, but there is no clear method to solve the modal truncation problem at present. To determine the contribution of different modes to the whole system, a new mode truncation method based on perturbation theory is proposed in this paper. The modes are subjected to perturbation parameters during discretization, and using norm error analysis on the stiffness matrix in different degrees of freedom (DOFs) systems confirms the model number of the continuum structure system. The results show that the DOF identified by the modal perturbation method is related to the perturbation parameter, and the smaller the perturbation parameter is, the fewer modes need to be considered. When the perturbation parameter is large enough, the response of the system can only be accurately explained by truncation to higher-order modes. Finally, the perturbation parameter is fixed to 1, and the traditional Galerkin method is connected to the modal perturbation, making traditional discretization a unique case for the modal perturbation method. This method can significantly reduce the modal truncation error, which is of great significance to the dynamic analysis of engineering applications.

Functionally graded materials（FGMs） are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties. This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory, the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach. Then, the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method（TMM）. Two models,i.e., a three-level FGM stepped beam and a five-level FGM stepped beam, are considered,and their natural frequencies and mode shapes are presented. To demonstrate the validity of the method in this paper, the simulation results by ABAQUS are also given. On this basis, the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out. The results show the accuracy and efficiency of the TMM.

Synchronization is a very important phenomenon in the nervous system, which is closely related to the encoding, integration and transmission of information. In this paper, synchronization and transition of a two-compartment respiratory neuron model under transcranial magnetic stimulation (TMS) are studied from the perspective of synchronization degree for the first time. We are established the correlation degree with synchronization, and discussed the firing mode and transition rule of the neurons in the two-compartment compartment pre-Bötzinger complex (PBC) by means of bifurcation theory and Lyapunov index. The results show that the synchronization of neurons has a great influence under TMS, which is embodied in the fact that the somatic will experience a peak firing and a transition from bursting to resting under the magnetic stimulation,which was a phenomenon never before shown in PBC neurons. These results fully reveal the dynamic behavior of PBC nervous system under TMS, and provide theoretical value for further understanding of respiratory rhythm.

The nonlinear dynamics and control of micro electromechanical systems (MEMS) is an important topic at present and homotopy analysis method (HAM) is an effective semi-numerical and semi-analytical method for solving strongly nonlinear problems. In this paper, the parametric resonance of MEMS with multi-frequency excitation is studied by HAM. Firstly, the differential equation of electrostatically driven microbeam is processed by Taylor expansion, and it is transformed into the parametric motion model with multi-frequency excitation. Then, the approximate solution and amplitude–frequency response equation of the system are obtained by HAM, and compared with the numerical solution. Finally, the influence of direct current (DC) and alternating current (AC) on principal parametric resonance and superharmonic resonance is discussed. The results show that HAM is an effective method to analyze the parametric vibration of multi-frequency excitation system, and the amplitude–frequency response curve of microbeam about parametric motion depends on the time scale in high DC and high AC state. This study effectively extends the application of HAM in parametric resonance, which is of great significance to the study of nonlinear vibration of MEMS.

类似光滑系统的余维二分岔的分类方法,余维二擦边分岔被划分为三种类型,分别是擦边点退化、退化环擦边(非双曲)以及两个擦边事件同时发生.分析了一个二自由度对称约束的碰撞振动系统,得到了该系统第二类余维二擦边分岔的存在条件.考虑双侧擦边周期运动,理论推导出双侧擦边周期运动的存在性条件;利用不连续映射方法,得出1/1/n碰撞周期运动发生鞍结分岔和倍周期分岔的解析表达式;结合双擦边周期运动的存在性条件和1/1/n碰撞周期运动的分岔条件,推导出发生余维二擦边分岔时满足的解析表达式,并以周期1运动为例,给出了余维二擦边分岔点的分布.

Pre-Bötzinger complex (PBC) neurons located in mammalian brain are the necessary conditions to produce respiratory rhythm, which has been widely verified experimentally and numerically. At present, one of the two different types of bursting mechanisms found in PBC mainly depends on the calcium-activated of non-specific cation current (I CaN ). In order to study the influence of I CaN and stimulus current I exc in PBC inspiratory neurons, a single compartment model was simplified, and firing patterns of the model was discussed by using stability theory, bifurcation analysis, fast, and slow decomposition technology combined with numerical simulation. Under the stimulation of different somatic applied currents, the firing behavior of neurons are studied and exhibit multiple mix bursting patterns, which is helpful to further understand the mechanism of respiratory rhythms of PBC neurons.

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.

当擦边分岔和光滑分岔同时发生时,非光滑系统会发生一类余维二擦边分岔.一类二自由度碰撞振动系统的此类余维二擦边分岔及余维二擦边点附近的动力学行为得到研究.讨论了擦边周期运动的存在性条件.利用不连续映射方法构造了1/n碰撞周期运动的全局庞加莱映射,并得出1/n碰撞周期运动的分岔条件.结合擦边周期运动条件和碰撞周期运动分岔条件推导出擦边分岔和光滑分岔同时发生时满足的解析表达式,并数值分析了不同周期下系统余维二擦边分岔点的分布情况.通过对比分别由全局庞加莱映射和原系统得到的余维二擦边点附近的分岔图,验证了理论分析的有效性.