Bifurcation Value Research Articles

Hopf bifurcation in delayed nutrient-microorganism model with network structure

In this paper, we introduce and deal with the delayed nutrient-microorganism model with a random network structure. By employing time delay τ as the main critical value of the Hopf bifurcation, we investigate the direction of the Hopf bifurcation of such a random network nutrient-microorganism model. Noticing that the results of the direction of the Hopf bifurcation in a random network model are rare, we thus try to use the method of multiple time scales (MTS) to derive amplitude equation and determine the direction of the Hopf bifurcation. It is showed that the delayed random network nutrient-microorganism model can exhibit a supercritical or subcritical Hopf bifurcation. Numerical experiments are performed to verify the validity of the theoretical analysis.

Stability and Bifurcation Analysis of a Fractional‐Order Food Chain Model with Two Time Delays

In this study, the stability and bifurcation problems of a fractional food chain system with two kinds of delays are studied. Firstly, the nonnegative, bounded, and unique properties of the solutions of the system are proved. The asymptotic stability of the equilibrium points of the system is discussed. Furthermore, the global asymptotic stability of the positive equilibrium point is deduced by using Lyapunov function method. Secondly, the system takes two kinds of time delays as bifurcation parameters and calculates the critical values of Hopf bifurcation accurately. The results show that Hopf bifurcation can advance with increasing fractional order and another delay. In conclusion, numerical simulation verifies and illustrates the theoretical results.

DYNAMICS OF A DELAYED PREDATOR-PREY MODEL WITH CONSTANT-YIELD PREY HARVESTING

<abstract> In this paper, we study a delayed predator-prey model of Holling and Leslie type with constant-yield prey harvesting, in which two types of delays caused by maturation time of prey and the gestation time of predator are considered. We mainly investigate the local dynamics of the model with emphasis on the impact of delays. The stability of equilibrium and the existence conditions of Hopf bifurcation are discussed. First, based on the different values of delays, five cases of Hopf bifurcation are studied in detail. The critical values of Hopf bifurcation for each case are presented. In addition, we explore the properties of Hopf bifurcation. The direction of Hopf bifurcation and the stability of periodic solutions by using the normal form theory and central manifold theorem are determined. The qualitative analyses have demonstrated that the values of time delays can affect the stability of equilibrium and induce small amplitude period oscillations of population densities. Numerical simulations are carried out for illustrating the theoretical results. Meanwhile, we further investigate the effects of delay on the period of periodic solutions and the influence of the harvesting term on the stability of the equilibrium with time delays. </abstract>

Stability of multi-hump localized solutions in the Holstein model for linear acoustic and soft nonlinear optical interactions

We consider the energy/electron transport problem along a lattice holding linear acoustic and soft cubic non-dispersive interactions in the long wave limit for Holstein’s approach. It is found that the model equations, consisting of the coupling between the linear Schrödinger and a nonlinear Klein–Gordon equations, support one- and two-hump solutions of the first and second kind. We derive these multi-humped solutions by using a compatible reduction of the equations of motion and by means of an adaptation of the direct method, both in the traveling frame. Remarkably, the solution profiles for the mechanical deformation recall the one and two soliton solutions of the Korteweg–de Vries equation. Then, based on the Fourier collocation method, we perform a numerical linear stability analysis of these solutions at the steady state to find the parameters regime for stability. Additionally, we are able to implement the Vakhitov–Kolokolov criterion to get the bifurcation value for the one-hump solution both of the first and second kind. We show that the two-hump solutions are linearly unstable.

The effect of the filling degree of the tumbling mill chamber on the bifurcation value of the froude number

The influence of the filling degree of the drum chamber on the bifurcation value of the Froude number in the case of self-excitation of self-oscillations of a two-fraction granular charge with a minimum swing is considered. Such a pulsating mode of the charge movement is used in the self-oscillating grinding process in a tumbling mill. The coarse fraction that simulated the milling bodies consisted of spherical particles of an incoherent granular material with a relative size in the chamber ψb=0.00733. The fine fraction, simulating the material to be ground, was cement with a relative particle size ψm≈0.092∙10-3. The value of the adopted analogue of the kinematic viscosity of the two-fraction granular loading approached the value of 10-3 m2/s. The main variable factor in experimental studies was the filling degree of the drum chamber with loading at rest κb=0.25, 0.35, and 0.45. An additional factor was the degree of filling the gaps between the spherical particles of the coarse fraction with particles of the fine fraction κb=0, 0.25, 0.5, and 1. The method of visual analysis through the transparent end wall of the chamber of transient processes of the loading behavior with a smooth change and fixation of the velocity was applied. The bifurcation minimum value of the rotation speed was recorded, at which the steady-state circulation mode of the load movement turns into a transient pulsation one with a minimum swing. A decrease in the bifurcation values of the Froude number Frb on the cylindrical surface of the chamber with an increase in κb has been established. An increase in the intensity of the decrease in Frb with an increase in κm was revealed. The recorded effect is due to an increase in the connected properties of a two-fraction loading during self-excitation of self-excited oscillations with an increase in κb and κm. The numerical values of the boundaries of the range of bifurcation values of the Froude number for a tumbling mill Frb=0.0484–1.17 have been determined. The obtained Frb range corresponds to the Reynolds value in the range Re=40–197. The maximum Frb value is obtained with coarse grinding. An increase in the likelihood of self-excitation of self-oscillations of the intra-chamber loading with a decrease in the fineness of grinding was revealed

Stagnation point flow of a micropolar fluid filled with hybrid nanoparticles by considering various base fluids and nanoparticle shape factors

PurposeThe boundary-layer analysis is required to reveal the fluid flow behavior in several industrial processes and enhance the products’ effectiveness. Therefore, this research aims to investigate the buoyancy or mixed convective stagnation-point flow (SPF) and heat transfer of a micropolar fluid filled with hybrid nanoparticles over a vertical plate. The nanoparticles silver (Ag) and titanium dioxide (TiO2) are scattered into various base fluids to form a new-fangled class of (Ag-TiO2/various base fluid) hybrid nanofluid along with different shape factors.Design/methodology/approachThe self-similarity transformations are used to reformulate the leading requisite partial differential equations into renovated non-linear dimensionless ordinary differential equations. The numerical dual solutions are gained for the transmuted requisite equations with the help of the bvp4c built-in package in MATLAB software. The results are validated by comparing them with previously available published data for a particular case of the present study.FindingsThe impact of various pertaining parameters such as nanoparticle volume fraction, material parameter, shape factor and mixed convective on temperature, heat transfer, fluid motion, micro-rotation and drag force are visualized and scrutinized through tables and graphs. It is observed that dual or non-uniqueness outcomes are found for the case of buoyancy assisting flow, whereas the solution is unique in the buoyancy opposing flow case. Additionally, the fluid motion and micro-rotation profiles decelerate in the presence of nanoparticle volume fraction, while the temperature augments.Originality/valueThe mixed convective stagnation point flow conveying TiO2/Ag hybrid nanofluid with micropolar fluid with various shape factors is the significant originality of the current investigation where multiple outcomes are obtained for the assisting flow. The various base fluids such as glycerin, water and water–ethylene glycol (50%:50%) are considered in the present problem. The bifurcation values of the considered problem do not exist, probably because of various base fluids. To the best of the authors’ knowledge, this work is new and original which were not previously reported.

Exotic Bifurcations in Three Connected Populations with Allee Effect

We consider three connected populations with strong Allee effect, and give a complete classification of the steady state structure of the system with respect to the Allee threshold and the dispersal rate, describing the bifurcations at each critical point where the number of steady states change. One may expect that by increasing the dispersal rate between the patches, the system would become more well-mixed, hence simpler. However, we show that it is not always the case, and the number of steady states may (temporarily) go up by increasing the dispersal rate. Besides sequences of pitchfork and saddle-node bifurcations, we find triple-transcritical bifurcations and also a sun-ray shaped bifurcation where twelve steady states meet at a single point then disappear. The major tool of our investigations is a novel algorithm that decomposes the parameter space with respect to the number of steady states and finds the bifurcation values using cylindrical algebraic decomposition with respect to the discriminant variety of the polynomial system.

A degenerate planar piecewise linear differential system with three zones

In (Euzébio et al., 2016 [10] ; Chen and Tang, 2020 [8] ), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F ( x ) − y , y ˙ = g ( x ) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincaré disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve.

Bifurcation of singularities of fluctuational paths for a noise-driven overdamped two-well system.

Noise-induced escape in a 2D generalized Maier-Stein model with two parameters μ and α is investigated in the weak noise limit. With the WKB approximation, the patterns of extreme paths and singularities are displayed. By employing the Freidlin-Wentzell action functional and the asymptotic series, critical parameters α inducing singularity bifurcation are determined analytically for μ=1. The switching line will appear with singularities and is equivalent to the sliding set in the Filippov system. The pseudo-saddle-node bifurcation on the switching line is found. Then, when -1<μ<1, it is found that all bifurcation values α will decrease as μ decreases and the second-order bifurcation values are bigger than all first-order ones. In addition, the variation of the switching line is also analyzed and a new switching line will emerge when the location of the minimum quasi-potential on the boundary changes. At last, when the noise is anisotropic, only the noise intensity ratio will affect the bifurcation value α.

Numerical and bifurcation analysis of SIQR model

In this mathematical research paper, we analyze in detail the basic SIQR epidemic model. We calculate its reproductive value R0, equilibrium points and analyze the stability of the SIQR system by using Routh-Hurwitz criterion in detail. We also find out the bifurcation value of the SIQR epidemic model by using Routh-Hurwitz criterion. Also, SIQR system is solved numerically by using four different mathematical techniques that are forward Euler scheme, Runge-Kutta (RK-4) method, variational iteration method and nonstandard finite difference scheme (NSFD). Analytical and graphical calculations show that the NSFD method preserves all the important conditions of the basic SIQR epidemic model while the rest three techniques fail to preserve the essential conditions of the system. Convergence analysis of the NSFD scheme has also been performed.

Analisis Bifurkasi Persamaan Longitudinal pada Gerak Roket Tiga Dimensi Tipe RKX-Lapan

This article discusses about longitudinal equation for the three dimensional rocket motion of RKX-Lapan. The RKX-Lapan’s rocket is guided rocket which was designed using two phases, boosting and sustaining. The rocket has six degrees of freedom movement. The freedom movements consists of three translational motions and three rotational motions which causess unstable rocket movement. Therefore, stable rocket motion system is needed. The rocket motion system is three dimensional of nonlinear equation, thus linearization process of rocket motion equation is requaired. One of this linearization process is longitudinal motion which consists of two translational motions and one rotational motion. The equation of longitudinal motion has stabilization system analysist on sustaining phase. Bifurcation analysis of longitudinal equation for the three dimensional rocket motion of RKX-Lapan has been created in this article. It determined equilibria and its equation stability character. Furthermore, bifurcation analysist on equilibria and numerical simulation have been done to find out that bifurcation value indicated a topologically nonequivalent.

Quantitative Geomorphological Parameters Analysis for the Aynalem- Illala Streams, Tigray, Northern Ethiopia

Morphometric analysis is the measurement and mathematical analysis of the configuration of the surface, shape, and dimension of landforms. The objective of this study is to characterize the Aynalem and Illala streams using the morphometric parameter. The topographic map at a scale of 1:50,000 taken from the Ethiopian National Mapping Agency was used to characterize the linear and areal aspects. ASTER Digital Elevation Model with 10m resolution was used to characterize the relief aspect. The Arc GIS 10.4.1 was used during the morphometric analysis. The analysis result of the streams is summarized based on the linear, areal, and relief aspects. The area is characterized by a dendritic drainage pattern which is characteristics of massive hard rock terrain. The Aynalem and Illala streams are 4th and 5th order streams. Considering the number of streams in the Aynalem (75.81%) and Illala (74.66%) is composed of first-order streams that indicate a flashy flood and the mean bifurcation value of Aynalem (6.8) and Illala (4.7) shows that the Aynalem area is more structurally affected than Illala but both show less stream integration. The analysis of areal aspects such as elongation ratio, circularity ratio, and form factor has indicated that both streams are characterized as elongated streams, this implies that both streams are flowing in heterogeneous rock material, presences of structural effect, and slow runoff discharge. The other areal aspect such as drainage density, stream frequency, infiltration number, and length of overland flow all show smaller values in both streams. This implies that the streams are characterized by a relatively permeable rock material with a higher infiltration capacity. The relief aspect of the Aynalem and Illala was also analyzed using basin relief, relief ratio, ruggedness number, hypsometric curves, and Hypsometric integral. The streams are characterized by a lower relief ratio and ruggedness number which implies a relatively flat slope and lower relief. The hypsometric curves and the Hypsometric Integral of the streams indicate that the Aynalem and Illala are at the maturity stage. This shows the area is characterized by higher erosion but less affected by recent structures. Based on the morphometric parameter analysis result it is possible to conclude that the stream development is dependent on the topography and geology of the study area and both streams show similar morphometric character.

Quantitative evaluation of drainage attributes to infer hydrologic and morphological characteristics of upper Beas Basin, Himachal Pradesh: A GIS-based approach

ABSTRACT Drainage morphometric analysis is a quantitative indexing of linear, areal, and relief attributes of a watershed, which is an essential prerequisite to understand the ongoing geomorphological, geological, and tectonic processes in a watershed that modifies its landscape. The present study is an investigation of the morphometric parameters of upper Beas basin. ALOS PALSAR digital elevation model available at 12.5 m resolution from Alaska Satellite Facility has been used for the study in Arc GIS 10.2 environment. The findings of the study categorize upper Beas as a sixth order drainage basin with a total area of 4958.98 km2. High bifurcation values infer an accelerated state of erosion and underline the mountainous origin and late youthful stage of river Beas and its tributaries. Geometrically, the basin has an elongated shape indicating lower peak flow but for a longer duration making flood management ambiguous. Upper Beas basin has a coarse drainage density and texture, suggesting high infiltration and less surface water availability. Tectonically, studyarea has some prominent faults and thrusts (MBT, MCT, and STD), making the region susceptible to water infiltration. These observations unquestionably indicate a strong tectonic and lithological control over the evolution of drainage patterns in the study area.

Dynamic analysis of 1-dof and 2-dof nonlinear energy sink with geometrically nonlinear damping and combined stiffness

Nonlinear energy sink (NES) refers to a typical passive vibration device connected to linear or weakly nonlinear structures for vibration absorption and mitigation. This study investigates the dynamics of 1-dof and 2-dof NES with nonlinear damping and combined stiffness connected to a linear oscillator. For the system of 1-dof NES, a truncation damping and failure frequency are revealed through bifurcation analysis using the complex variable averaging method. The frequency detuning interval for the existence of the strongly modulated response (SMR) is also reported. For the system of 2-dof NES, it is reported in a similar bifurcation analysis that the mass distribution between NES affects the maximum value of saddle-node bifurcation. To obtain the periodic solution of the 2-dof NES system with the consideration of frequency detuning, the incremental harmonic balance method (IHB) and Floquet theory are employed. The corresponding response regime is obtained by Poincare mapping, it shows that the responses of the linear oscillator and 2-dof NES are not always consistent, and 2-dof NES can generate extra SMR than 1-dof NES. Finally, the vibration suppression effect of the proposed NES with nonlinear damping, and combined stiffness is analyzed and verified by the energy spectrum, and it also shows that the 2-dof NES system demonstrates better performance.

Hopf bifurcation in a generalized Logistic reaction–diffusion population model with instantaneous and delayed feedback

This article considers a generalized Logistic reaction–diffusion population model with mixed instantaneous and delayed feedback control and subject to the homogeneous Dirichlet boundary condition on the one-dimensional bounded spatial domain (0,π). Existence and asymptotic expression of the small enough positive steady state bifurcating from the zero solution are given in virtue of the implicit function theorem. By choosing the delay τ as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation of the small enough positive steady state are obtained. Particularly, the normal form for Hopf bifurcations is derived according to the normal form method and the center manifold theory for partial functional differential equations (PFDEs). It is demonstrated that when the delayed feedback is dominant, the model can undergo a forward Hopf bifurcation at the positive steady state as the delay crosses through monotonically increasing a sequence of critical values and all the bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. Specifically, the bifurcating periodic solutions from the first bifurcation value are orbitally asymptotically stable in the entire phase space. The reasonability of the main analytical conclusions is also verified by carrying out some numerical simulations.

Mathematical modeling of tumor surface growth with necrotic kernels

A two‐dimensional tumor‐immune model with the time delay of the adaptive immune response is considered in this paper. The model is designed to account for the interaction between cytotoxic T lymphocytes (CTLs) and cancer cells on the surface of a solid tumor. The model considers the surface growth as a major growth pattern of solid tumors in order to describe the existence of necrotic kernels. The qualitative analysis shows that the immune‐free equilibrium is unstable, and the behavior of positive equilibrium is closely related to the ratio of the immune killing rate to tumor volume growth rate. The positive equilibrium is locally asymptotically stable when the ratio is smaller than a critical value. Otherwise, the occurrence of the delay‐driven Hopf bifurcation at the positive equilibrium is proved. Applying the center manifold reduction and normal form method, we obtain explicit formulas to determine the properties of the Hopf bifurcation. The global continuation of a local Hopf bifurcation is investigated based on the coincidence degree theory. The results reveal that the time of the adaptive immune system taken to response to tumors can lead to oscillation dynamics. We also carry out detailed numerical analysis for parameters and numerical simulations to illustrate our qualitative analysis. Numerically, we find that shorter immune response time can lead to longer patient survival time, and the period and amplitude of a stable periodic solution increase with the increasing immune response time. When CTLs recruitment rate and death rate vary, we show how the ratio of the immune killing rate to tumor volume growth rate and the first bifurcation value change numerically, which yields further insights to the tumor‐immune dynamics.

Hopf bifurcation of a delayed reaction–diffusion model with advection term

This article focus on the Hopf bifurcation of a delayed reaction–diffusion equation with advection term subject to Dirichlet boundary and no-flux boundary conditions in a bounded domain, respectively. It is shown that the existence of spatially non-homogeneous steady-state solutions will be obtained when the parameter λ of the model (9) closes to the principle eigenvalue λ1 of the elliptic operator Lλ. Moreover, a supercritical Hopf bifurcation occurs near the non-homogeneous positive steady-state at a series critical time delay values. Finally, we elucidate the effect of advection on Hopf bifurcation values. It is worth noting that the advective effect has accelerated the generation of Hopf bifurcation to a certain extent.

Drainage Morphometric Analysis of Shope watershed, Rift Valley, Ethiopia: Remote sensing and GIS-based approach

The quantitative drainage morphometric analysis at the watershed level is essential for understanding its geo-hydrological nature. Drainage morphometric evaluation has been conducted to identify the predominant geological divergence, topographical details, structural arrangement, and interrelationship in the Shope watershed Rift Valley, Ethiopia. The Remote Sensing Data and Geographical Information System (GIS) tools have delineated the Shope Watershed and calculated various morphometric attributes. The Shope watershed exhibits a dendritic drainage pattern very clearly. The watershed’s drainage pattern also suggested that the region lacked geological structural control over the drainage pattern growth. Shope watershed is categorized as a 5th order drainage watershed. The bifurcation ratio ranges between 1.11 and 2.28 (with an average of 1.72) between various stream orders. Low bifurcation values area is more prone to flooding. Mean drainage length and length ratio values indicating the present study area’s undulating topography and steep slope. The Shope watershed’s circularity and elongation ratio values are 0.7 and 0.63. Both values indicate that the present watershed is elongated and has steep slopes, and more prone to erosion and flooding. The higher drainage density values 6.75 km/km2 indicating low infiltration and high runoff. The Shope watershed’s relative relief value is 1757 m, and this high value indicating the study area’s proneness to the flooding and soil erosion. The results of the current drainage morphometric analysis of the Shope watershed are essential for assessing and managing soil and water resources and selecting suitable sites for artificial recharge structures and soil conservation structures for future soil and water management in the present study area.

Mathematical analysis of harvested predator-prey system with prey refuge and intraspecific competition

In this paper, a predator-prey relationship in the presence of prey refuge was studied. The analysis of the dependence of locally stable equilibrium points on the parameters of the problem was carried out. Bifurcation and limit cycles for the model were analyzed to show the dynamical behaviour of the system. The results showed that the system is stable at a constant prey refuge m = 0.3 and prey harvesting rate H = 0.3. However, increasing m and decreasing H or vice versa, the predator-prey system remains stable. It was further observed that for a constant prey refuge m ≥ 0.78, the predator population undergoes extinction. Therefore, m was found to be a bifurcation parameter and m = 0.78 is a bifurcation value.&#x0D; Keywords: Prey refuge, bifurcation, harvesting, intraspecific competition, phase portrait

Collective Oscillations in Coupled-Cell Systems

We investigate oscillations in coupled systems. The methodology is based on the Hopf bifurcation theorem and a condition extended from the Routh–Hurwitz criterion. Such a condition leads to locating the bifurcation values of the parameters. With such an approach, we analyze a single-cell system modeling the minimal genetic negative feedback loop and the coupled-cell system composed by these single-cell systems. We study the oscillatory properties for these systems and compare these properties between the model with Hill-type repression and the one with protein-sequestration-based repression. As the parameters move from the Hopf bifurcation value for single cells to the one for coupled cells, we compute the eigenvalues of the linearized systems to obtain the magnitude of the collective frequency when the periodic solution of the coupled-cell system is generated. Extending from this information on the parameter values, we further compute and compare the collective frequency for the coupled-cell system and the average frequency of the decoupled individual cells. To compare these scenarios with other biological oscillators, we perform parallel analysis and computations on a segmentation clock model.