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In this paper, the Bogdanov-Takens (B-T) and triple zero bifurcations are investigated for coupled van der Pol-Duffing oscillators with symmetry, in the presence of time delays due to the intrinsic response and coupling. Different from previous works, third order unfolding normal forms associated with B-T and triple zero bifurcations are needed, which are obtained by using the normal form theory of delay differential equations. Numerical simulations are also presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper the Bogdanov-Takens (BT) bifurcation of an 2 m coupled neurons network model with multiple delays is studied, where one neuron is excitatory and the next is inhibitory. When the origin of the model has a double zero eigenvalue, by using center manifold reduction of delay differential equations (DDEs), the second-order and third-order universal unfoldings of the normal forms are deduced, respectively. Some bifurcation diagrams and numerical simulations are presented to verify our main results. [ABSTRACT FROM AUTHOR]

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003; Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie-Gower functional response as well as that of Beddington-DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E* and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E*. At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values. [ABSTRACT FROM AUTHOR]

Abstract: This paper continues the work contained in two previous papers of the authors, devoted to the qualitative study of the dynamical system generated by a delay differential equation that models leukemia. The problem depends on five parameters and has two equilibria. As already known, at the non-zero equilibrium solution, for certain values of the parameters, Hopf bifurcation occurs. Our aim here is to investigate the Bautin bifurcation for the considered model. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We explore the space of parameters in a zone of biological interest and find, by direct computation, points where the first Lyapunov coefficient equals zero. These are candidates for Bautin type bifurcation. For these points we compute the second Lyapunov coefficient, that determines the type of Bautin bifurcation. The computation of the second Lyapunov coefficient requires a fourth order approximation of the center manifold, that we determine. The points with null first Lyapunov coefficient obtained are given in tables and are also plotted on surfaces of Hopf bifurcation obtained by fixing two parameters. Next we vary two parameters around a point in the space of parameters with and numerically explore the behavior of the solution. The results confirm the Bautin type-bifurcation. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a model for gene expression, unlike the models mathematically analyzed previously we have both transcriptional and translational time delays. The stability and Hopf bifurcation of the equilibrium point are investigated. Different to previous papers, a multiple time scale (MTS) technique is employed to calculate the normal form on the center manifold of system of delay differential equations, which is much easier to implement in practice than the conventional method, center manifold reduction. Our results show that when time delay is small the equilibrium is stable, when it is at its critical value Hopf bifurcation happens and while for very large value of time delay the oscillation sustains, which has been confirmed by the published data and proved mathematically by using the global continuity of the Hopf bifurcation in this paper. [Copyright &y& Elsevier]

Abstract: Mathematical analysis of the Hes1 genetic oscillator system is presented. The considered model is a modification of the model proposed by Monk [N.A. Monk, Oscillatory expression of Hes1, p53, and NF- B driven by transcriptional time delays, Curr. Biol. 13 (2003) 1409–1413]. In this paper we model separately time delays connected with transcription and translation processes. Global existence, uniqueness and non-negativity of solutions are proved. The existence of a unique positive steady state is shown. The stability of the steady state is studied and it is shown that it depends on the sum of time delays. Conditions guaranteeing the occurrence of the Hopf bifurcation as well as its stability are proved. Presented analysis partially confirms the results obtained numerically by Jensen et al. [M. Jensen, K. Sneppen, G. Tiana, Sustained oscillations and time delays in gene expression of protein Hes1, FEBS Lett. 541 (2003) 176–177]. [Copyright &y& Elsevier]

Abstract: In this paper, we study a retarded functional differential equation modeling a single neuron with inertial term subject to time delay. Bogdanov–Takens bifurcation is investigated by using center manifold reduction and the normal form method for RFDE. We get the versal unfolding of the norm forms at the B–T singularity and show that the model can exhibit saddle-node bifurcation, pitchfork bifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation. Some numerical simulations are given to support the analytic results. [Copyright &y& Elsevier]

Abstract: In this paper, we study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays . Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of . The function describing stability of HB is periodic with respect to . Within one period of length 4 five changes of HB stability are observed. We also introduce the second model with two delays which has a better biological interpretation than the first one. In that model several stability switches can occur, depending on the model parameters. We illustrate analytical results on the example of tumour growth model with parameters estimated on the basis of experimental data. [Copyright &y& Elsevier]

A FitzHugh-Nagumo neuron model with cubic nonlinearity and discrete delay is considered, in which the time delay is regarded as a parameter. The effect of time delay on the linear stability and Hopf bifurcation of the model is studied. The existence, stability and direction of the local and global Hopf bifurcation are derived. Some numerical simulations are employed to validate the main results of this work. [ABSTRACT FROM AUTHOR]

In this paper, a model of predator-prey with disease in food chain is investigated - where, prey is infected by bacteria and then the infected prey in turn infects predator, but the disease does not spread among predators. The law for disease development and biodiversity conservation are the focus. Stability and persistence are deduced in terms of system parameters. Next, time required delay is incorporated into the model. Stability and bifurcation analysis of the delay differential equation model are carried out. Furthermore, stability and direction of the bifurcating periodic solutions are performed by the normal form theory and the center manifold argument. Finally, numerical simulations are included for illustrating the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: This paper develops a continuous-time neoclassical growth model with time delay. Despite of its simple structure, the resulting dynamic system shows emergence of erratic fluctuations in the capital accumulation process when the production function is unimodal and the delay in production is explicitly considered. It complements the seminal paper of in which a discrete-time neoclassical growth model displayed chaotic behavior for some configurations of the propensity to save, the growth rate of labor and the capital depreciation rate. Our analysis has at least two implications. First, nonlinearities and delay matter for a birth of aperiodic fluctuations of national product in the continuous-time model. Second, comparing the effects caused by two different time delays, fixed and continuously distributed time delays, reveals that the continuous-time model with the former can generate complex fluctuations while the one with the latter generates simple fluctuations when the shape parameter of the density function is small and complex fluctuations when large. [Copyright &y& Elsevier]

Abstract: We show that the Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an -dimensional dynamical system. This extends results of Fiedler et al. [B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, E. Schöll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett. 98(11) (2007) 114101], who demonstrated that such a feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. The Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two-parameter plane. In each case, the Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the feedback phase angle satisfies a certain restriction. [Copyright &y& Elsevier]

The dynamics of a linearly coupled Mackey-Glass system with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and obtain the bifurcation set in the parameter plane. The explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold. The global existence of periodic solutions is established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson's criterion for higher dimensional ordinary differential equations due to [Li & Muldowney, 1993]. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we construct a two-layer feedback neural network to theoretically investigate the influence of symmetry and time delays on patterned rhythmic behaviors. Firstly, linear stability of the model is investigated by analyzing the associated transcendental characteristic equation. Next, by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delays of signal transmission on the pattern formation, but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Thirdly, based on the normal form approach and the center manifold theory, we derive the formula to determine the bifurcation direction and stability of Hopf bifurcating periodic solutions. Finally, some numerical examples and the corresponding numerical simulations are used to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper studied the stability and Hopf bifurcation of a type of protein synthesis system with time delay and negative feedback. Firstly, it is proved theoretically that the time delay, nonlinearity in the protein production and the cooperativity in the negative feedback are key factors to generate circadian oscillation; Taking time delay as a parameter, we obtained the critical value of the time delay that Hopf bifurcation generates. Secondly, based on the center manifold and normal form theorem, we derived the formulas for determining the stability of bifurcating periodic solutions and the supercritical or subcritical Hopf bifurcation. Finally, the matlab program is used to simulate the results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a physiological model of invasive blood-glucose (BG) measurement is employed to consider the diabetic treatment by the external auxiliary system, i.e., artificial pancreas (AP). For such system, there are two time delays, i.e., technological and liver''s physiological delay, where the former comes from external auxiliary system with the active pancreas inputting. The technological delay and the infection degree of patients are considered as two controlled parameters to regulate the BG level of patients. This two parameters can also lead to the non-resonant double Hopf bifurcations. The classification and unfolding for the non-resonant double Hopf bifurcation are performed in terms of non-linear dynamics. The results show that such controlled parameters are very important. They can determine the efficiency for the diabetic treatment. It implies whether the diabetic patients recover or are still tormented by the simple or complex glucose fluctuation. The results have also been promising applications on analyzing, predicting and optimizing the medical outcome, evaluating the medical risk and feasibility. The physiological meaning in this paper is that one is able to achieve the better medical outcomes for the different patients by controlling the technological delay qualitatively. [Copyright &y& Elsevier]

A closed-form calculation is presented for the analysis of the period-doubling bifurcation in the time-periodic delay-differential equation model of interrupted machining processes such as milling where the nonlinearity is essentially nonsymmetric. We prove the subcritical sense of this period-doubling bifurcation and approximate the emerging period-two oscillations by the Lyapunov--Perron method for computing the center manifold and by calculating the Poincaré--Lyapunov constant of the bifurcation analytically at certain characteristic parameter values. The existence of the unstable period-two oscillations around the stable stationary cutting is confirmed using a numerical continuation algorithm developed for time-periodic delay- differential equations. [ABSTRACT FROM AUTHOR]

Abstract: A differential delay equation model with a discrete time delay and a distributed time delay is introduced to simulate zooplankton–nutrient interaction. The differential inequalities’ methods and standard Hopf bifurcation analysis are applied. Some sufficient conditions are obtained for persistence and for the global stability of the unique positive steady state, respectively. It was shown that there is a Hopf bifurcation in the model by using the discrete time delay as a bifurcation parameter. [Copyright &y& Elsevier]

Abstract: In this paper we develop a mathematical model for understanding the effect of time lags in the insect predation benefits for spiders in agroecosystems. Bifurcation of equilibria giving stable limit cycles are shown to occur as functions of the time delay. [Copyright &y& Elsevier]

Abstract: This paper focuses on the delay induced Hopf bifurcation in a dual model of Internet congestion control algorithms which can be modeled as a time-delay system described by a one-order delay differential equation (DDE). By choosing communication delay as the bifurcation parameter, we demonstrate that the system loses its stability and a Hopf bifurcation occurs when communication delay passes through a critical value. Moreover, the bifurcating periodic solution of the system is calculated by means of the perturbation method. Discussion of stability of the periodic solutions involves the computation of Floquet exponents by considering the corresponding Poincaré–Lindstedt series expansion. Finally, numerical simulations for verifying the theoretical analysis are provided. [Copyright &y& Elsevier]

In this paper, the effects of harvesting and time delay on two different types of predator-prey systems with delayed predator specific growth and Holling type II functional response are studied by applying the normal form theory of retarded functional differential equations developed by Faria andMagalhãaes [ J. Differential Equations, 122 (1995), pp. 181-200, J. Differential Equations, 122 (1995), pp. 201-224]. Hopf bifurcations are demonstrated in models with harvesting of the prey at a constant rate by taking the delay as a bifurcation parameter, and numerical examples supporting our theoretical prediction are also given. Furthermore, bifurcation analysis indicates that delayed predator-prey systems with predator harvesting exhibit Bogdanov-Takens bifurcation. The versal unfoldings of the models at the Bogdanov-Takens singularity are obtained, and numerical simulations and bifurcation diagrams are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Abstract: In a previous paper [R. Miyazaki, K. Tchizawa, Bifurcation delay in a delay differential equation, Nonlinear Analysis 63 (2005) e2189–e2195], the influence of the delay feedback on the bifurcation delay was discussed. However, the result of numerical experiments is not completely consistent with the theory. In this paper, the structure which was given by numerical experiments was described. The theory on the structure is based on fundamental theorems of the differential-difference equations. The delay feed back made bifurcation delay shorter. We also showed a new result on the stability of bifurcation delay with delay time . [Copyright &y& Elsevier]

Abstract: A mathematical model of the glucose–insulin control mechanism which incorporates the role of -cells and time-delays is investigated. An analysis of the model is carried out by relying on the Hopf bifurcation theory in order to derive the conditions under which the system permits different dynamical behavior including the existence of periodic solutions. A genetic algorithm is utilized to find an optimal set of parametric values for which the simulated curve best fits our experimental glycemia data. [Copyright &y& Elsevier]