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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.

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.