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A reaction-diffusion predator-prey model with the dormancy of predators is considered in this paper. We are concerned with the long-time behaviors of the solutions of this system. We divided our investigations into two cases: for the ODEs system, we study the existence and stability of the equilibrium solutions and derive precise conditions on system parameters so that the system can undergo Hopf bifurcations around the positive equilibrium solution. Moreover, the properties of Hopf bifurcation are studied in detail. For the reaction-diffusion system, we are able to derive conditions on the diffusion coefficients so that the spatially homogeneous Hopf bifurcating periodic solutions can undergo diffusion-triggered instability. To support our theoretical analysis, we also include several numerical results.

This paper is devoted to a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The biomass specific growth rate is presented as sum kinetics with interaction parameters (SKIP). A discrete time delay is introduced and incorporated into the biomass growth response. These two aspects—the mutual influence of the two substrates and the natural biological time delay in the biomass growth rate—are new in the scientific literature concerning bioreactor (chemostat) models. The equilibrium points of the model are determined and their local asymptotic stability as well as the occurrence of local Hopf bifurcations are studied in dependence on the delay parameter. The existence and uniqueness of positive solutions are established, and the global stabilizability of the model dynamics is proved for certain values of the delay. Numerical simulations illustrate the global behavior of the model solutions as well as the transient oscillations as a result of the Hopf bifurcation. The performed theoretical analysis and computer simulations can be successfully used to better understand the biodegradation dynamics of the chemical compounds in the bioreactor and to predict and control the system behavior in real life conditions.

There are significant concerns regarding the stability of increased wind power generation in weak power grids. This paper investigates and improves the stability of Wind Turbine Squirrel Cage Induction Generators (WT-SCIGs) with series compensation and weak interconnections to the power grid. Detailed time-domain and state-space modeling have revealed new bifurcations and oscillatory modes for a WT-SCIG connected radially to a weak grid through a series compensated line. The stability domain analyses are carried out by computing bifurcations in the system by analyzing eigenvalues of the linearized system. The analyses demonstrate for the first time how the degree of compensation at which the Hopf bifurcation occurs depends on the X/R ratio of the line, operating slip of the induction generator, and voltage regulator parameters as well as the time delays in measurements. A new damping controller is proposed, which greatly improves the dynamic stability of the WT-SCIG and eliminates destructive Hopf bifurcations in weak grids for a wide range of series compensation. This allows for a much larger percentage of series compensation than what is usually possible, while avoiding instabilities, thereby maximizing the power transfer capability.

This paper discusses how a semi-analytical approach can be used alongside the reversible Schnakenberg model in a reaction-diffusion cell. It is worth noting that semi-analytical techniques are aligned to the Galerkin method, which offers an approximation of the governing partial differential equations through a system of ordinary differential equations. The bifurcation diagrams, steady-state curves and regions of the parameter space where bifurcations take place are part of the semi-analytical solutions and the numerical solution.

In this paper, five different models for five different kinds of diseases occurring in wildlife populations are introduced. In all models, a logistic growth term is taken into account and the disease is transmitted to the susceptible population indirectly through an environment reservoir. The time evolution of these diseases is described together with its spatial propagation. The character of spatial homogeneous equilibria against the uniform and non-uniform perturbations together with the occurrence of Hopf bifurcations are discussed through a linear stability analysis. No Turing instability is observed. The partial differential field equations are also integrated numerically to validate the stability results herein obtained and to extract additional information on the temporal and spatial behavior of the different diseases.

In this paper, we investigate the dynamics of a diffusive Gompertz population model with nonlocal delay effect and Dirichlet boundary condition. The stability of the positive spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcations with the change of the time delay are discussed by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. Then we derive the stability and bifurcation direction of Hopf bifurcating periodic orbits by using the normal form theory and the center manifold reduction. Finally, we give some numerical simulations.

In this paper, we analyze a mathematical model for an inflammatory response to bacterial infection of homogeneous tissues. Specifically, we provide a detailed analysis of the Lauffenburger-Kennedy bacterial infection model and show that the model exhibits three possible equilibria corresponding to a bacteria-free and two endemic compromised steady states. Asymptotic results of the steady states along with the existences of saddle-node connection Hopf bifurcations are shown under certain conditions of the parameters. Within the biological ranges of the parameter values, we observe that the system can exhibit both forward and backward bifurcation. In addition, in both cases, the larger compromise bacterial infection steady state can either approach an equilibrium or can oscillate around it via Hopf bifurcation depending on the value of the ratio of leukocyte mortality to phagocytosis rates. Numerical results are used to provide illustrative examples of these different dynamical patterns observed in the model.

In this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, which can be applied to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, are obtained. Numerical simulations are performed to illustrate the theoretical analysis results.

We consider a nonlinear model of a continuously stirred bioreactor and study the stability of the equilibrium points with respect to practically important model parameters. We determine regions in the parameter space where the steady states undergo transcritical and Hopf bifurcations. In the latter case, the stability of the emerged limit cycles is also studied. Numerical simulations in the computer algebra system Maple are presented to illustrate the theoretical results., This paper is partially supported by the Bulgarian Science Fund under grant Nr. DO 02– 359/2008.

In this paper, we consider a regulated logistic growth model. We first consider the linear stability and the existence of a Hopf bifurcation. We show that Hopf bifurcations occur as the delay τ passes through critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit algorithm determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, numerical simulation results are given to support the theoretical predictions.

In this paper, we consider a model described the survival of red blood cells in animal. Its dynamics are studied in terms of local and global Hopf bifurcations. We show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay crosses some critical values. Using the reduced system on the center manifold, we also obtain that the periodic orbits bifurcating from the positive equilibrium are stable in the center manifold, and all Hopf bifurcations are supercritical. Further, particular attention is focused on the continuation of local Hopf bifurcation. We show that global Hopf bifurcations exist after the second critical value of time delay.

The paper explores the effects of markup variability, associated with free entry of Cournotian firms, on the occurrence of local indeterminacy, local Hopf bifurcations and multiplicity of steady states. It uses an overlapping generations model with capital, in which differentiated intermediate goods are produced with constant marginal cost plus a positive fixed cost. If the markup factor were constant (the case of no free entry or the case of zero fixed costs leading to the perfectly competitive outcome), no local endogenous fluctuations or multiplicity of steady states can emerge, unless factors of production are complementary. On the contrary, with any positive (variable) market power, indeterminacy and multiplicity of steady states is possible for any value of the elasticity of technological substitution, and local Hopf bifurcations may also emerge, provided (variable) market power is high enough.

This paper offers a unified perspective of the analytical detection of Hopf bifurcation, which is a crucial tool in dynamic economic modelling. We clarify the relations between stability theorems and the notions of simple and general Hopf bifurcations. A Lienard–Chipart-type theorem for detecting bifurcations, which appears of considerable usefulness in applications, is proved. Subsequently we show how to use the notions of ‘stability boundary’ and ‘bifurcation boundary’, providing a new, surprisingly straightforward, tool for detecting bifurcations in economics. An economic illustration is given by two models with time-delay: a Solow-type demo-economic model and a Kaleckian extension of the Lotka–Volterra–Goodwin model.

This paper pursues the study carried out by the authors in {\it Stability and Hopf bifurcation in the Watt governor system} \cite{smb}, focusing on the codimension one Hopf bifurcations in the centrifugal Watt governor differential system, as presented in Pontryagin's book {\it Ordinary Differential Equations}, \cite{pon}. Here are studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, illustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found a region in the space of parameters where an attracting periodic orbit coexists with an attracting equilibrium., 34 pages, 9 figures

In this paper, bifurcations of limit cycles close to certain singularities of the vector fields are explored using an algorithm based on the harmonic balance method, the theory of nonlinear feedback systems and the monodromy matrix. Period-doubling, pitchfork and Neimark–Sacker bifurcations of cycles are detected close to a Gavrilov–Guckenheimer singularity in two modified Rössler systems. This special singularity has a zero eigenvalue and a pair of pure imaginary eigenvalues in the linearization of the flow around its equilibrium. The presented results suggest that the proposed technique can be promising in analyzing limit cycle bifurcations arising in the unfoldings of other complex singularities. Fil: Robbio, Federico I.. Universidad Nacional del Sur. Departamento de Ingeniería Eléctrica y de Computadoras; Argentina Fil: Alonso, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Ingeniería Eléctrica y de Computadoras; Argentina Fil: Moiola, Jorge Luis. Universidad Nacional del Sur. Departamento de Ingeniería Eléctrica y de Computadoras; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina

The objective of this paper is to present an analytical/numerical analysis of the phenomenon of limit-cycle stability reversal (from unstable to stable, and vice versa ). A singular perturbation technique, the method of the normal form (in the asymptotic- expansion version), is utilized. The number of equations is then reduced to a “minimal set”, for which the results are in good agreement with those from the original equations. This minimal set is determined by the amplitude of the λ-points (a concept closely related to the small divisors in the KAM theory). This set is larger than that corresponding to the zero real-part eigenvalues (center-manifold theorem). The method is applied to a specific problem: an aeroelastic section with cubic free-play non-linearities where the parameter μ is the flight speed. Numerical studies have been performed to show the dependence of the Hopf bifurcation characteristics upon the structural and geometric properties of the wing section. Plots depicting amplitudes and frequency versus flight speed are presented.

This paper describes a full micromagnetic characterization of the magnetization dynamics driven by spin-polarized current in anisotropic spin-torque oscillators (STOs). For field angles approaching the hard in-plane axis, the excited mode is uniform and a super-critical Hopf-bifurcation takes place at the critical current density JC. For field angles close to the easy axis of the free layer, the excited mode is localized (non-uniform) and a sub-critical Hopf-bifurcation occurs at JC. In this latter region, a hysteretic behaviour is, therefore, found. We demonstrate numerically that the non-linearities of the STO are strongly reduced when the oscillation frequency at the critical current is near the ferromagnetic resonance (FMR) frequency computed at zero bias current, and in particular, this condition corresponds to the field orientation at which a minimum in the FMR-frequency is achieved.