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With the wide applications of social networks, government and individuals increasingly emphasize information networks security. This paper is devoted to investigating a reaction–diffusion malware propagation model with mixed delays to describe the process of social networks. Applying matrix theory for characteristic values, we establish the local stability conditions of a positive equilibrium point. Based on the linear approximation method of nonlinear systems, the Hopf bifurcation at the positive equilibrium point is considered. Additionally, we identify some sensitive parameters in the process of malware propagation that are significant for control theory. Finally, numerical simulations are performed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a novel computer virus propagation model with dual delays and multi-state antivirus measures is considered. Using theories of stability and bifurcation, it is proven that there exists a critical value of delay for the stability of virus prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the explicit formulas determining the stability and direction of bifurcating periodic solutions are obtained by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to verify the theoretical analysis. The conclusions of this paper can contribute to a better theoretical basis for understanding the long-term actions of virus propagation in networks. [Copyright &y& Elsevier]

Abstract: The paper presents a detailed study of a four component population dynamical model consisting of the biomasses of carbon of phytoplankton , nitrogen of phytoplankton , carbon of zooplankton and total free nutrient . In this model, both the influence of the internal (carbon) and external (nitrogen) nutrient on the growth of phytoplankton is included. This is usually termed a variable yield model in contrast to the constant yield model analyzed by J. Monod [Recherches sur la croissance des cultures bacteriennes, Herman, Paris, 1942. ]. The existence and stability of various steady states are discussed analytically and numerically using a computer simulation (AUTO) designed by Doedel [E.J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer. 30 (1981) 265–284; E. Doedel, X. Wang, T. Fairgrieve, AUTO: Software for continuation and bifurcation problems in ordinary differential equations, in: Applied Mathematics Reports, California Institute of Technology, 1994]. Various bifurcation diagrams so obtained support the analytical results and identified complex behavior which cannot be obtained analytically. The purpose of this paper is as a preliminary step to determine the change in the richness of behavior as complexity, such as spatial compartments, is added, which in limiting form would be a continuous spatial dependence (as a partial differential equation model). [Copyright &y& Elsevier]

In the paper, a model of multicellular spheroids dynamics is studied. It can be also interpreted as avascular tumour growth. The model is based on the diffusion of nutrient and mass conservation for two processes cell proliferation and apoptosis. Two types of apoptosis is taken into account-underlying and regulatory apoptosis. It is assumed that the process of regulatory apoptosis is delayed compared to proliferation and underlying apoptosis.Nonnegativity of solutions, steady state stability, and existence of periodic solutions are studied in the paper. Computer simulations of solutions are presented. Some conclusions for special values of the model parameters are stated. [Copyright &y& Elsevier]

In the paper, a simple model of solid avascular tumour growth is studied. The model is based on the reaction-diffusion dynamics and mass conservation law and is considered with delay in cell proliferation process. In the case considered in this paper, the model reduces to one ordinary differential equation with time delay which describes the evolution of tumour radius.Basic properties of the model are studied in the paper. Steady-state analysis is presented with respect to the magnitude of delay. Existence of periodic solutions is proved for some parameter values. Global stability is proved for other ones. Results are illustrated by computer simulations. [Copyright &y& Elsevier]

In this paper, a higher-order accuracy method is proposed for the solution of time-dependent nature convection problems based on the stream function-vorticity form of Navier–Stokes equations, in which an optimized third-order upwind compact scheme (Opt-UCD3) with high resolution is proposed to approximate the nonlinear convective terms, the fourth-order symmetrical Padé compact scheme is utilized to discretize the viscous terms, the fourth-order compact scheme on the nine-point 2D stencil is used for approximating the stream-function Poisson-type equation and the third-order TVD Runge–Kutta method is employed for the time discretization. To assess numerical capability of the newly proposed algorithm, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Moreover, the nature convection flows in the square cavity with adiabatic horizontal walls and differentially heated vertical walls are also computed for the wide range of Rayleigh numbers ( 10 3 < Ra < 10 10 ). The characteristic parameters such as Nusselt number, velocity, and streamline show excellent agreement with benchmark solutions and some accurate results available in the literature. Additionally, the detailed features of flow phenomena for the higher Rayleigh numbers ( 10 8 < Ra < 10 10 ) are delineated. The results show that the natural convection flow looses stability firstly via a Hopf bifurcation at Ra c 1 to the periodic flow regime, and then undergoes second bifurcation at a critical Rayleigh number Ra c 2 to quasi-periodic flow regime, and eventually transits to turbulent through a further bifurcation. In the periodic regime, there exist at least two branches of solutions. All of the results are agree well with ones in the literature and show the capabilities of the present method to properly simulate the unsteady nature convection problems. [ABSTRACT FROM AUTHOR]

A predator–prey model is investigated in which the predator population is assumed to have an age structure. We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness of positive age-dependent equilibrium for the model. Bifurcation analysis indicates that the predator–prey system with age structure exhibits Hopf bifurcation which is the main result of this paper. Numerical simulations are given to illustrate the given results. [ABSTRACT FROM AUTHOR]

A kind of stage-structured predator–prey model with Beddington–DeAngelis functional response incorporating a prey refuge is investigated in this paper. By analyzing the corresponding characteristic equations, the local stability of the equilibria is investigated. Moreover, Hopf bifurcations occur at the positive equilibrium as the delay τ crosses some critical values. Further, the influence of prey refuge on densities of predator species and prey species is investigated. Numerical simulations are carried out to illustrate our main results. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with spatially homogeneous and inhomogeneous autocatalysis models with arbitrary order. For the spatially homogeneous model, the existence and stability of Hopf bifurcation surrounding the interior equilibrium are considered. For the model subject to zero-flux boundary conditions, Turing instability of the interior equilibrium is firstly discussed and Turing instability region regarding the diffusion coefficients is established. Then by the center manifold theory and normal form method, the existence of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained, which is much more difficult and complicated than dealing with the spatially homogeneous case. Finally, to verify our analytical results, some examples of numerical simulations are also included. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the effect of delayed feedbacks on the 3-D chaotic system only with two stable node-foci by Yang et al. The stability of equilibria and the existence of Hopf bifurcations are considered. The explicit formulas determining the direction, stability and period of the bifurcating periodic solutions are obtained by employing the normal form theory and the center manifold theorem. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodic orbit in the chaotic system with direct time delay feedback. We also find that the control law can be applied to the chaotic system only with two stable node-foci for the purpose of control and anti-control of chaos. Finally, some numerical simulations are given to illustrate the effectiveness of the results found. [Copyright &y& Elsevier]

Abstract: In this paper, a stage-structured predator–prey system with Holling type-III functional response and time delay due to the gestation of the predator is investigated. By analyzing the characteristic equations, the local stability of each of feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main theoretical results. [Copyright &y& Elsevier]

Abstract: Rotors supported by journal bearings may become unstable due to self-excited vibrations when a critical rotor speed is exceeded. Linearised analysis is usually used to determine the stability boundaries. Non-linear bifurcation theory or numerical integration is required to predict stable or unstable periodic oscillations close to the critical speed. In this paper, a dynamic model of a short journal bearing is used to analyse the bifurcation of the steady state equilibrium point of the journal centre. Numerical continuation is applied to determine stable or unstable limit cycles bifurcating from the equilibrium point at the critical speed. Under certain working conditions, limit cycles themselves are shown to disappear beyond a certain rotor speed and to exhibit a fold bifurcation giving birth to unstable limit cycles surrounding the stable supercritical limit cycles. Numerical integration of the system of equations is used to support the results obtained by numerical continuation. Numerical simulation permitted a partial validation of the analytical investigation. [Copyright &y& Elsevier]

Abstract: In this paper, considering full Logistic proliferation of CD4+ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results. [Copyright &y& Elsevier]

Abstract: Digital filters, frequently used in active control of mechanical systems, enable one to improve the signal-to-noise ratio and the control performance, but introduce group delays into the control loops simultaneously. In order to gain an insight into the effects of a digital filter on a controlled mechanical system, this paper presents the stability switches and the corresponding Hopf bifurcations of a double pendulum system with the linear quadratic control having a digital filter via theoretical analysis, numerical simulations and experiments. In this study, the digital filters are used to remove the undesired noise of high frequency, which is embedded in the control signal, and are modeled as the components of pure time delay during the theoretical analysis and numerical simulations. The study shows that a digital filter with moderate specifications can not only improve the vibration reduction effectively, but also save the energy consumption of the servo-motor remarkably. However, over demanding specifications will make the group delay of the filter exceed a critical value and cause either a divergent motion or a self-excited vibration through a Hopf bifurcation, the occurrence of which depends on both the stability and the size of the basin of attraction of the bifurcating periodic motion. The experimental results well coincide with the theoretical and numerical ones, and strongly support the simplification of the digital filters as the components of pure time delay. Finally, some suggestions are made to avoid the group delay induced instability. [Copyright &y& Elsevier]

In this paper, we investigate the spatial pattern formation of a predator–prey system with prey-dependent functional response Ivlev-type and reaction-diffusion. The Hopf bifurcation of the model is discussed, and the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained. Based on this, we perform the spiral and the chaotic spiral patterns via numerical simulation, i.e., the evolution process of the system with the initial conditions which was small amplitude random perturbation around the steady state. For the sake of learning the pattern formation of the model further, we perform three categories of unsymmetric initial condition, and find that with these special initial conditions the system can emerge not only spiral pattern but also target pattern and so on, and the effect of these special conditions on the formation of spatial patterns is less and less with more and more iterations but the effect does not decay forever. This indicates that for prey-dependent type predator–prey system, pattern formations do depend on the initial conditions, while for predator-dependent type they do not. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study a non-Kolmogrov type prey-predator system. First, we investigate the linear stability of the model by analyzing the associated characteristic equation of the linearized system. Second, we show that the system exhibits the Hopf bifurcation. The stability and direction of the Hopf bifurcation are determined by applying the norm form theory and center manifold theorem. Finally, numerical simulations are performed to illustrate the obtained results. [Copyright &y& Elsevier]