Limit cycles of a class of hybrid piecewise differential systems with a discontinuity line of L shape

Limit cycles for a class of piecewise linear systems with three zones separated by two parallel lines

Rhythmic gene expression underlies core physiological processes across organisms, from circadian timekeeping to stress responses. Recent experiments suggest that the regulation of such rhythmic dynamics involves protein compartmentalization mediated by liquid-liquid phase separation (LLPS), yet the mechanisms by which LLPS feeds back onto oscillatory behaviour remain unclear. Here we develop a minimal two-phase gene-expression model in which proteins are synthesized in the dilute phase, reversibly partition into a protein-dense droplet phase, and repress their own production via condensate-mediated regulation. In the deterministic limit, LLPS does not generate limit cycles; instead, nonlinear partitioning and timescale separation between phase separation and protein turnover convert purely relaxational dynamics into damped oscillatory transients, altering the approach to equilibrium without producing sustained oscillations. In the stochastic regime, intrinsic noise interacting with this near-focus dynamics is amplified into noise-sustained, near-periodic fluctuations with a characteristic timescale, as revealed by the power spectral density and autocorrelation functions. These results show how LLPS reshapes oscillatory signatures by encoding and filtering temporal signals in phase-specific ways, providing a quantitative framework for interpreting LLPS-rhythm coupling and for engineering biomolecular systems with tunable dynamic behaviour.

Lithium-oxygen batteries (LOBs) offer combustion fuel-like energy densities but remain constrained by low efficiency, limited cycle life, and coupled degradation pathways linking the electrochemical growth and decay of the reaction product (Li2O2) and associated generation of reactive oxygen species, electrolyte and electrode instabilities, and lithium dendrite growth. Here, we introduce a hybrid materials-informatics framework that integrates structured query learning with retrieval-augmented generation (RAG) to systematically analyze the full-text corpus of 3134 peer-reviewed articles in LOBs. Unlike conventional artificial intelligence (AI) tools, which learn from unstructured literature and risk factual drift, the present approach forms a relational performance-validated database, enabling evidence-traceable comparison of cathode architectures, catalyst types, electrolytes, redox mediators, and lithium protection strategies. The analysis reveals composition-dependent performance hierarchies and exposes interdependencies among Li2O2 morphology, singlet-oxygen formation, overpotentials, and solid electrolyte interface disruption, as reported under their documented experimental conditions. Using this method, we identified the catalyst-electrolyte-anode configurations capable of reducing charge polarization by 0.3-0.6 V and extending cycling stability to 100-200 cycles under standard cycling conditions reported in the source studies. This data-driven roadmap establishes a quantitative foundation for translating LOBs from laboratory demonstrations to deployable high-energy systems and demonstrates how materials informatics can accelerate electrochemical materials synthesis and device design.

LiNiO2 (LNO), a promising candidate for the commercialization of high-energy-density lithium-ion batteries, offers high reversible capacity but suffers from limited cycle life due to anisotropic lattice distortion and the resulting chemical and mechanical degradation during repeated cycling. To address these challenges, we propose a nano-sol infusion doping strategy that overcomes the inherent limitations of conventional solid-state doping, namely poor dopant homogeneity throughout secondary particles and particle agglomeration caused by high-temperature calcination. This process enables nanoscale (∼10nm) dopant precursors to be uniformly infused into the cathode precursor using only a small amount of solvent and simple equipment, facilitating stable dopant incorporation and homogeneous distribution during subsequent calcination. The infusion-doped LNO (ID-LNO) synthesized by this approach delivered a capacity retention of 86.06% after 100 cycles at 1 C, outperforming both undoped and solid-state doped LNO. The results clearly demonstrate that ID-LNO possesses superior structural stability, characterized by the suppression of microcrack formation and mitigation of c-axis contraction during the H2-H3 phase transition. This study demonstrates that nano-sol infusion doping is a novel synthesis strategy capable of fundamentally alleviating the structural degradation of LNO and suggests it as a viable approach with potential applicability to various high-valence dopants and ternary layered oxide compositions (e.g., NCM and NCA) for the development of long-life, high-rate, and high-energy-density lithium-ion batteries.

With the growing emphasis on developing clean fuels to reduce emissions, hydrogen has become a critical focus due to its potential for lowering environmental impact. However, the operational challenges in lean fuel systems, particularly combustion instability, highlight the necessity of parametric studies to ensure stable performance with alternative fuels. This study numerically examines the thermoacoustic instability of pure hydrogen and methane in a Rijke tube combustor using the Unsteady Reynolds-Averaged Navier-Stokes method. The investigation focuses on two distinct instability behaviors: beating and limit cycle oscillations. Key findings reveal that for both fuels, increasing the equivalence ratio toward richer mixtures tends to damp thermoacoustic instabilities. It was observed that a 20% increase in fuel flow rate caused a transition from beating to limit cycle oscillations. A comparative analysis shows that lean methane flames produce stronger fluctuations, whereas hydrogen has a wider instability range under fuel-rich conditions. Advanced modal analysis using Dynamic Mode Decomposition identifies the dominant longitudinal acoustic modes and localized oscillations near the fuel nozzle that influence flame structure. These results provide critical insights into the unique thermoacoustic characteristics of hydrogen and methane, which can inform the design of stable, next-generation combustion systems.

The development of high-energy-density sodium-ion batteries (SIBs) is limited by the structural instability of O3-type layered oxide cathodes at high voltages. In this study, we present a comprehensive mechanistic analysis of the structural evolution in Na[Ni0.5Mn0.5]O2 (NM55) and its derivatives, utilizing operando high-resolution X-ray diffraction to elucidate the complex phase transition sequence and how structural changes impact electrochemical performance. We identify a cascade of transformations-O3-O'3-P'3-P3-P'3'-P3'-O3'-O3'-marked by abrupt lattice distortions, sodium/vacancy ordering, and significant c-axis contraction at high states of charge. These structural dynamics directly contribute to capacity fading, mechanical degradation, and limited cycle life. Building on these insights, we explore systematic doping strategies with Fe, Ti, and Ca. Fe substitution suppresses monoclinic distortions and alters high-voltage transitions, while co-doping with Ti and Ca further stabilizes the lattice, inhibits undesirable phase evolution, and preserves interface integrity. The optimized quaternary composition, Na0.96Ca0.02[Ni0.4Fe0.2Mn0.375Ti0.025]O2, achieves a stable capacity retention of 70.18% after 600 cycles at 1C. This work establishes a vital link between specific phase transition mechanisms in O3-type cathodes and their electrochemical durability, providing crucial guidance for the rational design of next-generation SIB cathode materials.

Supercapacitors are emerging as efficient energy storage devices due to their high power density and rapid charge–discharge capabilities. Their performance, however, is often restricted by electrode materials with low capacitance and limited cycling stability. To address this challenge, we synthesized and optimized TiO 2 –MnO 2 nanocomposites with improved colloidal stability and reduced particle size for enhanced electrochemical behavior. The synergistic interaction between TiO 2 and MnO 2 at a 25:75 ratio significantly improves ion diffusion, redox activity, and electrical conductivity, resulting in efficient charge storage. Structural characterization confirms the formation of a well‐optimized nanostructure that supports fast electron transport and stable electrochemical response. Electrochemical tests reveal a high specific capacitance of 1749 F/g at 3.3 A/g, strong rate capability, and minimal capacitance degradation during long‐term cycling. The optimized nanocomposite outperforms the unoptimized material and previously reported systems in both energy and power density. Long‐cycle testing further demonstrates excellent stability with significant capacitance retention. Additionally, density functional theory (DFT) confirms the strong interaction of TiO 2 –MnO 2 with imidazolium‐based electrolytes, indicated by a negative Gibbs free energy (−128.67 kcal/mol) and reduced bandgap (0.484 eV). Overall, the optimized TiO 2 –MnO 2 composite is a promising electrode material for next‐generation high‐performance supercapacitors.

Motivated by important applications in cognitive processes, we explore the constructive role of noise in systems with sequential dynamics. As a conceptual model, we use the well-known May-Leonard model, which describes the dynamics of three populations under competition. For this model, noise-induced phenomena are studied for three important cases. When three axial saddle equilibria are connected by a homoclinic cycle, random noise stabilizes the frequency of stochastic oscillations. In the case where axial equilibria are stable, random disturbances generate stochastic oscillations in the form of sequential dynamics with a temporary slowdown near these equilibria. In the extended version of the model, taking into account a positive constant influx, we reveal the most susceptible parts of the limit cycles. In the analysis of sequential behavior depending on system parameters, scaling laws are identified and stochastic sensitivity technique is used.

The growing demand for higher-energy lithium-ion batteries, encompassing consumer electronics, stationary grid storage, and electric mobility to specialized sectors like aerospace, medical devices, and industrial robotics, requires cathode materials that offer higher capacity while remaining cost-effective. This trend has intensified the development of nickel-rich LiNi1−x−yMnxCoyO2 (NMC) systems. However, high-Ni NMCs such as LiNi0.9Mn0.05Co0.05O2 (NMC90) suffer from limited thermal and cycling stability. Core–shell architectures using LiNi0.6Mn0.2Co0.2O2 (NMC622) as a shell can partially alleviate these drawbacks, but structural degradation caused by interdiffusion between the core and shell persists as a major challenge. This study investigates whether a tungsten oxide interlayer can act as a protective barrier that suppresses interdiffusion, stabilizes the crystal structure, and improves long-term electrochemical performance. In this work, NMC cathode powders were synthesized via a one-pot oxalate co-precipitation route, followed by structural characterization using X-ray diffraction (XRD), scanning electron microscopy (SEM), X-ray photoelectron spectroscopy (XPS), and ion scattering spectroscopy (ISS). Electrochemical performance, including capacity retention, cycling stability, and internal resistance, was evaluated through galvanostatic charge–discharge (GCD) testing and electrochemical impedance spectroscopy (EIS). The core–shell configuration delivered higher specific discharge capacity compared to the individually synthesized core-only and shell-only reference materials, and the incorporation of a tungsten oxide interlayer resulted in a twofold increase in cycle life. These results demonstrate that tungsten oxide effectively enhances cycling stability by inhibiting core–shell interdiffusion, offering a promising pathway toward more durable high-Ni NMC cathodes.

ABSTRACT Thin specimens exhibit limited signal cycles excitation during nonlinear ultrasonic longitudinal wave testing, resulting in insufficient spectral resolution and low excitation energy. The standing wave method is proposed for testing thin specimens. Among the same batch of specimens, minor length differences during the processing may make it difficult for the same frequency signals to stably form standing waves, thus affecting the detection effectiveness. To address this issue, variable-frequency standing wave method is further proposed. Simulation and experimental validations using 6061-T6 thin specimens with different tensile stresses demonstrate that this method can more effectively excite second harmonics. The ultrasonic nonlinearity coefficient shows more pronounced variation trends with the evolution of early-stage damage, enabling effective detection of early damage in thin specimens. In addition, to address spectral leakage and the fence effect caused by non-integer period truncation of standing wave signals, matrix pencil method based on time-domain feature extraction is proposed and compared with the FFT method. Simulation results demonstrate that this method can control the error of the ultrasonic nonlinearity coefficient within 5%. Experimental results further validate its superiority in signal amplitude extraction accuracy.

Abstract The friction-induced vibration between the brake pad and brake disc is a critical factor that significantly influences brake performance. The method for suppressing friction-induced vibration via expanding the stable region of the equilibrium point while reducing the vibration amplitude of the limit cycle is proposed. A friction-induced vibration model with nonlinear vibration energy harvester(NVEH) is modeled, from which the stability boundary of the equilibrium point is derived. The analysis reveals that the NVEH significantly widens the stable region of the equilibrium point. Furthermore, the stability boundary of the equilibrium point is found to be primarily governed by the linear parameters of the NVEH. A significant expansion of the stable region of the equilibrium point is achieved via optimizing the linear parameters of the NVEH. The optimization of the NVEH's nonlinear parameters is employed to mitigate the vibration amplitudes of the limit cycle. The trade-off between vibration suppression and energy harvesting is addressed which indicates that an increase in the nonlinear parameters improves vibration suppression performance while reducing energy harvesting efficacy. Conversely, a decrease in these parameters enhances energy harvesting efficacy at the cost of reduced vibration control. Compared to the pure mechanical vibration absorber there may exists an additional pathway for dissipating power when the electromechanical coupling effect is considered - transforming relatively complex vibrations into simpler ones, thereby dissipating vibrational energy. This research can provide a theoretical basis for the parameter design of NVEH intended for suppressing friction-induced vibration in brake system.

Abstract The saturation effect, characterized by inherent non-smoothness, plays a pivotal role in shaping the qualitative behavior of nonlinear dynamical systems. To investigate the impact of saturation nonlinearity on compound bursting, a modified Lorenz-like system incorporating a saturation function and slow-varying excitation is considered. Numerical simulations demonstrate that variations in the saturation threshold govern an ordered evolution of compound bursting patterns, manifested as a stepwise increase and reorganization of bursting clusters. A systematic bifurcation analysis of the fast subsystem reveals that saturation fundamentally restructures the equilibrium configuration, stability properties, and bifurcation topology. In addition to classical fold and Hopf bifurcations, various non-smooth bifurcations about equilibrium points and limit cycles are identified. Based on slow–fast analysis, the underlying mechanisms of compound bursting are systematically elucidated. In particular, it is found that the non-smooth homoclinic bifurcation serves as the organizing center, leading to the generation and rearrangement of multiple bursting clusters. Furthermore, the slow passage effect suppresses further transitions near critical bifurcations, resulting in the saturation of the number of bursting clusters. These findings provide new insight into compound bursting in piecewise-smooth slow–fast systems and highlight the essential role of saturation-induced non-smoothness in bursting dynamics.

Lithium-oxygen (Li-O2) batteries offer ultrahigh theoretical energy density, but suffer from limited cycle life and high overpotentials, particularly in LiOH-based systems. While LiOH chemistry provides superior environmental tolerance compared to Li2O2 systems, the inherent four-electron redox process creates substantial charging overpotentials that compromise performance. Here, we tailor electrolyte activity to enable an efficient LiOH redox process by integrating 1-phenylpyrrolidine (PPD) as a redox mediator within an ionic liquid electrolyte. PPD possesses an optimal oxidation potential and stable p-π conjugation, enabling homogeneous chemical decomposition of LiOH and overcoming electrode-electrolyte contact limitations. The ionic liquid 1-propyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide (C3C1im TFSI) is engineered to regulate water reactivity and maintain hydrogen-bond networks, thereby promoting selective LiOH formation over Li2O2 during discharge, while providing high oxidative stability to suppress mediator degradation-an issue prevalent in ether-based electrolytes. This electrolyte-mediator synergy shifts the charging mechanism from sluggish interfacial charge transfer to a fast, solution-mediated chemical route, delivering 180 stable cycles with markedly reduced overpotentials and ∼10× longer cycle life. This work offers molecular-level design principles for tailoring electrolyte activity to achieve high-efficiency and durable Li-O2 batteries based on LiOH chemistry.

This paper presents a new methodology for parameter estimation of models with limit cycle and Hopf bifurcation. By calculating average cycle values for the process signal, a convex objective function can be produced using a maximum likelihood approach. This methodology allows for the proper statistical analysis of the results and the usage of local optimization algorithms for fast and efficient estimation. A case study is presented for a simplified well model of slugging oil systems, for which parameter estimation is crucial to correctly represent any system of interest. Two other objective functions from the literature are compared through Particle Swarm Optimization. The proposed method outperformed those of previous works as determined by a rigorous statistical assessment.

Purpose For the purpose of studying the stability and stick-slip vibration characteristics of the coke pushing system, a dynamic model used to describe the friction self-excited vibration system is established in the article. Design/methodology/approach The Stribeck friction model is introduced into the established dynamic model and the theoretical calculation and numerical simulation are carried out. The critical instability speed of the coke pushing system is 0.45 by theoretical calculation, and the numerical simulation results verify the correctness of the theoretical calculation. In order to control the stability and stick-slip vibration of the coke pushing system, a linear and nonlinear state feedback controller is designed in this paper. Findings Stick-slip vibration emerges in the coke pushing system when its operating speed is below the critical instability speed of 0.45, while the system stabilizes at speeds above this threshold. The linear gain of the state feedback controller can reduce the critical instability speed of the coke pushing device to achieve stable operation at lower operating speeds. Meanwhile, its nonlinear gain can reduce the limit cycle size, suppress stick-slip vibration amplitude and significantly improve the vibration state of the whole system. Originality/value (1) A dynamic model used to describe the friction self-excited vibration of the coke pushing system is established. (2) Design a linear and nonlinear state controller to control the stability and stick-slip vibration of the coke pushing system. (3) Investigate the effect of linear and nonlinear state controllers’ linear gain on the critical instability speed and nonlinear gain on the vibration limit cycle size of the coke pushing system.

Objective: The Hodgkin-Huxley model is one of the most influential mathematical models in neuroscience, describing how electrical activity is generated and propagated in neurons. In this work, bifurcation analysis and Multiobjective Nonlinear Model Predictive Control are performed on the dynamic Hodgkin-Huxley model.Methods: The MATLAB program MATCONT was used to perform the bifurcation analysis. The MNLMPC calculations were performed using the optimization language PYOMO in conjunction with the state-of-the-art global optimization solvers IPOPT and BARON. Results: The bifurcation analysis revealed the existence of Hopf bifurcation points and limit points. The MNLMC converged on the Utopian solution. Hopf bifurcation points, which cause unwanted limit cycles, are eliminated using an activation function based on the tanh function. Conclusion: The limit points (which cause multiple steady-state solutions from a singular point) are very beneficial because they enable the Multiobjective Nonlinear Model Predictive Control calculations to converge to the Utopia point (the best possible solution) in the model. The tanh activation function is highly effective at eliminating Hopf Bifurcations.

This study develops a nonlinear analytical model of a spacecraft with flexible solar arrays to investigate complex vibration phenomena occurring within the thermal flutter regime, particularly during the satellite's transition through the penumbra phase. The model captures dynamic behaviors driven by transient thermal gradients and structural flexibility key contributors to self-excited oscillations in orbit. Two major nonlinear phenomena are examined: limit cycle oscillations (LCOs) self-sustained periodic motions arising under subcritical thermal conditions, and internal resonance, which emerges within the flutter velocity range and strongly influences modal interactions. Special attention is given to the Hubble Space Telescope (HST), where nonlinearities stem from large elastic deformations and thermally induced loads and torques. A combined analytical-numerical approach is adopted, incorporating nonlinear dynamic modeling and time-domain simulations. The focus is on the penumbra phase, where the solar radiation rate undergoes significant variations. The system's coupled bending-bending-twisting motion equations include quadratic and cubic nonlinearities. Through model reduction, a tractable set of nonlinear ordinary differential equations is derived. Using the method of multiple scales (MMS), internal resonance conditions are analytically identified to characterize energy exchanges among modes. To further assess the system's dynamic behavior, phase portraits, Poincaré sections, and bifurcation diagrams are employed. These analyses elucidate the onset and evolution of LCOs, offering critical insights into the coupled thermomechanical instabilities of flexible space structures and contributing to the design of more robust spacecraft systems under dynamic thermal environments.

ABSTRACT Zinc‐anode electrochromic windows capable of actively controlling light and heat transfer on demand, have been emerged as an intriguing technology for indoor thermal management. However, their practical development has been hindered by slow switching speeds, limited cycling stability and unclear operating mechanisms. Herein, we present a high‐performance Zn‐anode dual‐band electrochromic device utilizing tungsten oxide quantum dot (WO 3 QD) cathode, and reveal their detailed charge transfer mechanism. This device not only can control the visible light and near‐infrared effectively and independently through bright, cool and dark modes, but also shows excellent electrochromic properties with a high optical modulation (78.8% at 633 nm), fast response and long‐term cycling stability (92.1% capacity retention after 10,000 cycles). The ultra‐small size and large hexagonal tunnel of WO 3 QDs notably enhance ion diffusion kinetics and structure stability during cycling. Furthermore, we reveal a synergistic co‐insertion mechanism of Zn 2+ and H + with a molar ratio of 2:1 in nonaqueous electrolytes. Outdoor tests and simulation results confirm the higher energy‐saving performance of our device than the commercial low‐emissivity glass in most climate zones around the world. This work paves the way toward designing high‐performance electrochromic smart windows for future zero‐carbon buildings.

Nickel-based hydroxides are promising pseudocapacitive materials, yet their poor conductivity, limited cycling stability, and severe self-discharge hinder applications. Here, a facile one-step electrochemical co-deposition strategy, combining continuously varied metal-ion ratios with multi-current steps (ISTEP), was developed to directly construct NiCo hydroxide nanosheet arrays with dual gradients in composition and channel size (DG). The DG electrode delivers 2200 F g-1 at 1 A g-1, 45.1% retention at 20 A g-1, and 88% after 10000 cycles, outperforming the traditional non-gradient NiCo hydroxide (NG) by 74%, 9%, and 16%, respectively. Under open-circuit conditions, the voltage drop within 2 h is only 140 mV, compared with 190 mV for NG. The assembled DG||AC asymmetric supercapacitor achieves 525 F g-1 at 1 A g-1, retains 60% at 50 A g-1, and delivers 146 Wh kg-1 at 750 W kg-1, outperforming NG||AC and most reported hydroxide-based devices. More importantly, theoretical calculations and finite-element simulations reveal that the compositional gradient enhances intrinsic conductivity, while the channel-size gradient suppresses ion migration during self-discharge. This facile and scalable one-step strategy, together with the excellent pseudocapacitive performance, highlights the considerable application potential of DG-based ASC and provides new insights for advanced green energy storage.