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In this paper, a nonlinear phenomenological model of two coupled oscillators is presented, which is able to describe the rich nonlinear behaviour stemming from an unstable pure Intrinsic Thermo- Acoustic (ITA) mode of a simple combustion system. In a bifurcation analysis of a pure ITA mode, it was observed that for increasing mean upstream velocity the flames loose stability through a supercritical Hopf bifurcation and subsequently exhibit limit cycle, quasi-periodic and period-2 limit cycle oscillations. The quasi-periodic oscillations were characterized by low frequent amplitude and frequency modulation. It is shown that a phenomenological model consisting of two coupled oscillators is able to reproduce qualitatively all the different experimentally observed regimes. This model consists of a nonlinear Van der Pol oscillator and a linear mass-spring-damper oscillator, which are nonlinearly coupled to each other. Furthermore, a parameter identification of the model parameters is executed, which shows a good quantitative match between the model and the experimental data. Hence, the presented phenomenological dynamical model accurately describes the nonlinear self-excited acoustic behaviour of premixed flames. [ABSTRACT FROM AUTHOR]

The paper deals with a delayed version of prey predator model with the incorporation of modified Holling-Tanner functional response in predator population. The model highlights the loss in prey population via migration and the natural death rate due to age factor and other reasons such as prenatal death, infection etc. The model also considers the significance of delay in the predator population which is taken as a parameter of bifurcation. Local stability analysis is done in the paper and it is examined that delay is important in attaining system's stability. Hopf bifurcation is discussed in brief along with the stability of bifurcated type of periodic solutions stability by considering normal form and central manifold theories is applied. Numerical simulations asserts the theory and concludes the paper. [ABSTRACT FROM AUTHOR]

This paper concerns with the dynamic behavior of a prey-predator model with Holling type II functional responses and the environmental carrying capacity of prey. The carrying capacity of prey grows logistically with the parameter η being the maximum carrying capacity of prey. We assume that predators only eat prey, where predation follows a type II Holling functional response. The local stability analysis shows that there are three equilibrium points. One equilibrium point represents both species that do not persist always unstable, the other points prey persists but the predator cannot defend and the coexistence point are stable under some certain conditions. Several numerical results are presented to illustrate the analytical findings. The dynamic behavior of the system changes quantitatively at the bifurcation point when the parameter (η) is varied, a stable equilibrium becomes unstable and vice versa. The result of numerical continuity of the parameters of maximum prey carrying capacity causes changes in the instability of the system solution so that a Hopf bifurcation occurs at the value of η = 4.13068. [ABSTRACT FROM AUTHOR]

This paper reveals the stability analysis of the equilibrium points of a fractional-order predator-prey model involving Beddington-DeAngelis functional response and stage-structured on predator using an extension of the Lyapunov direct method for the Atangana-Baleanu derivative in the Caputo sense (ABC). Here, we use the ABC fractional derivative with the Mittag-Leffler kernel since it has the non-singular type of kernel which is better know to describe real-world problems mathematically. The Mittag-Leffler function helps to preserve and store model information over time. The proposed study considers linear, quadratic, and Volterra-type to construct the Lyapunov functions. Furthermore, to illustrate the validity of the results, some numerical simulations are carried out. The existence of Hopf bifurcation is also shown numerically which is driven by the order of the fractional derivative as the index of the memory. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an IS-LM macroeconomic model with delay-dependent coefficients. The model assumes that the tax revenues have two complementary components, one based on the current income and the other on a past income where the fixed discrete time delay arises. Moreover, we assume that the tax share on each component depends on the time delay, and thus the resulting model is a system of delay differential equations with delay-dependent coefficients. Previously, De Cesare and Sportelli (2005) studied a similar model but the tax shares that they used were fixed constants. They showed that the equilibrium point may undergo a sequence of alternated stability switches. This can be observed for example in the case with low tax rate and low share of delayed tax revenue where loci of Hopf bifurcation points form closed curves with some of these curves overlap, and the equilibrium is unstable inside these loci. Our proposed model introduces a new parameter μ in the tax share terms, and the case where μ = 0 captures the previously studied model with constant tax shares. We showed that increasing this new parameter results to decoupling and shrinking of the 'islands of instability', and eventually results to the absolute stability of the equilibrium. [ABSTRACT FROM AUTHOR]

This paper discussed the transmission dynamics of discrete SEIR model. It is possible to determine the prerequisites for the endemic and disease-free equilibria, as well as their local stability. In addition to demonstrating the validity of our findings, the numerical simulations also display more sophisticated dynamical characteristics such as the chaos phenomena, flip bifurcation, and Hopf bifurcation. These findings demonstrate that, as compared to models for both continuous and discrete epidemics exhibits far richer dynamical features. [ABSTRACT FROM AUTHOR]

In this paper, we study the mathematical model associated to an economic growth process with logistic population, CES utility function and delay. We have that the model is described by dynamical system of differential functions equations have the steady state. This steady state exhibits the Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical properties of a discrete time prey predator model under imprecise biological parameters. We consider refuge for prey and cost of fear into prey reproduction. The equilibria of the model are obtained and the dynamic behaviors of the proposed system are examined. Hopf Bifurcation of the proposed prey predator model are analysis. Simulations of the model are performed for different parameters of the model. Numerical simulations show that the proposed discrete model exhibits rich dynamics of a chaotic and complex nature. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the paper [1]. Some basic dynamical analysis will be done using the Routh-Hurwitz Criterion. Furthermore, this paper will determine the Hopf bifurcation of the three-dimensional modified Lorenz system by means of the first Lyapunov coefficient. The result shows that Hopf bifurcations occurred at two equilibrium points and are subcritical. [ABSTRACT FROM AUTHOR]

In this paper the semi-analytical solution is investigated for the delayed diffusive neural network model. Delay partial differential equations are approximated to the delay ordinary differential equation systems, by using Galerkin technique method. The main aim of this work is to determine the effect of diffusion and delay parameters and exploring the full map of stability analysis of the system. Stability analysis and bifurcation maps are discussed, as well, and in order to further explain certain concepts. The effect of diffusion parameter and delay values is comprehensively studied and as a result both variables can destabilize or stabilize the system. Illustrated examples of the unstable and stable limit cycles, and the Hopf bifurcation points are shown to prove the formerly revealed outcomes in the Hopf bifurcation map. In addition, comparisons between the numerical results and semi-analytical outcome provide comparison for all figures shown in the work. [ABSTRACT FROM AUTHOR]

In this paper, we work on predator-prey Model of fractional order. The system of the model in [1] is extending in the sense of Caputo fractional derivatives. More specifically, the study discussed the fractional predator-prey model that rely on biotic resources existence. The idea of Caputo derivative was employed in defining the fractional ordinary differential equations. The fractional models' stability analysis for the models equilibrium points were also presented. Adams-type predictor-corrector (ATPC) scheme is applied to compute an approximation to the solution of the model of fractional order. Furthermore, we investigated the Hopf bifurcation analysis. The result of the experiment show that, for certain values, the model undergo Hopf bifurcation, and further confirmed that the choice of an appropriate figure of the fractional α ∈ (0, 1] increase the region of the stability for the equilibrium points. [ABSTRACT FROM AUTHOR]

Natural convection is happening due to the temperature field variations in the river. Owing to the presence of various plants the river bottom becomes hot, the plume arises owing to the buoyancy effect and transports vertically. In this paper, transition to chaos in a V-shaped cavity is numerically studied by considering a large range of Rayleigh numbers (Ra = 100 – 106) with the Prandtl number, Pr = 7.0 and the aspect ratio, A = 0.5. A set of bifurcations can happened with the increase of Ra, where a Pitchfork bifurcation from symmetric to asymmetric state and a Hopf bifurcation from steady to unsteady state are included. The critical Rayleigh numbers for all the bifurcations have been confirmed by a great number of numerical tests.The spectral analysis with fundamental frequency has been investigated. [ABSTRACT FROM AUTHOR]

This paper discusses a prey-predator model with a Type II functional response and environmental suitability components. The stability of equilibrium points and the emergence of bifurcation are analyzed using techniques from dynamical systems. It is discovered that the transcritical and Hopf bifurcations appear as some ecologically- relevant parameters change. In particular, our two-parameter bifurcation analysis shows how the interaction of transcritical and Hopf bifurcations can lead to different dynamics of species interaction e.g. coexistence of species, species exclusion and oscillatory dynamics. Overall, our findings suggest significant roles of predation and abiotic environments in mediating species biodiversity. [ABSTRACT FROM AUTHOR]

Harvesting policy is an important issue in maintaining the existence of a population. This paper is focused on studying the effects of continuous predator threshold harvesting policy on the dynamical behavior of a fractional-order Gause-type predator-prey system. This policy is applied to ensure that harvesting does not occur when the population density is less than a specified threshold. The dynamical analysis is done to study the local stability of equilibrium points and the existence of Hopf bifurcation. By using a fractional-order predictor-corrector method, the numerical results are shown to illustrate the analytical result. [ABSTRACT FROM AUTHOR]

In the present paper, a predator-prey model with holling type IV functional response in the presence of weak Allee effect and non-linear harvesting in prey species has been proposed and analyzed. The local stability, existence of a Hopf bifurcation, direction of Hopf-bifurcation and the stability of the bifurcating periodic solution at the positive interior equilibrium point for the temporal model has been studied. The stability of positive constant equilibrium, Hopf-bifurcations, and diffusion-driven Turing instability for the spatio-temporal model under the Neumann boundary conditions has also been analyzed. Numerical simulations have been carried out to validate the analytical findings. [ABSTRACT FROM AUTHOR]

Species interactions such as predation have long been perceived as major factors shaping community compositions within ecology. In the main, the study of ecological dynamics involving prey-predator interactions with group defence has received significant attention in ecology due to the influential role that biotic interactions (i.e., interaction between species) played in the development of ecological theory. In this paper, we study a Lotka-Volterra two-species model consisting of the death rate of the predator and the search efficiency of predator for prey species. The existence and stability of steady states are determined to illustrate different possible outcomes of species interactions with group defence. We also examine the joint influences of the death rate of the predator and the search efficiency of predator for prey on community dynamics; in particular, the effects of varying these two ecologically-relevant parameters on species coexistence and the occurrence of limit cycle are explored. In addition, we also observe the emergence of local bifurcations in this model i.e., transcritical bifurcation and Hopf bifurcation, which has interesting implications on ecological dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and study a time-delayed SIS epidemic model with latency and nonlinear incidence rate, where the susceptible host population satisfies the logistic equation and the incidence rate is of saturated form with the susceptible. By regarding the time lag as bifurcation parameter, the local stability of the endemic equilibrium is investigated and sufficient conditions on the occurrence of stability switches through Hopf bifurcations are obtained. Further, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined by using the center manifold reduction and the normal form method. Numerical simulations are carried out to illustrate theoretical results. [ABSTRACT FROM AUTHOR]

Specific activator and repressor transcription factors which bind to specific regulator DNA sequences, play an important role in gene activity control. Interactions between genes coding such transcripion factors should explain the different stable or sometimes oscillatory gene activities characteristic for different tissues. In this paper, the dynamic P53-Mdm2 interaction model with distributed delays is investigated. Both weak and Dirac kernels are taken into consideration. For Dirac case, the Hopf bifurcation is investigated. Some numerical examples are finally given for justifying the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we consider a model with time delay, which describe tumor growth depending on vascular mass and regulation of new vessel formation through a key angiogenic factor. This model presents a Hopf bifurcation. Also, the stochastic approach is taken into consideration. [ABSTRACT FROM AUTHOR]

In this paper, we study an eco-epidemiology model which is derived from S I epidemic model with bilinear incidence rate and modified Leslie Gower predator-prey model with harvesting on susceptible prey. Existence condition and stability of all equilibrium points are discussed for the proposed model. Furthermore, we show that the model exhibits a Hopf bifurcation around interior equilibrium point which is driven by the rate of infection. Our numerical simulations using some different value of parameters confirm our analytical analysis. [ABSTRACT FROM AUTHOR]

In this paper we study the effect of prey infection on the modified Leslie-Gower predator-prey model with saturated incidence rate. The model will be analyzed dynamically to find the equilibria and their existence conditions as well as their local stability conditions. It is found that there are six type of equilibria, namely the extinction of both prey and predator point, the extinction of infective prey and predator point, the extinction of predator point, the extinction of prey point, the extinction of infective prey point and the interior point. The first four equilibrium points are always unstable, while the last two equilibria are conditionally stable. We also find that the system undergoes Hopf bifurcation around the interior point which is controlled by the rate of infection. To illustrate our analytical results, we show some numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider delayed three-species predator-prey model with non-monotonic functional response where two predator populations feed on a single prey population. Response function in both predator populations includes a time delay which represents the gestation period of the predator populations. We call a positive equlibrium solution of the form E * S = ( x * , y * , y * ) as a symmetric equilibrium. The goal of this paper is to determine the effect of the difference in gestation periods of predator populations to the local dynamics of symmetric equilibria. Our results include conditions on the existence of equilibrium solutions, and stability and bifurcations of symmetric equilibria as the gestation periods of predator populations are varied. A numerical bifurcation analysis tool is also used to illustrate our results. Stability switch occurs at a Hopf bifurcation. Moreover, a branch of stable periodic solutions, obtained using numerical continuation, emerges from the Hopf bifurcation. This shows that the predator population with longer gestation period oscillates higher than the predator population with shorter gestation period. [ABSTRACT FROM AUTHOR]

A high order unstructured spectral element method with a domain decomposition Stokes solver is presented for the hydrodynamic instability and bifurcation analysis. A Jacobian-Free Inexact-Newton-Krylov algorithm with a Stokes time-stepping preconditioning technique is introduced for the steady-state solution of incompressible flow. A matrix-free arc-length approach with Householder transformation is used for the numerical continuation near a turning point. An Arnoldi method is utilized to calculate the leading eigenvalues and their corresponding eigenvectors for the big system of linearized incompressible Navier-Stokes equations, which are responsible for initiating the hydrodynamic instability. The new method can do the steady and unsteady simulations in the similar way without time-splitting divergence error, it do not form the Jacobian matrix, which can reduce the memory allocation, decrease the computation cost, and speed up the convergence rate. The symmetric-breaking Hopf and Pitchfork bifurcations are considered in the flow passed a circular cylinder between two parallel plates. An antisymmetric sinusoidal velocity driven cavity problem is considered and the stable and unstable patterns are analyzed by checking the leading eigenvalues of their steady states. Besides the stable patterns of steady symmetric and steady asymmetric solutions£a new pair of unsteady asymmetric solutions are found depending on the different initial conditions. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of PIO (pilot induced oscillations) in a longitudinal flight. The phenomenon of PIO is the interaction between aircraft's motion and the pilot. PIO are present more often in terminal flight conditions. Using the qualitative theory of delay differential equations, a conventional time-delay model is considered and the stability of the equilibrium is analyzed. A limit cycle is shown to appear by a Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we will analyze a mathematical model associated to a cardiovascular process. This process is described in [1]. Mathematical modeling of this process leads to a system of differential equations with delay. We prove the existence of equilibrium point for this system and we show that the equilibrium point exhibits the Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper we introduce some methods on the problem of distinguishing between a center and a focus and the theory of Hopf bifurcation for polynomial systems on the plane, and pose some open problems on the number of limit cycles. [ABSTRACT FROM AUTHOR]

Motivated by the observation that Faraday waves can develop on the interfaces of the columnar structures that emerge after the frozen wave instability in microgravity, we undertake a theoretical investigation of interacting Faraday waves in alternating multi-layer systems without gravity. A linear stability analysis is used to study the primary subharmonic instability and extend the results of Labrador et al.1 to a general number of columns. For an even number n of columnar interfaces (an odd number of columns), Faraday waves appear via a (multiple) Hopf bifurcation, with n/2 distinct frequencies. For an odd number interfaces (an even number of columns), the primary bifurcation is a (multiple) Hopf-pitchfork with (n− 1)/2 distinct frequencies where the pitchfork (0 eigenvalue) is associated with a mode (eigenvector) with alternating excited and quiescent interfaces; the uncoupled dynamics in this subspace is equivalent to that of a single interface, but is likely unstable in the presence of nonlinearities. The theoretical results are complemented by numerical simulations for the relatively simple cases of n=2 and 4 and parameters characteristic of the immiscible liquids FC-40 and 20 cSt silicone oil. For the two-interface system, the simulations confirm a primary Hopf bifurcation that can either be subcritical or supercritical depending on the applied forcing frequency (detuning). In the subcritical case, this primary Hopf bifurcation is preceded by a saddle-node bifurcation that creates the associated unstable modulated solutions and another finite-amplitude branch of stable modulated solutions that is later observed to disappear in a global saddle-node heteroclinic bifurcation, giving way to purely subharmonic (unmodulated) Faraday waves. [ABSTRACT FROM AUTHOR]

Rumor is a social phenomenon in which the same comment spreads quickly on a large scale via a chain of com-munication. Rumors are an important part of social interaction, and they have a large impact on everyone's work and daily life. Misinformation can have a devastating effect on people, especially when they are confronted with a religious or political crisis. It is necessary to understand the dynamic characteristics of rumor spread in order to reduce and avoid the dangers of rumor spread in online social networks. The model is similar to a disease spread model. This occurrence merits further investigation or research. As a result, the researcher is curious about the rumor spreading model with delay time as a saturation control function. Analyzing the mathematical model of rumor spread's local stability of the equilibrium point and the Hopf Bifurcation, as well as numerical simulations of the model of rumor spread. [ABSTRACT FROM AUTHOR]

A predator-prey model with proportional prey harvesting and square root functional response in prey has been studied. The existence and the local stability of equilibria of the system has been analyzed. By analysing the model it is observed that the dynamics of the model depends on the parameter values. It is examined the solutions near the interior equilibrium point the system exhibits Hopf bifurcation. The calculation of the Lypunov number shows that the Hopf bifurcation is super-critical. Various numerical simulations are carried out to justify the analytical results. [ABSTRACT FROM AUTHOR]

Allee effect and harvesting are two important objects in the ecological system since they are directly connected to the existence of biological resources. Here, we study the impact of the strong Allee effect in prey and Michaelis-Menten type of harvesting in predators on the dynamics of a Gause-type predator-prey model. To involve the influence of the memory effect, the Caputo fractional-order derivative is applied. As a preliminary analysis, we obtain three types of equilibrium points namely the origin point, a pair of the predator extinction points, and the co-existence point. Some interesting dynamics are shown such as the local stability for each equilibrium point, the existence of transcritical bifurcation around the predator extinction points, and the occurrence of Hopf bifurcation around the co-existence point. Furthermore, some numerical simulations are performed to reinforce the theoretical findings. [ABSTRACT FROM AUTHOR]

In this article we deal with the Hutchinson's equation with distributed delay dx(t)/dt= rx(t)(1-a1x(t)-a2(t-τ)-a3∫t8734f(t-s)x(s)ds where r,τ, a1,a2,a3 are positive constants and f is the delay kernel function. By analyzing the associated characteristic equation, the local stability of positive equilibrium and Hopf bifurcation are investigated. The bifurcation here is controlled by the time delay. Some numerical simulations are performed to verify and illustrate the analytical findings. [ABSTRACT FROM AUTHOR]

The analysis of mutual competition of the growth of plant with particular emphasis on time reliant variations in their densities has been done in our present work. We are proposing a mathematical model of plant population growth with competitive and allelopathic impacts that are stimulating as well as inhibitory on each other. Equilibrium points are calculated and stability analysis is performed about non-zero equilibrium point. Delay parameter destabilizes the system when allelopathic effect is supposed to be of stimulatory in nature. The system shows asymptotic stability when delay parameter having value below the critical point. Hopf bifurcation is observed when the value of delay parameter crosses the critical point. The numerical results are substantiated using MATLAB. [ABSTRACT FROM AUTHOR]

Competition between species with asymmetrical interaction strength is thought to have a significant force in determining species biodiversity. Generally, competitive interaction has received a lot of attention among ecologists due to the important influences that interspecific competition played in the advancement of ecological theory and the development of population dynamics field. Mathematical modelling has been employed widely in the study of population dynamics and this tool can be used to better understand the influential roles of asymmetrical interaction in structuring community dynamics. To achieve this aim, we examine a multispecies Lotka-Volterra competition model consisting of asymmetrical interactions and environmental carrying capacity components. The significant influences of competitive interactions on multispecies community dynamics are investigated; in particular, we examine how the variation in competitive strength between multiple species leads to different outcomes of biotic interactions. To do this, we conduct a detailed bifurcation analysis of the model, as the strength of asymmetrical interaction varies, by means of dedicated continuation software. This allows us to discover regions of different types of attractors (e.g., limit cycles, alternative stable state communities, species coexistence and species exclusion) in the parameter space and to understand its bifurcation structures. Transcritical and Hopf bifurcations are observed in this ecological system and these bifurcations correspond to threshold values for asymmetrical interactions, which can lead to certain dynamics occur and disappear. Our bifurcation analysis also explains several previously published results, highlights parameter space with qualitatively different dynamics and uncovers some intriguing ecological phenomena concerning the interplay between asymmetrical competition and the maintenance of species biodiversity. [ABSTRACT FROM AUTHOR]

We capture the evolution in competition of healthy and leukemic cells in Chronic Myelogenous Leukemia (CML) taking into consideration the response of the immune system. Delay-differential equations in a Mackey-Glass approach are used. We start with the study of stability of the equilibrium points of the system. Conditions on parameters for the local stability are given. Oscillatory behaviors occur naturally in biological phenomena. Thus, we investigate the periodic behavior of solutions and we obtain conditions for periodic solutions to appear through a Hopf bifurcation. [ABSTRACT FROM AUTHOR]

We capture the evolution in competition of healthy and leukemic cells in Chronic Myelogenous Leukemia (CML) taking into consideration the response of the immune system. Delay-differential equations in a Mackey-Glass approach are used. We start with the study of stability of the equilibrium points of the system. Conditions on parameters for the local stability are given. Oscillatory behaviors occur naturally in biological phenomena. Thus, we investigate the periodic behavior of solutions and we obtain conditions for periodic solutions to appear through a Hopf bifurcation. [ABSTRACT FROM AUTHOR]

We consider a model of a tetrad of semiconductor lasers with unidirectional coupling. This system is described by Lang-Kobayashi (LK) rate equations, which is a system of delay differential equations with one fixed delay. Basic solutions to this system are called compound laser modes (CLMs). We classify the CLMs of this four-laser system according to their symmetry. The symmetric CLMs are identified by looking at the isotropy subgroups of the symmetry group of this system. Numerical continuation in DDE-Biftool generates a branch of symmetric CLMs. We then find steady-state and Hopf bifurcations along the branch of fully symmetric CLMs and identify those that are symmetrybreaking and symmetry-preserving. Finally, we use the Equivariant Branching Lemma and Equivariant Hopf Theorem to establish the existence of emanating branches of solutions from symmetry-breaking bifurcations. [ABSTRACT FROM AUTHOR]

In this article a multi-team delayed predator-prey model has been considered. There are two preys and one predator species in this model and the time delay appears for gestation of the predator. The essential mathematical features of the proposed model around the interior equilibrium point are studied in terms of local asymptotic stability by constructing a suitable Lyapunov functional and the condition for existence of Hopfbifurcation is derived. By the assumption that the prey teams may help each other the effect of the rate of cooperation on the stability of the predator-prey model has been observed. Numerically a critical value for the delay parameter is obtained as a condition for Hopf-bifurcation. [ABSTRACT FROM AUTHOR]

The nonlinear analysis of a forest-grassland ecosystem is performed as the main system parameters are changed. The model consists of a couple of nonlinear ordinary differential equations which include dynamically the human perceptions of forest/grassland value. The system displays multiple steady states corresponding to different forest densities as well as periodic regimes characterized by oscillations in time. We performed the bifurcation analysis of the system as the parameter relative to the human opinions influence is changed. We found that the main mechanisms which regulate the transitions occurring between different states or the appearance of new steady and dynamic regimes are transcritical, saddle/node and Hopf bifurcations. [ABSTRACT FROM AUTHOR]