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A control strategy combining cubic displacement feedback nonlinear control and proportional-differential (PD) linear control is used to control the vibration performance of the maglev system. The maglev system is divided into positive stiffness maglev system, quasi-zero stiffness maglev system and negative stiffness maglev system according to the linear stiffness value of maglev system. Firstly, an improved multi-scale method is used to analyze the vibration characteristics of suspension in the positive stiffness state of the maglev system. Secondly, the influence of control parameters on train vibration amplitude and vibration center displacement under quasi-zero stiffness is studied. Finally, the vibration characteristics of the train when the maglev system is in negative stiffness are analyzed by numerical simulation. The maglev system exhibits the worst vibration performance under negative stiffness compared with positive stiffness and quasi-zero stiffness. The suspension frame is easy to enter the chaotic motion state, and its vibration center is easy to deviate from the equilibrium position and produce large displacement when the maglev system is in the negative stiffness state. The control results show that the control strategy combining the cubic displacement feedback nonlinear control with the PD linear control can make the maglev system exhibit better vibration characteristics under positive and quasi-zero stiffness. [ABSTRACT FROM AUTHOR]

During the bearing operation, surface waviness can cause severe nonlinear vibrations and reduce the life and reliability of bearing-rotor systems. A temperature increase can deform the surface waviness, which can exacerbate this unstable vibration. To address this problem, a dynamic waviness model considering thermal deformation is proposed, and the dynamic support stiffness is calculated and introduced into the dynamic model of a 12-DOF full-ceramic bearing-rotor system. The Newton–Raphson and Newmark-β nested iterative solution method combines the quasi-static and dynamical models. The bifurcation, maximum Lyapunov exponent, and Poincaré mapping analysis methods serve to analyse how thermal deformation, waviness amplitude and other parameters affect the system nonlinear vibration. Experimental measurements are conducted to verify the model accuracy and reveal that a larger waviness amplitude causes a delayed motion state and expands the influence of the thermal deformation. The wavenumber is close to an integer multiple of the ball, and the time-varying displacement excitation curve shows a monotonic variation trend, which makes the system violently vibrate. The model effectively reveals the characteristics of the failure frequency and provides important theoretical support for the fault detection of full-ceramic bearings. [ABSTRACT FROM AUTHOR]

This article explores the examination of the widely employed zig-zag optical lattice model for cold bosonic atoms, which is commonly utilized to depict nonlinear wave in fluid mechanics and plasma physics. The focus is on obtaining soliton solutions in optics and investigating their physical properties. A wave transformation is initially applied to convert a partial differential equation (PDE) into an ordinary differential equation (ODE). Soliton solutions are subsequently obtained through the application of two distinct methods, namely the generalized logistic equation method and the Sardar sub-equation method. These solutions include bright, dark, combined dark-bright, chirped type solitons, bell-shaped, periodic, W-shape, and kink solitons. In this paper, the solutions derived from two analytical approaches were compared to enhance the understanding of the behavior of the discussed nonlinear model. The obtained solutions have significant implications across various fields such as plasma physics, fluid dynamics, optics, and communication technology. Furthermore, 3D and 2D graphs are generated to depict the physical phenomena of the derived solutions by assigning appropriate constant parameters. The qualitative evaluation of the undisturbed planar system involves the analysis of phase portraits within bifurcation theory. Subsequently, the introduction of an outward force is carried out to induce disruption, and chaotic phenomena are unveiled. The detection of chaotic trajectory in the perturbed system is achieved through 3D plots, 2D plots, time scale plots, and Lyapunov exponents. Furthermore, stability analysis of the examined model is addressed under distinct initial conditions. Finally, the sensitivity assessment of the model under consideration is carried out using the Runge–Kutta method. The results of this study are innovative and have not been previously investigated for the system under consideration. The results obtained underscore the reliability, simplicity, and effectiveness of these techniques in analyzing a variety of nonlinear models found in mathematical physics and engineering disciplines. [ABSTRACT FROM AUTHOR]

Background: Onchocerciasis infection is one of the neglected tropical diseases targeted for eradication by 2030. The disease is usually transmitted to humans through the bites of black flies. These black flies mostly breed near well-oxygenated fast-running water bodies. The disease is common in mostly remote agricultural villages near rivers and streams. Objective: In this study, a deterministic model describing the infection dynamics of human onchocerciasis disease with control measures is presented. Methods: We derived the model's reproductive number and used a stability theorem of a Metzler matrix to show that disease-free equilibrium is both locally and globally asymptotically stable whenever the reproductive number is less than one. Parameter contribution was conducted using sensitivity analysis. The model endemic equation is shown to be a cubic polynomial in the presence of infected immigrants and a quadratic form in their absence. Results: When the inflow of infected immigrants is null, the model endemic equation may admit a unique equilibrium if the reproductive number is greater than one, or admits multiple endemic equilibria if the reproductive number is less than unity. We carried out a sensitivity analysis to identify the significant parameters that contribute to onchocerciasis spread. Conclusion: Onchocerciasis disease can be eradicated if the importation of infected immigrants is properly monitored. The integration of the One Health concept in the public health system is key in tackling the emergence and spread of diseases. [ABSTRACT FROM AUTHOR]

The broaching that follows the surf-riding is a dangerous phenomenon that can lead to the capsizing of a vessel due to its violent yaw motion. Most of the previous studies on surf-riding phenomena in irregular waves have been conducted by replacing irregular waves with regular waves. In contrast, this study provides suggestions on how to directly calculate nonlinear surge motion in irregular seas. In this study, the statistical aspects of the surf-riding phenomenon are first presented. Then, under several approximations, we show how to calculate the probability density function theoretically. Although the results obtained are based on strong approximations, it is found that the nonlinear surge oscillations in irregular following seas can be explained from a qualitative point of view. [ABSTRACT FROM AUTHOR]

Biological invasions significantly impact native ecosystems, altering ecological processes and community behaviors through predation and competition. The introduction of non-native species can lead to either coexistence or extinction within local habitats. Our research develops a lizard population model that integrates aspects of competition, intraguild predation, and the dispersal behavior of intraguild prey. We analyze the model to determine the existence and stability of various ecological equilibria, uncovering the potential for bistability under certain conditions. By employing the dispersal rate as a bifurcation parameter, we reveal complex bifurcation dynamics associated with the positive equilibrium. Additionally, we conduct a two-parameter bifurcation analysis to investigate the combined impact of dispersal and intraguild predation on ecological structures. Our findings indicate that intraguild predation not only influences the movement patterns of brown anoles but also plays a crucial role in sustaining the coexistence of different lizard species in diverse habitats. [ABSTRACT FROM AUTHOR]

A modified incremental harmonic balance (IHB) method is used to determine periodic solutions of wave propagation in discrete, strongly nonlinear, periodic structures, and solutions are found to be in a two-dimensional hyperplane. A novel method based on the Hill's method is developed to analyze stability and bifurcations of periodic solutions. A simplified model of wave propagation in a strongly nonlinear monatomic chain is examined in detail. The study reveals the amplitude-dependent property of nonlinear wave propagation in the structure and relationships among the frequency, the amplitude, the propagation constant, and the nonlinear stiffness. Numerous bifurcations are identified for the strongly nonlinear chain. Attenuation zones for wave propagation that are determined using an analysis of results from the modified IHB method and directly using the modified IHB method are in excellent agreement. Two frequency formulae for weakly and strongly nonlinear monatomic chains are obtained by a fitting method for results from the modified IHB method, and the one for a weakly nonlinear monatomic chain is consistent with the result from a perturbation method in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a nonlocal reaction–diffusion–advection model with nonlinear boundary conditions arising from phytoplankton species is investigated, where the species depends solely on light for its metabolism. Firstly, we prove that a single phytoplankton model with nonlinear boundary conditions is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determine the global dynamics of single phytoplankton species in terms of boundary reaction functions. Secondly, we obtain the existence and uniqueness of positive steady states in terms of the critical death rate of the phytoplankton species by using the Lyapunov–Schmidt reduction method. Thirdly, we analyze the effects of advection rate, diffusion rate and water column depth on critical death rate under some conditions. Finally, we investigate the asymptotic behaviors of positive steady state for large advection rate. [ABSTRACT FROM AUTHOR]

It has been observed in experiments that, for some materials, when a sheet is subjected to increasing equal biaxial tensile forces on its edges, its deformation can change from homogeneous equal biaxial extension to homogeneous unequal biaxial extension. Such response has been analyzed in the literature as a bifurcation from the base deformation to a new deformation. The analyses have used a constitutive equation that expresses the stress tensor as a function of a deformation tensor. The present work is concerned with analyzing this bifurcation using a new class of constitutive equations in which the deformation tensor is expressed as a function of the stress tensor. Using such a model for an incompressible isotropic nonlinear elastic material, a condition is derived for determining when during an equal biaxial extension of a sheet, there is a bifurcation into an unequal biaxial extension. An example is provided using a constitutive equation that has been fit to experimental data. [ABSTRACT FROM AUTHOR]

Introduction: The aim of this study was to examine the relationship between the morphological parameters of retinal arteriolar bifurcations and coronary artery disease (CAD). Methods: In this cross-sectional observational study, fundus photography was conducted on 444 participants to capture retinal arteriolar bifurcations. A total of 731 fundus photographs yielded 9625 measurable bifurcations. Analyzed bifurcation parameters included the diameters of the parent vessel (d0), the larger branch (d1), and the smaller branch (d2), as well as the angles (θ1) and (θ2) representing the orientation of each branch in relation to the parent vessel, respectively. Additionally, theoretical optimal angles ( θ 1 ′ ) and ( θ 2 ′ ), calculated from the measured parameters, provided a benchmark for ideal bifurcation geometry. The study assessed the variation in these parameters across different levels of coronary atherosclerosis severity. Results: After adjusting for anatomical characteristics including the asymmetry ratio, area ratio, and distance to the optic disc, we observed that patients with severe coronary artery stenosis had significant deviations from the theoretical optimal bifurcation angles, with a decrease in ( θ 1 ′ ) and an increase in ( θ 2 ′ ) compared to those with moderate stenosis. Conclusion: The findings suggest a clear alteration in retinal arteriolar bifurcation morphology among patients with severe CAD, which could potentially serve as an indicator of disease severity. [ABSTRACT FROM AUTHOR]

The global impact of high unemployment rates has significant economic and social consequences. To overcome this, various skill development programs are initiated by governments of developing countries. But the problem of unemployment is still increasing day by day. So, there is a pressing necessity to revise the current policies and models. Therefore, this research proposes a fractional‐order mathematical model that examines the impact of various skill development programs for youths. The proposed model incorporates fractional‐order differential equations to capture the complex dynamics of unemployment. The main objective of this research is to examine the impact of training programs aimed at enhancing the abilities of unemployed individuals, with the ultimate goal of reducing the overall unemployment rate. The reproduction number is calculated using the next‐generation matrix approach, which is crucial for both the existence and stability analysis of the equilibria. When the reproduction number is less than 1, the employment‐free equilibrium is locally and globally asymptotically stable. The employment‐persistence equilibrium point emerges only when the reproduction number exceeds one. We also explore the possibility of transcritical bifurcation and investigate the impact of skill development on the unemployment rate. We conduct numerical simulations to validate our analytical findings, further supporting our qualitative conclusions. These simulations help illustrate the unemployment dynamics and confirm the stability and behavior of the equilibrium points predicted by the mathematical model. [ABSTRACT FROM AUTHOR]

We study the class of symmetric Dziobek configurations where there are p + q bodies with non-zero masses at the vertices of a regular (p + q - 1) -simplex and one more body with zero mass on the symmetric line going through the centers of a subsimplex with p vertices and its complementary subsimplex with q vertices. All bodies from the same subsimplex have equal masses and let μ ≠ 0 be the ratio of masses from different subsimplices. We consider such central configurations for homogeneous force laws with exponent a ≠ 0 . The Newtonian case of a = - 3 2 with μ > 0 is studied more intensively. Using elementary methods, we obtain results for general values of the parameters a, p, q, and μ . For a = - 3 2 , we count the real roots of a polynomial system in two variables with three parameters. We are able to reduce the number of parameters required in our symbolic computation to two or one and use some techniques to count roots rigorously for all parameter values. Our results extend many previously known cases. For example, the known three positions of the zero mass in the case of a = - 3 2 , p = q = 1 , μ > 0 are at the Lagrange points L 1 , L 2 , and L 3 discovered by Euler. [ABSTRACT FROM AUTHOR]

Summary: Trigger waves, propagating in a loop of a one‐dimensional lattice of tunnel‐diode (TD) oscillators, can be effectively applied to voltage‐controlled oscillation with high tuning gain. The lattice under investigation can generate both phase and trigger waves by varying the bias voltage to the TD. We demonstrate a voltage‐controlled oscillator (VCO) achieving high tuning gain through the utilization of rotating trigger waves. As the loop size decreases, the trigger wave becomes bound to itself or to adjacent waves. At this point, the excitation motion of the individual lattice cell is constrained by the global interaction caused by the trigger wave(s), leading to coherent behavior. Consequently, the bifurcation curve of the trigger wave seamlessly connects to that of the phase wave with changes in the TD bias voltage. Owing to the trigger wave's property of significantly increasing velocity with the TD bias voltage, a VCO with very high tuning gain can be developed, leveraging the seamless dependence of the oscillation frequency on the TD bias voltage. [ABSTRACT FROM AUTHOR]

In this paper, we consider two-dimensional periodic capillary-gravity waves traveling under the influence of a vertical electric field. The full system is a nonlinear, two-layered, free boundary problem. The interface dynamics are derived by coupling Euler equations for the velocity field of the fluid with voltage potential equations governing the electric field. We first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. We then prove the existence of small-amplitude electrohydrodynamic waves with constant vorticity using local bifurcation theory. Moreover, we show that these electrohydrodynamic waves are formally stable in the linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch, consisting of ripple solutions on the interface. As far as we know, such solutions in electrohydrodynamics are established for the first time. It is worth noting that the electric field E 0 plays a key role in controlling the shapes and types of waves on the interface. [ABSTRACT FROM AUTHOR]

This paper studies a dynamical system that models the free recall dynamics of working memory. This model is an attractor neural network with n modules, named hypercolumns, and each module consists of m minicolumns. Under mild conditions on the connection weights between minicolumns, the authors investigate the long-term evolution behavior of the model, namely the existence and stability of equilibria and limit cycles. The authors also give a critical value in which Hopf bifurcation happens. Finally, the authors give a sufficient condition under which this model has a globally asymptotically stable equilibrium consisting of synchronized minicolumn states in each hypercolumn, which implies that in this case recalling is impossible. Numerical simulations are provided to illustrate the proposed theoretical results. Furthermore, a numerical example the authors give suggests that patterns can be stored in not only equilibria and limit cycles, but also strange attractors (or chaos). [ABSTRACT FROM AUTHOR]

This study investigates the influence of fear, refuge, and migration in a predator–prey model, where the interactions between the species follow an asymmetric function response. In contrast to some other findings, we propose that prey develop an anti-predator response in response to a concentration of predators, which in turn increases the fear factor of the predators. The conditions under which all ecologically meaningful equilibrium points exist are discussed in detail. The local and global dynamics of the model are determined at all equilibrium points. The model admits several interesting results by changing the rate of fear of predators and predator aggregate sensitivity. Numerical simulations have been performed to verify our theoretical findings. We found that under certain conditions, the system appears to be losing the stability to acquire the periodic attractor when the trait-mediated direct effect of one of each (prey growth, competition, fear, and prey death). [ABSTRACT FROM AUTHOR]

Background: Recurrent intracranial aneurysms present a significant clinical challenge, demanding innovative and effective treatment approaches. The Woven EndoBridge (WEB) device has emerged as a promising endovascular solution for managing these intricate cases. This study aims to assess the safety and efficacy of the WEB device in treating recurrent intracranial aneurysms. Methods: We conducted a comprehensive search across multiple databases, including PubMed, Scopus, Embase, and Web of Science, from inception to June 5, 2023. Eligible studies focused on evaluating WEB device performance and included a minimum of five patients with recurrent intracranial aneurysms. The complete and adequate occlusion rates, neck remnant rates, and periprocedural complication rates were pooled using SATA V.17. Results: Our analysis included five studies collectively enrolling 73 participants. Participant ages ranged from 52.9 to 65 years, with 64.4% being female. Aneurysms were wide-necked and predominantly located in the middle cerebral artery, basilar artery, and anterior cerebral artery. Previous treatments encompassed coiling, clipping, and the use of WEB devices. Our study found an overall adequate occlusion rate of 0.80 (95% CI 0.71–0.89), a complete occlusion rate of 0.39 (95% CI 0.28–0.50), and a neck remnant rate of 0.38 (95% CI 0.27–0.48). Periprocedural complications were reported at a rate of 0%, although heterogeneity was observed in this data. Notably, evidence of publication bias was identified in the reporting of periprocedural complication rates. Conclusion: Our findings suggest that the WEB device is associated with favorable outcomes for treating recurrent wide-neck intracranial aneurysms. [ABSTRACT FROM AUTHOR]

In this paper, we studied the Klausmeier-Gray-Scott model with non-diffusive plants, which is a coupled ordinary differential equation-partial differential equation (ODE-PDE) system. We first established the critical conditions for instability of the constant steady state in general coupled ODE-PDE activator-inhibitor systems. Turing instability does not occur when the self-activator is diffusive and the self-inhibitor is non-diffusive. Conversely, Turing instability occurred and was caused by the continuous spectrum and infinitely many eigenvalues in the right half of the complex plane. In addition, the local structure of the nonconstant steady state and the condition to determine the local bifurcation direction were obtained. However, the nonconstant steady state was unstable. The models with non-diffusive plants exhibit spike spatial patterns with vegetation concentrated on small areas, which cannot be explained by the bifurcated steady-state solutions. Thus, we investigated the spatial pattern of the model with slowly diffusive plants to understand the formation of the spike pattern. Specifically, the Turing bifurcation curves and nonconstant steady states for the model with diffusive plants were characterized. The vegetation distribution became more uneven and even formed a spike-like spatial pattern as plants dispersed slower. Through analysis, we found that the spike-like vegetation pattern approximates the superposition of multiple Turing patterns with specific wave frequencies, which provided a certain explanation for the spike pattern. [ABSTRACT FROM AUTHOR]

Magnetically coupled resonant wireless power transfer (MCR‐WPT) system with constant power load (CPL), finds extensive applications in military, industrial, and medical treatment. However, this system can easily exhibit complex non‐linear behaviors under certain parameter conditions due to the influences of constant power loads, switching devices, and feedback controllers. These behaviors limit the effectiveness of controllers and reduce the efficiency and stability of the system. It is necessary to study the non‐linear characteristics of the MCR‐WPT system with CPL. The stroboscopic discrete mapping of the MCR‐WPT system with CPL is established. Then the system's non‐linear dynamics are analyzed theoretically using a bifurcation diagram, the maximal Floquet multipliers, and the maximal Lyapunov exponent spectrum obtained from the proposed discrete mapping. The results show that the MCR‐WPT system with CPL will exhibit rich non‐linear dynamics with the variation of the power of CPL, such as cyclic fold bifurcation, Neimark–Scaker bifurcation, border collision bifurcations, chaos, etc. The excellent alignment of experimental and theoretical outcomes in corresponding states confirms the accuracy of the proposed discrete mapping and the nonlinear analysis of the system. The results of this study can provide a reference for selecting parameters for an actual MCR‐WPT system with CPL. [ABSTRACT FROM AUTHOR]

ABSTRACT An ecosystem shifts to an alternative stable state when a threshold of accumulated pressure (i.e. direct impact of environmental change or human activities) is exceeded. Detecting this threshold in empirical data remains a challenge because ecosystems are governed by complex interlinkages and feedback loops between their components and pressures. In addition, multiple feedback mechanisms exist that can make an ecosystem resilient to state shifts. Therefore, unless a broad ecological perspective is used to detect state shifts, it remains questionable to what extent current detection methods really capture ecosystem state shifts and whether inferences made from smaller scale analyses can be implemented into ecosystem management. We reviewed the techniques currently used for retrospective detection of state shifts detection from empirical data. We show that most techniques are not suitable for taking a broad ecosystem perspective because approximately 85% do not combine intervariable non‐linear relationships and high‐dimensional data from multiple ecosystem variables, but rather tend to focus on one subsystem of the ecosystem. Thus, our perception of state shifts may be limited by methods that are often used on smaller data sets, unrepresentative of whole ecosystems. By reviewing the characteristics, advantages, and limitations of the current techniques, we identify methods that provide the potential to incorporate a broad ecosystem‐based approach. We therefore provide perspectives into developing techniques better suited for detecting ecosystem state shifts that incorporate intervariable interactions and high‐dimensionality data. [ABSTRACT FROM AUTHOR]

This piece of work circles around the investigation of a predator–prey model that incorporates nonlinear harvesting and group defense, both of which are important phenomena occurring in the ecosystem. To analyze the proposed model, both analytical and numerical methods are employed. The model experiences a series of bifurcations, including saddle-node, Hopf and Bogdanov–Takens. The system exhibits bifurcations of codimension one and two but no bifurcation of codimension three emerged. Phase portrait diagrams are also presented to confirm the validity of both the analytical and numerical findings. [ABSTRACT FROM AUTHOR]

In this research work, the authors have developed a mathematical model to examine the interaction between dystrophin protein and tumor. The authors formulated a system of ordinary differential equations to describe the dynamics of the dystrophin-tumor interaction system. Jacobian matrix and Routh–Hurwitz stability techniques were used to determine equilibrium points, perform stability and bifurcation analysis, and establish the conditions required for the stability of the proposed model. Numerical simulations are performed using Euler's method to investigate the temporal evolution of the proposed model under different parameter values, such as tumor growth rate and feedback strength of dystrophin protein. The numerical results are presented in tables, and corresponding to each table, a graphical analysis is done. The graphical analysis includes creating phase portraits to visually represent stability regions around the equilibrium points, bifurcation diagrams to identify critical points, and time series analysis to highlight the behavior of the proposed model. The authors explore how variations in dystrophin expression impact tumor progression, identifying potential therapeutic implications of maintaining higher dystrophin levels. This comprehensive analysis enhances our understanding of the dystrophin-tumor interaction, providing a basis for further experimental validation and potential therapeutic strategies. [ABSTRACT FROM AUTHOR]

In this work, we study the Nanoparticles (NPs) impact on a phytoplankton–zooplankton interaction model with Ivlev-like and Holling type-II functional responses. We found that the growth rate of phytoplankton reduces due to NPs. In the non-spatial model, we investigated boundedness, stability, bifurcation and chaos. The stability criteria is determined using the Routh–Hurwitz criterion. Hopf bifurcation is demonstrated with parameter K, which represents the NPs carrying capacity while interacting with phytoplankton. The normal theory is used to examine the Hopf bifurcation direction and the stability of bifurcating periodic solutions. Moreover, the stability of non-hyperbolic equilibrium points have been determined using the Center Manifold theorem. Also, the parameter β, which represents the interaction rate between NPs and phytoplankton, exhibits chaotic behavior. Furthermore, we also investigated Hopf bifurcation and Turing instability in spatial model systems. This study demonstrates that NPs can influence the dynamics of the system in a balanced environment. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of reaction–diffusion equations arising from the competition of two competing species for a single limited nutrient with flocculation in an unstirred chemostat. By the conservation principle, we reduce the dimension of the system by eliminating the equation for the nutrient. Then the global structure of the reduced system is studied by the bifurcation theory in its feasible domain. Finally, we use numerical simulation to verify and supplement our theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a new discrete form of the classical continuous Rosenzweig–MacArthur predator–prey model and the biological implications of the dynamics. The discretization method used here is to modify the continuous model to another with piecewise-constant arguments and then to integrate the modified model, which is quite different from the traditional Euler discretization method. First, the existence and local stability of fixed points have been thoroughly discussed. Then, all codimension-1 bifurcations have been studied, including the transcritical bifurcations at the trivial fixed point and the boundary fixed point, and the Neimark–Sacker bifurcation at the unique positive fixed point. Furthermore, it is proved that there are no codimension-2 bifurcations. Finally, the control of the bifurcations is studied. Regarding the biological significance, we have considered the following four aspects: When no harvesting effort is applied to both predators and prey, it is observed that providing more resources to the prey species leads to predator extinction, causing the ecosystem to lose stability. This implies that the paradox of enrichment occurs. When predators are harvested, the system exhibits the hydra effect, i.e. increasing the harvesting effort on predators will lead to an increase in the mean population density of predators. When the prey is harvested, numerical examples indicate that increasing the harvesting effort initially reduces prey density. Upon reaching the bifurcation point, prey density stabilizes while the predator density continues to decline. When both prey and predators are harvested, the biological significance is similar to the previous case. The outcomes of this paper have significant theoretical meaning in the study of population ecology. By MATLAB, the bifurcation diagrams, maximal Lyapunov exponent diagrams, phase portraits and mean population density curves are plotted. Numerical simulations not only show the correctness of the theoretical findings but also reveal many new and interesting dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel two-delayed tri-neuron neural network (NN) with no connection between the first and third neurons. Neural networks with incomplete connections offer a range of advantages, including improved efficiency, generalisation, interpretability, and biological plausibility, making them useful in various applications across different domains. Such kinds of NNs exist in some diseases, such as epilepsy, Alzheimer's, and schizophrenia, where the neuron's connections can be broken. Our NN is defined in two different forms: one with integer-order derivatives and another with Caputo fractional derivatives. The fundamental results of existence, uniqueness, and boundedness of the solution for the proposed NN are derived. We perform the bifurcation analysis along with the stability of the initial state of the fractional-order NN, considering self-connection delay and communication delay as bifurcation parameters, respectively. The proposed NN is numerically solved by using a recently proposed L1-predictor-corrector method with its error analysis. The theoretical proofs are verified through graphical simulations. [ABSTRACT FROM AUTHOR]

The bursting with long burst duration of the pancreatic β-cells related to the diabetes is helpful to maintain the normal blood glucose concentration. Then, nonlinear dynamics underlying burst duration is an important issue, which is investigated in a theoretical model containing two slow variables (s and z) in the present paper. For the deterministic case, different shapes and sizes of the bursting trajectory show that the slow bursting is modulated by s, z, and s-nullcline in wide ranges, the medium bursting by s and z in medium ranges, and the fast bursting mainly by s in narrow ranges. Through two-parameter bifurcation analysis, the burst of the three bursting patterns terminates at saddle homoclinic orbit (SH) bifurcation curve. For the slow and medium bursting patterns, the latter part of the burst mainly runs along z-direction and closely to the saddle surface. Then, noise can induce the burst to terminate earlier than the SH via passage through the saddle surface, resulting in decreased burst duration. Compared with the drastic decrease for the medium bursting, slow bursting exhibits small reduction, since the large size induces two phases insensitive to noise. One is the latter part of the quiescent state, which runs along the s-nullcline to exhibit extra-stable dynamics, and the other is the former part of the burst, which runs far from the saddle surface. However, the fast bursting mainly runs along s-direction and far from the saddle surface exclusive the SH, then, noise can only induce small changes. Then, the different changing trends of the stochastic slow, medium, and fast bursting patterns are well explained with the different stochastic responses around different bifurcation points or critical phases. The results present dynamical mechanism for the long burst duration which is favor for the maintenance of the normal blood glucose concentration. [ABSTRACT FROM AUTHOR]

In this work, we first discuss the positive equilibrium point of the continuous predator–prey system and its stability, and we discuss the parameter conditions under which the continuous system undergoes a cusp bifurcation (Bogdanov–Takens bifurcation) of codimension two bifurcation at the positive equilibrium point. Then, we provide an insightful study of discrete predator–prey systems by the use of Euler's method, which includes square-root function responses and nonlinear prey harvesting. By synthesizing the new standard form of differential-algebraic systems, the central manifold theorem, and the bifurcation theory, we identify the specific conditions under which the system may undergo flip bifurcation and Neimark–Sacker bifurcation. In addition, codimension-two bifurcations associated with 1:2 strong resonances are analyzed by using a series of affine transformations and bifurcation theory. Through numerical simulations, we not only verify the validity and correctness of our findings, but also elucidate the frequency of trajectory bifurcations in the intervals of 2, 4, and 8 and chaotic phenomena. These findings reveal a richer and more diverse dynamic behavior of discrete differential-algebraic bioeconomic systems, which is of great theoretical and practical significance to the fields of mathematics and biology. [ABSTRACT FROM AUTHOR]

Mitigating traffic jams is a critical step for the betterment of the urban transportation system, which comprises a large number of interconnected routes to form an intricate network. To study the distinct features and complexities of vehicular flow, a network consisting of multiple-loops with a single intersection is considered a directed weighted graph. The governing equations for individual loop densities are derived using the principle of mass conservation, considering a data-driven flow-density relationship. Firstly, we examine the stability of a double-loop network and explore the traffic behavior using the macroscopic fundamental diagram (MFD) for different sets of parameters. Utilizing popular techniques of nonlinear dynamics, the existence of bistability, bifurcations, oscillations, and switching dynamics is demonstrated in the case of the triple-loop network. Further, bistability is characterized by plotting the basin of attraction diagram for the coexisting attractors. Our study reveals that an increase in the number of loop lines enriches the dynamical properties of vehicular traffic flow. It is observed that depending upon the initial density configuration, a loop network can show various phases, namely free flow, stop-and-go traffic jams, and loop congestion. Additionally, we show the presence of period-2 orbits in the case of the quadruple-loop system. [ABSTRACT FROM AUTHOR]

This contribution concerns studying a realistic predator–prey interaction, which was achieved by virtue of formulating a modified Leslie–Gower predator–prey model under the influence of the double Allee effect and fear effect in the prey species. The initial theoretical work sheds light on the relevant properties of the solution, presence, and local stability of the equilibria. Both analytic and numerical approaches were used to address the emergence of diverse bifurcations, like saddle-node, Hopf, and Bogdanov–Takens bifurcations. It is noteworthy that while making the assumption that the characteristic equation of the Jacobian matrix J has a pair of imaginary roots C (ρ) ± ι D (ρ) , it is sufficient to consider only C (ρ) + ι D (ρ) due to symmetry. The impact of the fear effect on the proposed model is discussed. Numerical simulation results are provided to back up all the theoretical analysis. From the findings, it was established that the initial condition of the population, as well as the phenomena (fear effect) introduced, played a crucial role in determining the stability of the proposed model. [ABSTRACT FROM AUTHOR]

In online social networks (OSNs), the spread of negative information can cause social shocks, endanger public safety and harm public interests. In this paper, to reduce the impact of negative information on the network ecosystem as much as possible, we consider a competitive strategy of mainstream media coverage and construct a competitively regulated reaction–diffusion model of negative information transmission. For this model, we obtain the basic reproduction number of information propagation, the invade numbers of negative information and positive information. Besides, the local and global stabilities of equilibrium states are discussed. Further, steady bifurcation analyses from a positive information equilibrium and a negative information equilibrium are carried out. Finally, we present some numerical simulations to verify the theoretical analysis results and discuss how spatial heterogeneity affects the process of information diffusion. The conclusions are as follows: Increasing the spread of positive information, controlling the spatial heterogeneity of the transmission rate and reducing the diffusion coefficient when spatial heterogeneous transmission rate occurs are beneficial for suppressing the spread of negative information. [ABSTRACT FROM AUTHOR]

This paper studies the steady states to a diffusive Lotka–Volterra cooperative model with population flux by attractive transition. The first result gives many bifurcation points on the branch of the positive constant solution under the weak cooperative condition. The second result shows every steady state approaches a solution of the scalar field equation as the coefficients of the flux tend to infinity. Indeed, the numerical simulation using pde2path exhibits the global bifurcation branch of the cooperative model with large population flux is near that of the scalar field equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the influence of the nonlocal prey competition on the spatio-temporal dynamics for a generalist predator–prey system. The condition of stability and bifurcations is clearly determined. Our results show that when the prey spreads quickly, the nonlocal intraspecific competition of the prey does not affect the dynamics, however, when the prey spreads slowly, it can affect the dynamics. Besides, no Hopf bifurcation occurs if the ratio of the growth rate of the predator to prey is larger, otherwise, system has Hopf bifurcation and Hopf Bogdanov–Takens bifurcation and so on. It is surprised that the system only with the nonlocal prey competition has more rich dynamics than the system with the nonlocal competitions in both the prey and the predator. The coexistence of bistable spatially inhomogeneous steady states is also found. [ABSTRACT FROM AUTHOR]

Population ecology theory is replete with density-dependent processes. However, trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems, where the non-consumptive behavioral fear effect and the Allee effect, a density-dependent process, are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter. Interestingly, we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear. There are three parametric regimes of interest in the fear parameter. For small and intermediate amounts of fear, the Allee + fear effect opposes dynamics driven by the fear effect. However, for large amounts of fear the Allee + fear effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end, the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity. [ABSTRACT FROM AUTHOR]

This paper investigates a discrete conformable fractional order bacteria population model in a microcosm. When the parameters of this discrete model satisfy some conditions, its positive fixed point has a double eigenvalue - 1 with geometric multiplicity 1, corresponding to a 1:2 strong resonance. Applying the normal form theory and the approximation theory by a flow, we firstly transform this discrete model into a two-dimensional ordinary differential system. Then we use the bifurcation theory to analyze the qualitative properties of this ordinary differential system near the degenerate equilibrium. Therefore, the qualitative properties of the corresponding fixed point are obtained, which show that this discrete model undergoes the period-doubling bifurcation, the Neimark-Sacker bifurcation, the homoclinic bifurcation and the double closed orbit bifurcation. We also add a hybrid controller to this model such that the fixed point is locally asymptotically stable. Finally, numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy. [ABSTRACT FROM AUTHOR]

We present an example of a monotone two-parameter family of vector fields on a torus whose bifurcation diagram we demonstrate to be in the class of 'simplest' diagrams proposed by Baesens and MacKay (2018 Nonlinearity 31 2928–81). This shows that the proposed class is realisable. [ABSTRACT FROM AUTHOR]

This study aims to investigate a financial system consisting of four ordinary differential equations associated with the rate of interest, investment demand, price index, and the density of financial information gained by the population. The equilibrium and local stability of the system are investigated numerically. The impact of saving amounts and the rate of investment demand increases after getting financial information on the system are discussed. The findings of the study are verified graphically. It is found that the system becomes stable if the rate of investment demand increases after getting financial information kept at a certain level, such that the savings amount is maintained at a higher level. Also, the bifurcation diagrams of the system for various significant parameters that affect the system's stability have been depicted. [ABSTRACT FROM AUTHOR]

Considering the impact of fear levels, Allee effects and hunting cooperation factors on system stability, a Leslie-Gower predator-prey model was formulated. The existence, stability and bifurcation analysis of equilibrium points were studied by use of topological equivalence, characteristic equations, Sotomayor's theorem, and bifurcation theory. The sufficient conditions of saddle-node, Hopf, and Bogdanov-Takens bifurcations were established, respectively. Numerically, the theoretical findings were validated and some complicated dynamical behaviors as periodic fluctuation and multistability were revealed. The parameter critical values of saddle-node, Hopf bifurcation, and Bogdanov-Takens bifurcations were established. Biologically, how these factors of fear, Allee effect, and hunting cooperation affect the existence of equilibria and jointly affect the system dynamics were analyzed. [ABSTRACT FROM AUTHOR]

This paper proposed two new generalizations of the logistic function, each drawing on non-extensive thermodynamics, the q-logistic Equation and the logistic Equation of arbitrary order, respectively. It demonstrated the impact of chaos theory by integrating it with logistics Equations and revealed how minor parameter variations will change system behavior from deterministic to non-deterministic behavior. Moreover, this work presented BifDraw – a Python program for drawing bifurcation diagrams using classical logistic function and its generalizations illustrating the diversity of the system’s response to the changes in the conditions. The research gave a pivotal role to the place of the logistic Equation in chaos theory by looking at its complicated dynamics and offering new generalizations that may be new in terms of thermodynamic basic states and entropy. Also, the paper investigated dynamics nature of the Equations and bifurcation diagrams in it which present complexity and the surprising dynamic systems features. The development of the BifDraw tool exemplifies the practical application of theoretical concepts, facilitating further exploration and understanding of logistic Equations within chaos theory. This study not only deepens the comprehension of logistic Equations and chaos theory, but also introduces practical tools for visualizing and analyzing their behaviors. [ABSTRACT FROM AUTHOR]

A paradoxical nonlinear phenomenon called post-inhibitory facilitation (PIF) has been observed in the superior olive neuron for precise sound localization, which means that an inhibitory synaptic current appearing earlier than an excitatory one does not suppress but facilitates a spike for a short interval between the two currents. Although the PIF is reproduced for type III excitability/non-bifurcation and mediated by a fast low-threshold potassium current (IKLT), contradictorily, explained with a negative threshold related to type II excitability/Hopf bifurcation, even if in the absence of the IKLT. In the present paper, explanations to the seemingly contradictory nonlinear and ionic current mechanisms for the PIF are presented in a modified model containing the IKLT. Firstly, the modified model exhibits Hopf bifurcation/type II for weak or zero IKLT, in addition to non-bifurcation/type III for strong IKLT which resembles the original models. Secondly, the PIF can be evoked for types III and II with a positive threshold, which presents novel conditions for the PIF. Thirdly, with the excitatory current considered, some traditional positive thresholds become novel negative thresholds, which presents successful explanations to the novel PIF modulated by the IKLT and the inhibitory and excitatory synapses. The novel PIF appears for a strong and relatively slow inhibitory synaptic current which can induce membrane potential reduced to run across the negative threshold, while not for weak or fast inhibitory current or not for novel positive thresholds related to fast excitatory synapses. Finally, the PIF with short time interval appears for type III with fast dynamics, which is mainly mediated by the fast IKLT, resembling the experiment. The PIF with wide interval appears for type II with slow dynamics, which is mediated by the weak IKLT and/or the slow leak current (IL), presenting a novel current to mediate the PIF. The bifurcation/excitability, novel threshold surface, IKLT and IL, and inhibitory and excitatory synapses present a comprehensive viewpoint on the PIF and the potential modulations to obtain the PIF with short interval for the precise sound localization. [ABSTRACT FROM AUTHOR]

This paper addresses the challenging issues of transmission quality degradation and difficulty in obtaining dynamic response characteristics caused by the nonlinear behavior of non-circular planetary gear systems (NPG). A dynamic model for NPG was developed, encompassing axial elastic displacement, backlash, tooth surface friction, time-varying meshing stiffness, and viscoelastic damping. Fourier fitting matrices for time-varying mesh stiffness and polynomial models for dynamic backlash in non-circular gears were acquired to enhance model precision. Various analysis techniques including phase trajectory diagrams, bifurcation diagrams, time history diagrams, Poincaré mapping diagrams, and phase amplitude frequency characteristic curves were used to evaluate the nonlinear behavior of NPGs. Research results indicate that increasing the damping ratio benefits frequency response bandwidth, reduces phase lag, and improves system stability. The friction coefficient on the surface of non-circular gears also plays a role in ensuring the stability and phase consistency of NGP, although excessive coefficients can induce chaos. The output solar gear is more sensitive to internal excitation, with higher internal excitation leading to stronger system chaos. [ABSTRACT FROM AUTHOR]

In the study of continuous amensalism systems, it has been widely accepted that Michaelis–Menten type harvesting has a significant impact on the survival and extinction of species. However, scholars have not yet studied discrete amenslism models that include Michaelis–Menten type harvesting. To characterize such dynamics, a discrete amensalism system with Michaelis–Menten type harvesting for the second species is investigated. Firstly, we study the existence and stability of all possible equilibrium points. Under different parameters, there are two stable equilibria, which means that the model is not always globally stable. Then, the conditions of various types of bifurcations likely: pitchfork bifurcations, transcritical bifurcations, fold bifurcations, and flip bifurcations have been established. In addition, a global dynamics analysis of the model is also conducted. Finally, the significance of Michaelis–Menten type harvesting in species relationships is shown by numerical simulations. Although proper harvesting reduces the density of the second species, it favors the stable coexistence of both species and excessive harvesting leads directly to the extinction of the second species. Therefore, the results of this paper can provide a reference for research on how to maximize harvesting without destroying the ecological balance of the species. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a class of discrete predator–prey model with alternative prey. We prove the boundedness of the solution, the existence and local/global stability of equilibrium points of the model, and verify the existence of flip bifurcation and Neimark-Sacker bifurcation. In addition, we use the maximum Lyapunov exponent and isoperimetric diagrams to verify the existence of periodic structures namely Arnold tongue and the shrimp-shaped structures in bi-parameter spaces of a class of predator–prey model. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the free boundary problem in three dimensions describing the growth of tumor cords. The model consists of a reaction-diffusion equation describing the concentration $ \sigma $ of nutrients and an elliptic equation describing the distribution of the internal pressure $ p $. The model is defined in a bounded domain in $ \mathbb{R}^{3} $ whose boundary consists of two disjoint closed curves, the known interior part and the unknown exterior part. The concentration of nutrients outside the tumor region is denoted by $ \bar{\sigma} $. We shall show that there is a positive integer $ n^{**} $ and a sequence $ \bar{\sigma}_{n} $ such that for each $ \bar{\sigma}_{n}(n>n^{**}) $, symmetry-breaking stationary solutions bifurcate from the annular stationary solution in the longitude direction. [ABSTRACT FROM AUTHOR]

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations. [ABSTRACT FROM AUTHOR]

Rotor systems are widely used in industrial power generation and propulsion. Once the nonlinear contact stiffness and oil film force are taken into account, the dynamics and stability of rotor systems become quite complex, often accompanied by super-harmonic and chaotic motions. Furthermore, conventional methods face limitations in real-time detection of the bifurcations and complex nonlinear motions. This research investigates the bifurcations and stability induced by nonlinear factors in a rotor-bearing system from an energy perspective. Dynamic equations of a rotor-bearing system considering cubic term stiffness are established, the steady responses are obtained by the fourth-order Runge–Kutta method. The relationship between the bifurcations and energy transfers is analyzed numerically, the proposed stability criterion is validated by comparing the Lyapunov exponents. The bistable phenomenon of period-3 m (m = 0 , 1) motion is discussed in terms of numerical results and experiments which corresponds to the asymmetric jumps of the generalized energy. It is found that bifurcations and unstable motions of the nonlinear system can be captured accurately by detecting the energy transfers, the proposed generalized energy curve exhibits more detailed information than the conventional speed-up curve. These findings provide a new perspective on studying bifurcations and stability of rotating systems, which can be further applied in the condition monitoring, stability prediction as well as the design of nonlinear energy sink, representing significant progress in converting theory into engineering applications for rotor systems. [ABSTRACT FROM AUTHOR]

In this paper, a single-species logistic model with both fear effect-type feedback control and additive Allee effect is developed and investigated using the new coronavirus as a feedback control variable. When the system introduces additive Allee effect and fear effect-type feedback control, more complicated dynamical behavior is obtained. The system can undergo transcritical bifurcation, saddle-node bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation. By numerical simulations, the system exhibits homoclinic bifurcation and saddle-node bifurcation of limit cycles as parameters are altered. Remarkably, it is the first time that two limit cycles have been discovered in a single-species logistic model with the Allee effect. Further, stronger Allee effect or stronger fear effect can lead to the extinction of the species population. [ABSTRACT FROM AUTHOR]

This study delves into the often‐overlooked impact of immature mosquitoes on the dynamics of malaria transmission. By employing a mathematical model, we explore how these aquatic stages of the vector shape the spread of the disease. Our analytical findings are corroborated through numerical simulations conducted using the Runge–Kutta fourth‐order method in MATLAB. Our research highlights a critical factor in malaria epidemiology: the basic reproduction number R0. We demonstrate that when R0 is below unity R0<1, the disease‐free equilibrium exhibits local asymptotic stability. Conversely, when R0 surpasses unity R0>1, the disease‐free equilibrium becomes unstable, potentially resulting in sustained malaria transmission. Furthermore, our analysis covers equilibrium points, stability assessments, bifurcation phenomena, and sensitivity analyses. These insights shed light on essential aspects of malaria control strategies, offering valuable guidance for effective intervention measures. [ABSTRACT FROM AUTHOR]