Limit Cycle Research Articles

Asymmetric limit cycles within Lorenz chaos induce anomalous mobility for a memory-driven active particle

On applying a small bias force, nonequilibrium systems may respond in paradoxical ways such as with giant negative mobility (GNM)—a large net drift opposite to the applied bias, or giant positive mobility (GPM)—an anomalously large drift in the same direction as the applied bias. Such behaviors have been extensively studied in idealized models of externally driven passive inertial particles. Here, we consider a minimal model of a memory-driven active particle inspired from experiments with walking and superwalking droplets, whose equation of motion maps to the celebrated Lorenz system. By adding a small bias force to this Lorenz model for the active particle, we uncover a dynamical mechanism for simultaneous emergence of GNM and GPM in the parameter space. Within the chaotic sea of the parameter space, a symmetric pair of coexisting asymmetric limit cycles separate and migrate under applied bias force, resulting in anomalous transport behaviors that are sensitive to the active particle's memory. Our work highlights a general dynamical mechanism for the emergence of anomalous transport behaviors for active particles described by low-dimensional nonlinear models. Published by the American Physical Society 2024

Multistable dynamics and chaos in a system consisting of an inertial neuron coupled to a van der Pol oscillator

Abstract We consider a dynamical system consisting of a van der Pol oscillator linearly coupled to an inertial neuron with two wells potential. Analytical studies are conducted focusing on the energy computation, the dissipation and symmetry, as well as the determination and characterization of the equilibrium points. We define the parameter ranges related to different types of oscillations in the coupled system in order to have an overall idea of the nature of the attractors (hidden or self-excited) that may exist. We apply numerical analysis techniques (2-parameter diagrams, bifurcation analysis, phase portraits, basins of attractions, etc) in accordance with the previous operating range in order to shed light on the plethora of competing dynamics of the model and possible forms of strange attractors as well. Another salient point of this work is the coexistence between five self-excited attractors (limit cycle and chaos) with a hidden attractor (limit cycle). We also examine the impact of symmetry breaking on the system response. An appropriate analog simulator of the coupled system is designed and simulated in PSpice in order to check the results reported during the theoretical analyses. We believe that the results of the present work complement and enrich previously published ones concerning the dynamics of a system composed of a van der pol oscillator coupled to a (non-oscillating) double-well oscillator.

Bifurcation analysis in a modified Leslie-type predator–prey system with Holling-type III functional response and special Allee effect induced by fear factors

In this paper, a modified Leslie-type predator–prey system with Holling-type III functional response is formulated to investigate the dynamics in the presence of special Allee effect induced by fear factors. First, we calculate the first three focal values by the method of successor function to ensure that the system has an unstable weak focus of multiplicity 3. A focus- or center-type degenerate Bogdanov–Takens singularity of codimension 3 and a weak focus of multiplicity 1 or 2 together with a cusp of codimension 2 are derived for the system. Then multiple bifurcations are explored. It is shown that the system undergoes Hopf bifurcation of codimension 3 and hence three limit cycles are generated. There exists another large stable limit cycle enclosing these three limit cycles by Poincaré–Bendixson theorem. We demonstrate that a degenerate focus-type Bogdanov–Takens bifurcation of codimension 3 can occur by calculating the universal unfolding. It is also shown that a subcritical, supercritical or degenerate Hopf bifurcation occurs together with a Bogdanov–Takens bifurcation. Finally, numerical simulations are used to support the theoretical results. The analysis results reveal that the special Allee effect is beneficial for the persistence and diversity of ecosystem.

From deterministic to stochastic: limits of extracting bifurcation diagrams for noisy bistable oscillators with the control-based continuation method

AbstractNoise limits the information that can be experimentally extracted from dynamical systems. In this study, we review the Control-based Continuation (CBC) approach, which is commonly used for experimental characterisation of nonlinear systems with coexisting stable and unstable steady states. The CBC technique, however, uses a deterministic framework, whereas in practice, almost all measurements are subject to some level of random perturbation, and the underlying dynamical system is inherently noisy. In order to discover what the CBC is capable of extracting from inherently noisy experiments, we study the Hopf normal form with quintic terms with additive noise. The bifurcation diagram of the deterministic core of this system is well-known, therefore the discrepancies introduced by noise can be easily assessed. First, we utilise the Step-Matrix Multiplication based Path Integral (SMM-PI) method to approximate the system’s steady state probability density function (PDF) for different intensity noise perturbations. We associate the local extrema of the resulting PDFs with limit cycles, and compare the resulting bifurcation diagram to those captured by CBC. We show that CBC estimates the bifurcation diagram of the noisy system well for noise intensities varying from small to moderate, and in practice, the amplitudes provided by CBC may be accepted as a ’best guess’ proxy for the vibration amplitudes characteristic to the near periodic solutions in a wide range of experiments.

Crystallographic Dimensionality Determines the Electrochemical Reaction Mechanism in Alkali Transition-Metal Chlorides.

Intense research efforts on transition metal chalcogenides (oxides and sulfides), pnictides (nitrides and phosphides), and fluorides have demonstrated the complex, intertwined effects of structural and chemical changes on their electrochemical response leading to intercalation, conversion, or displacement reactions when reacting with lithium. Prior efforts largely left halides unexplored due to their heightened solubility in classical liquid electrolytes. In this work, we employ superconcentrated electrolytes to demonstrate the composition- and structure-dependent electrochemical reactivity of A2MCl4 compounds (A = Li or Na and M = Cr, Mn, Fe, and Co). Comparing four lithiated compounds, we demonstrate that they all undergo conversion reactions when reacting with 2 Li+ per formula unit, associated with large polarization and limited cycling ability. Nevertheless, combining in situ XRD with post-mortem XPS and STEM/EDS analysis, we demonstrate that Li2CoCl4 first reacts with one Li+ following a displacement reaction providing a reversible capacity of 125 mAh g-1. This reaction is enabled by the formation of a Li6CoCl8 intermediate, which shares a similar anionic framework with pristine Li2CoCl4, ensuring the topotactic insertion of Li+ balanced by the Co2+/Co0 redox couple and the formation of metallic Co nanoparticles. Comparing these compounds, we propose that two criteria are necessary to trigger the displacement reaction in A2MCl4 compounds: the presence of 1D chains of edge-sharing octahedra to favor electronic delocalization and the availability of a metal-deficient intermediate. Screening numerous A2MCl4 compounds, we demonstrate the universality of these design principles, which extend to Na-ion materials by demonstrating a low-polarization, reversible displacement reaction for Na2MnCl4.

Impact of the Prey Refuge Parameter in the Presence of Fear Effect on the Fuzzy Two-Prey–One Predator Model with Switching Effect

It has been thought of as a class of models where a generalist predator feeds on two distinct prey species. We wish to examine how prey refuge factors affect prey–predator models when fear effects are present. In order to achieve this, a two-prey–one predator model with prey refuge has been taken into consideration in the presence of the predator’s fear effect on two prey species. Both prey species engage in intra-specific competition. We enhance our model by including a switching mechanism in predation. In order to account for the inherent imperfection of environmental conditions, some parameters have been taken as fuzzy numbers. Analytical and numerical results on the system have been examined in fuzzy sense. The system’s positivity, boundedness, and permanence are examined. The system’s local and global stability analyses have been investigated. Hopf bifurcation analysis around the positive interior equilibrium point has been explored. The stability of the limit cycle of our suggested fuzzy system has been discussed. The system’s numerical simulations have been investigated with pertinent tables and graphical illustrations. When the prey refuge parameter and fear parameter exceed the critical value, the system experiences Hopf bifurcation at the positive interior equilibrium point.

On a variant of Hilbert’s 16th problem

Abstract We study the number of limit cycles that a planar polynomial vector field can have as a function of its number m of monomials. We prove that the number of limit cycles increases at least quadratically with m and we provide good lower bounds for m ⩽ 10 .

Inverse stochastic resonance in adaptive small-world neural networks.

Inverse stochastic resonance (ISR) is a counterintuitive phenomenon where noise reduces the oscillation frequency of an oscillator to a minimum occurring at an intermediate noise intensity, and sometimes even to the complete absence of oscillations. In neuroscience, ISR was first experimentally verified with cerebellar Purkinje neurons [Buchin et al., PLOS Comput. Biol. 12, e1005000 (2016)]. These experiments showed that ISR enables a locally optimal information transfer between the input and output spike train of neurons. Subsequent studies have further demonstrated the efficiency of information processing and transfer in neural networks with small-world network topology. We have conducted a numerical investigation into the impact of adaptivity on ISR in a small-world network of noisy FitzHugh-Nagumo (FHN) neurons, operating in a bi-metastable regime consisting of a metastable fixed point and a metastable limit cycle. Our results show that the degree of ISR is highly dependent on the value of the FHN model's timescale separation parameter ε. The network structure undergoes dynamic adaptation via mechanisms of either spike-time-dependent plasticity (STDP) with potentiation-/depression-domination parameter P or homeostatic structural plasticity (HSP) with rewiring frequency F. We demonstrate that both STDP and HSP amplify the effect of ISR when ε lies within the bi-stability region of FHN neurons. Specifically, at larger values of ε within the bi-stability regime, higher rewiring frequencies F are observed to enhance ISR at intermediate (weak) synaptic noise intensities, while values of P consistent with depression-domination (potentiation-domination) consistently enhance (deteriorate) ISR. Moreover, although STDP and HSP control parameters may jointly enhance ISR, P has a greater impact on improving ISR compared to F. Our findings inform future ISR enhancement strategies in noisy artificial neural circuits, aiming to optimize local information transfer between input and output spike trains in neuromorphic systems and prompt venues for experiments in neural networks.

Ginkgo biloba-derived biogenic carbon quantum dots modified NiCo-LDH: significantly enhanced energy density and cycle stability

A significant development in the usage of hydroxides in energy storage applications is the creation of LDH materials with excellent electrical conductivity and structural stability. Due to their poor electrical conductivity, bimetallic layered double hydroxides have weak rate performance and limited cycling ability. To address these issues, carbon quantum dots derived from waste Ginkgo biloba and labeled as GBC were incorporated in the preparation of NiCo-LDH. With the many oxygen-containing functional groups on the surface, GBC increases the composite's wettability while maintaining the LDH lamellar structure. It can also create localized electron-rich regions, which give the material more space and channels for electron transport, improving electrical conductivity and rate performance. Furthermore, GBC is evenly dispersed throughout the LDH skeleton, giving the composites a homogenous surface state that can mitigate the structural collapse issue brought on by the volume change during cycling. The findings demonstrate that by modulating the amount of GBC, NiCo-LDH/GBC-20 performs better electrochemically than NiCo-LDH. The capacity was 276.1 mAh g-1 at 1 A g-1, an 84.1% improvement over NiCo-LDH. Finally, the energy density displayed by the NiCo-LDH/GBC-20//AC HSC device is 72.5 Wh kg-1 (at 798.7 W kg-1), and maintains 78.2% of its original capacity after 12,000 cycles.

Nano ZnO modified amorphous carbon materials enabling long-cycle performance and high-capacity for lithium/sodium-ion batteries

Zinc-modified carbon materials exhibit high specific capacity, good safety, and low self-discharge, making them promising anode materials for lithium-ion and sodium-ion batteries. However, the high cost of raw materials, complex fabrication processes, and limited cycle life remain significant challenges to their further development. In this study, we elucidated a two-step process, simple in technique and suitable for large-scale production, for in-situ zinc modification of amorphous carbon. The synthesized carbon material exhibited superior reversible specific capacity (maintaining 780.27 mAh·g−1 capacity after 80 cycles at 50 mA·g−1 current density), high cycling stability (maintaining Coulombic efficiency at 100.06 % after 15,000 cycles at 10 A·g−1 current density), and good application value (achieving a capacity of 89.67 mAh·g−1 after 1000 cycles at 200 mA·g−1 current density for the assembled MFAC-Zn||NVP full cell). Through our investigation, we discovered that Nano ZnO not only directly participate in electrochemical reactions but also effectively improve the microstructure of amorphous carbon, increasing its three-dimensional axial stacking and providing more reaction sites. The enrichment of open pores significantly enhances the diffusion efficiency of ions within the carbon-based matrix. The superior diffusion ability of alkali metal-ion in the electrode is the reason for the excellent electrochemical performance of Zinc-modified amorphous carbon. Moreover, the incorporation of Nano ZnO aids in the formation of a robust and thin SEI layer. This significantly enhances the stability of the electrode materials during cycling and improves charge transfer efficiency during the charge-discharge process. Our study provides a new strategy for adjusting the internal microstructure of amorphous carbon materials to achieve high-performance lithium/sodium storage.

Periodic Solutions and Bifurcations in a Long-Timescale Oceanic Carbon Cycle Model

We study the dynamics of a recent model of Rothman for long timescale carbon cycle. We reproduce and extend variousresults of the Rothman model. We present numerical results showing that the model exhibits both stable and unstable limitcycles via Hopf bifurcations as the parameters are varied. We numerically find normal forms of Bautin bifurcations to confirmtheir criticality. We also extend the analysis of the normal form coefficients to identify where the fold limit cycle bifurcationoccurs.

Predicting bifurcation and amplitude death characteristics of thermoacoustic instabilities from PINNs-derived van der Pol oscillators

Self-sustained thermoacoustic oscillations as observed in low-emission combustion- involved gas turbines and aero-engines involve complicated thermal fluid–acoustics interaction and rich nonlinear dynamics. Such pulsating oscillations are known as thermoacoustic instability. When it occurs, large-amplitude limit cycle oscillations (LCOs) of thermodynamic parameters are frequently observed. These LCOs could cause overheating, flame flashback, and even engine failures. Thus it is critical to understand and predict the generation mechanisms and nonlinear dynamics behaviours, and then develop corresponding control approaches to prevent or control the onset of such instabilities. In this work, we develop and extend the classical van der Pol oscillators by integrating a physics-informed neural networks (PINNs) algorithm with a modelled nonlinear Rijke-type thermoacoustic combustor. The theoretical Rijke tube system (with Galerkin expansion and modified King's law implemented) and a CFD simulation model are applied to provide ‘training/calibration data’ for the extended van der Pol (EVDP)-PINNs model. The optimized EVDP oscillators are confirmed to be capable of capturing the key nonlinear characteristics by comparing the transient growth behaviours of thermodynamic perturbations and LCO amplitude and frequency. Further investigations are conducted to obtain Hopf bifurcation and amplitude death (AD) characteristics. Comparison is then made to the coupled EVDP systems. Quite similar Hopf bifurcation features, but differences in regions of AD, are observed. In general, we demonstrate an applicable approach to intelligently ‘learn’ a nonlinear thermoacoustic system and to create reliable EVDP oscillator systems, which have great potential to contribute to the development and testing of control approaches, such as the coupling described in this work, which may replace costly experimental tests.

Hopf Bifurcation Analysis of a Nose Landing Gear System of Aircraft

The Hopf bifurcation problem of a nose landing gear system is investigated. Taking the nearly smooth landing gear system with high-order nonlinear terms as the research object, the nose landing gear system is simplified to a plane dynamic system considering nonlinear factors by using the center manifold simplification method. The first Lyapunov coefficient of the landing gear system is derived by using the classical bifurcation theory, and the Hopf bifurcation type of the system is determined according to its sign. The bifurcation diagram is obtained by solving the differential equations of motion numerically. It is revealed that there is a stable limit cycle in the neighborhood of the supercritical Hopf bifurcation point. The influence of velocity and other factors on the shimmy oscillations of the landing gear system is analyzed. The dynamical behavior in the neighborhood of the Hopf bifurcation point is explored theoretically and numerically. The research results can provide a theoretical basis for the optimal design of aircraft landing gear systems.

Destabilizing effect of Coulomb friction induced limit cycles on sampled-data linear systems

AbstractThis paper presents a counterintuitive effect of Coulomb friction on the dynamics of sampled-data linear systems. Due to dry friction, the motion becomes sensitive to the initial conditions, and the different possible motion trajectories are separated by limit cycles. This paper shows that the linearly stable behavior of the frictionless case degrades due to the presence of dry friction and how the stable domain of operation decreases. The results are illustrated through the example of a single-degree-of-freedom system model. The presented analysis of the nonlinear system gives insight into the intricate dynamics of sampled-data systems due to the effect of dry friction, and numerical simulations verify the results.

Influence of External Clearance Variation on Vibration Characteristics of Ship Electrically Assisted Turbocharger Rotor Bearing System

Electrically assisted turbocharging (EAT) technology is an important technical means to effectively solve the problems of insufficient intake air and poor transient response of traditional turbocharged engines at low speed or acceleration, and to realize the electrification of marine internal combustion engine power system. In the EAT design stage, the original rotor model of a marine turbocharger was established and verified. Subsequently, the EAT rotor dynamics design and analysis were carried out, and the influence of the unbalanced phase and external clearance on rotor vibration was discussed. The results show that when the external clearance is small, extensive sub-synchronous vibration occurs and bifurcation occurs at high speed. When the external clearance is large, the sub-synchronous vibration component almost disappears but high synchronous vibration occurs at a high speed. At the same time, the shaft is significantly affected by the gravity load at low speeds, and the rotor makes a balanced motion on the critical limit cycle at high speed. In order to minimize the vibration and noise of the same type of EAT, it is recommended to control the outer clearance of the EAT bearing in the range of 0.30–0.36[Formula: see text]mm. The research content provides a reference for the dynamic design and analysis of the EAT, which effectively helps the development of the original engine of the ship electric auxiliary turbocharger and improves the operation stability of the EAT.

North America’s Potential for an Environmentally Sustainable Nickel, Manganese, and Cobalt Battery Value Chain

The Detroit Big Three General Motors (GMs), Ford, and Stellantis predict that electric vehicle (EV) sales will comprise 40–50% of the annual vehicle sales by 2030. Among the key components of LIBs, the LiNixMnyCo1−x−yO2 cathode, which comprises nickel, manganese, and cobalt (NMC) in various stoichiometric ratios, is widely used in EV batteries. This review reveals NMC cathodes from laboratory research. Furthermore, this study examines the environmental effect of NMC cathode production for EV batteries (including coating technologies), encompassing aspects such as energy consumption, water usage, and air emissions. Although gaps persist in NMC cathode environmental assessments (NMC111, NMC532, NMC622, and NMC811), limited life cycle assessments “(LCA)” have been conducted. Most available data originate from Asia (primarily China), accounting for 85% of the production of EV LIB cathode materials. The concept of battery passports for data collection on LIB components has been proposed to facilitate material traceability as a system for ensuring a sustainable supply chain for critical minerals. The automotive industry’s shift to electrification necessitates a sustainable supply chain from mine to vehicle end-of-life. As the critical mineral supply moves from Asia to North America, environmentally friendly industrial methods must be studied to provide this supply chain direction.

PROFIL TEKNOLOGI BUDIDAYA BANDENG (Chanos chanos) DI PUJUT KABUPATEN LOMBOK TENGAH

Milkfish cultivation with traditional technology in Mertak Village only lasts one cycle per year. The limited cycle of milkfish cultivation causes it to be less desirable, even though the profit level is quite high. For this reason, this study aims to study the key variables of cultivation and estimate the feasibility of the milkfish cultivation business. This research was conducted in Mertak Village, Central Lombok. used in this research is descriptive survey, and data collection using census method. Data analysis uses Regression-Correlation which aims to predict the determination / influence of the independent variable (X) on the dependent variable (Y). The results of the technical evaluation of milkfish cultivation in the village of Mertak amounted to 563 kg/ha/cycle with the size of the milkfish harvesting 8 individuals/kg. In addition, averange body weight (ABW) of 125 grams/individual was obtained, survival rate (SR) reached 53.68%, and FCR 0.23. The results of the business feasibility analysis show the value of the Benefit-Cost Ratio (B/C Ratio) = 2.218, which means that the milkfish cultivation business is feasible to continue because the B/C ratio is >1.0. The milkfish cultivation activities in Mertak Village, Pujut Subdistrict, Central Lombok Regency have followed the rules of good aquaculture practice (GAP). The milkfish farmers consistently run their business to increase pond productivity.

Stability and Bifurcations in Time-Delay Systems: A Novel and Efficient Blend of Methods

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.

Refining habitat selection for sulfate-reducing bacteria: Evaluating suitability and adaptability for sulfate-metal wastewater treatment during anaerobic-to-aerobic transitions

Inoculating sulfate-reducing bacteria (SRB) habitats offers an eco-friendly method for treating sulfate-metal laden wastewater, characterized by high sulfate levels, low pH, and elevated heavy metals. This study optimizes source habitat selection of SRB by evaluating groundwater, sewage sludge, and lake sediment, focusing on their suitability and adaptability to aerobic-anaerobic transitions in industrial settings. Sewage sludge, with its slightly acidic pH, reducing environment, and high nutrient levels (Total organic carbon: 207.53 g kg−1, Total nitrogen: 47.12 g kg−1), provides robust SRB potential, as supported by its highest diversity index. However, heavy metals and polycyclic aromatic hydrocarbons pose application challenges. All habitats effectively reduced metal concentrations anaerobically, with Cu removal reaching 95%–99%, and groundwater achieved the highest chemical oxygen demand reduction (63.6%) aerobically. Sludge and sediment showed high biomass and extracellular polymeric substances (EPS) accumulation, while groundwater's nucleic acid-rich EPS enhanced metal immobilization, resulting in stable residual metal forms but with potential remobilization under oxidative conditions. Microbial analysis revealed that Proteobacteria and Firmicutes were key players during transitions, with the highest SRB abundance in groundwater. SRB composition varied across habitats, with Sedimentibacter (13.04%), Desulfovibrio (6.33%), and Desulfomonile (8.1%) dominating in groundwater, sludge, and sediment, respectively, during the anaerobic stage. Functional analysis highlighted sludge's persistence in sulfate reduction under aerobic conditions, while groundwater's limited nitrogen cycle involvement indicated broader biogeochemical limitations. Collectively, these findings highlight strengths and limitations of each habitat as SRB inoculum source, emphasizing the importance of tailored anaerobic-to-aerobic strategies for effective wastewater management.

Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse

Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful for enhancing the processing capacity and storage capabilities of neural networks. When integrated into systems like Hopfield neural networks, memristors enable the study of complex dynamic behaviors, such as chaos and multistability. Moreover, fractional calculus is significant for their ability to model memory effects, enabling more accurate simulations of complex systems. Fractional-order Hopfield networks, in particular, exhibit chaotic and multistable behaviors not found in integer-order models. By combining memristors with fractional-order Hopfield neural networks, these systems offer the possibility of investigating different dynamic phenomena in artificial neural networks. This study investigates the dynamical behavior of a fractional-order Hopfield neural network (HNN) incorporating a memristor with a piecewise segment function in one of its synapses, highlighting the impact of fractional-order derivatives and memristive synapses on the stability, robustness, and dynamic complexity of the system. Using a network of four neurons as a case study, it is demonstrated that the memristive fractional-order HNN exhibits multistability, coexisting chaotic attractors, and coexisting limit cycles. Through spectral entropy analysis, the regions in the initial condition space that display varying degrees of complexity are mapped, highlighting those areas where the chaotic series approach a pseudo-random sequence of numbers. Finally, the proposed fractional-order memristive HNN is implemented on a Field-Programmable Gate Array (FPGA), demonstrating the feasibility of real-time hardware realization.