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This study investigates the effect of variable spindle speed with sinusoidal modulation on chatter vibrations generated in turning of flexible parts. The Euler–Bernoulli beam theory is assumed for mathematical modeling of the slender workpiece that is clamped at the chuck and pinned at the tailstock. The induced force as interaction between tool and workpiece (A2024-T351) during chip formation is calculated using finite element simulation. Numerical-based semi-discretization approach is utilized to seek stability regions and chatter frequencies regarding coupled dynamic model of tool and workpiece, in the form of delay differential equations with time varying delay term. The simulated data are validated and compared with those for a rigid part. The results indicate the efficiency of periodic modulation in improvement of machining stability especially for low spindle speeds. The stability charts for different interaction location and also time history response are studied, extensively. Finally, detection of period-one, flip and secondary Hopf bifurcations for different spindle speeds at instability borders are analyzed and discussed by solving corresponding eigenvalue problem. [ABSTRACT FROM AUTHOR]

Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields. [ABSTRACT FROM AUTHOR]

We apply the 1 -equivariant degree method to a Hopf bifurcation problem for functional differential equations with a state-dependent delay. The formal linearization of the system at a stationary state is extracted and translated into a bifurcation invariant by using the homotopy invariance of 1 -equivariant degree. As a result, the local Hopf bifurcation is detected and the global continuation of periodic solutions is described. [ABSTRACT FROM AUTHOR]

Since the stochastic bifurcation theory is still in its infancy, we try to analyze some stochastic bifurcation phenomenon from a simple mathematical model. Thus, this paper mainly focuses on studying the stochastic bifurcation of a stochastic logistic model with distributed delay in the strong kernel case, which is affected by noise. Therefore, we use the intrinsic growth rate as a bifurcation parameter. First, we study the stochastic D-bifurcation and stochastic P-bifurcation for stochastic logistic model. Furthermore, by deriving the corresponding Fokker–Planck equation, we obtain the expression of the joint density function of the stochastic logistic system near the positive equilibrium point. Finally, some conclusions are given. [ABSTRACT FROM AUTHOR]

The Investment Savings-Liquidity preference Money supply (IS-LM) model is represented as a graph depicting the intersection of products and the money market. It elaborates how an equilibrium of money supply versus interest rates may keep the economy in control. In this paper, we combine the basic business cycle IS-LM model with Kaldor's growth model in order to create an augmented model. The IS-LM model, when coupled with a certain economics expansion (in our instance, the Kaldor–Kalecki Business Cycle Model), provides a comprehensive description of a developing but robust economy. Right after the introduction of capital stock into the system, it cannot be employed and also, while making some investment choices, this requires some time in execution, which ultimately alters resources, i.e. capital. Thus, in the capital accumulation, we will be incorporating double time delays in Gross product and Capital Stock. These time delays represent the time periods during which investment decisions were made and executed and the time spent in order for the capital to be put to productive use. After formulating a mathematical model using delayed differential equations, dynamic functioning of the system around equilibrium point is examined where three instances appeared based on time delays. These cases are: when both delays are not in action, when only one delay is in action and when both delays are in action. It is shown that time delay affects the stability of the equilibrium point and, as the delay crosses a critical point, Hopf bifurcation exists. It is observed that by using Kaldor type investment function, the delay residing in capital stock only will destabilize in less time as compared to when both the delays are present in the system. The system is sensitive to certain parameters which is also analyzed in this work. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a model called the k-nearest neighbor queue with the possibility of having delayed queue length feedback. We prove fluid limits for the stochastic queueing model and show that the fluid limit converges to a system of delay differential equations. Using the properties of circulant matrices, we derive a closed form expression for the value of the critical delay, which governs whether the delayed information will induce oscillations or a Hopf bifurcation in our queueing system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a delayed mathematical model describing oncolytic virotherapy treatment of a tumour that proliferates according to the logistic growth function, incorporating viral lytic cycle. The tumour population cells are divided into uninfected and infected cell sub-populations and the virus spreading is supposed to be in a direct mode (i.e. from cell to cell). Depending on the time delay, we analyze the positivity and boundedness of solutions and the stability of tumour, infected and uninfected free equilibria (TFE, IFE, UFE) and uninfected–infected equilibrium (UIE) is established. We prove that, delay can lead to "Jeff's phenomenon" observed in a laboratory which causes oscillations in tumour size whose phase and period change over time. With nonlinear dependence of UIE equilibrium on time delay, we develop a more general algorithm determining the stability/instability of the oscillating periodic solutions bifurcating from the UIE equilibrium. Finally, we present numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, Myc/E2F/miR-17-92 network under Gaussian white noise is studied. Taking the time delay as the parameter, the Hopf bifurcation of the system is obtained, which causes the protein concentration to oscillate periodically. Under the influence of time delay and noise, the stochastic D-bifurcation of the system is obtained. It is worth noting that the occurrence of stochastic P-bifurcation is successfully captured. Thus a pattern of coexistence of high and low protein concentrations is founded in the network. The specific research methods of this paper are as follows: firstly, the system is reduced to a finite dimensional system by using stochastic center manifold and normal form theory. Then, using the stochastic averaging method, the Fokker–Planck–Kolmogorov equation of the system is constructed in which the statistical response in the stationary state is the probability density. Finally, the stochastic bifurcation analysis and numerical simulation are carried out. The agreements between the analytical method and those obtained numerically validate the effectiveness of the analytical investigations. [ABSTRACT FROM AUTHOR]

Considering the mature condition of any individual to have eaten a specific amount of food during the entire period that it can spend at its immature stage, we propose a size-structured model by a first-order quasi-linear partial differential equation. The model can be firstly reduced to a single state-dependent delay differential equation and then to a constant delay differential equation. The state-dependent delay represents intra-specific competition among individuals for limited food resources. A complete analysis of the global dynamics on the positivity and boundedness of solutions, global stability for each equilibrium and Hopf bifurcation is carried out. Our results imply that the delay leads to instability that is shown by a simple example of a certain structured population model. [ABSTRACT FROM AUTHOR]

Giving customers queue length information about a service system has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze a two-dimensional deterministic fluid model that incorporates customer choice behavior based on delayed queue length information. Reports in the existing literature always assume that all queues have identical parameters and the underlying dynamical system is symmetric. However, in this paper, we relax this symmetry assumption by allowing the arrival rates, service rates, and the choice model parameters to be different for each queue. Our methodology exploits the method of multiple scales and asymptotic analysis to understand how to break the symmetry. We find that the asymmetry can have a large impact on the underlying dynamics of the queueing system. [ABSTRACT FROM AUTHOR]

We prove that all Hopf bifurcations in the Nicholson's blowfly equation are supercritical as we increase the delay. Earlier results treated only the first bifurcation point, and to determine the criticality of the bifurcation, one needed to substitute the parameters into a lengthy formula of the first Lyapunov coefficient. With our result, there is no need for such calculations at any bifurcation point. [ABSTRACT FROM AUTHOR]

To reduce the global burden of mosquito-borne diseases, e.g. dengue, malaria, the need to develop new control methods is to be highlighted. The sterile insect technique (SIT) and various genetic modification strategies, have a potential to contribute to a reversal of the current alarming disease trends. In our previous work, the ordinary differential equation (ODE) models with different releasing sterile mosquito strategies are investigated. However, in reality, implementing SIT and the releasing processes of sterile mosquitos are very complex. In particular, the delay phenomena always occur. To achieve suppression of wild mosquito populations, in this paper, we reassess the effect of the delayed releasing of sterile mosquitos on the suppression of interactive mosquito populations. We extend the previous ODE models to the delayed releasing models in two different ways of releasing sterile mosquitos, where both constant and exponentially distributed delays are considered, respectively. By applying the theory and methods of delay differential equations, the effect of time delays on the stability of equilibria in the system is rigorously analyzed. Some sustained oscillation phenomena via Hopf bifurcations in the system are observed. Numerical examples demonstrate rich dynamical features of the proposed models. Based on the obtained results, we also suggest some new releasing strategies for sterile mosquito populations. [ABSTRACT FROM AUTHOR]

Due to the unpredictable growth of tumor cells, the tumor–immune interactive dynamics continues to draw attention from both applied mathematicians and oncologists. Mathematical modeling is a powerful tool to improve our understanding of the complicated biological system for tumor growth. With this goal, we report a mathematical model which describes how tumor cells evolve and survive the brief encounter with the immune system mediated by immune effector cells and host cells which includes discrete time delay. We analyze the basic mathematical properties of the considered model such as positivity of the system and the boundedness of the solutions. By analyzing the distribution of eigenvalues, local stability analysis of the biologically feasible equilibria and the existence of Hopf bifurcation are obtained in which discrete time delay is used as a bifurcation parameter. Based on the normal form theory and center manifold theorem, we obtain explicit expressions to determine the direction of Hopf bifurcation and the stability of Hopf bifurcating periodic solutions. Numerical simulations are carried out to illustrate the rich dynamical behavior of the delayed tumor model. Our model simulations demonstrate that the delayed tumor model exhibits regular and irregular periodic oscillations or chaotic behaviors, which indicate the scenario of long-term tumor relapse. [ABSTRACT FROM AUTHOR]

The dynamics of two related models of second order delay differential equations with four bifurcating parameters are analyzed. Through a classical technique in the time domain which involves the location of the roots of an exponential polynomial equation, the areas of stability of the equilibrium are set. A frequency-domain methodology is applied to study the Hopf bifurcation phenomena and to describe the behavior of the emerging cycles completely via a feedback system approach. Certain types of singularities, which provoke fold bifurcations of cycles are detected precisely. Also, a complete picture of parameter configurations to produce resonances is established for both models. All results are checked with the software DDE-Biftool. [ABSTRACT FROM AUTHOR]

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method. [ABSTRACT FROM AUTHOR]

Time delay is inevitable in unidirectionally coupled drive-free vibratory gyroscope system. The effect of time delay on the gyroscope system is studied in this paper. To this end, amplitude death and Hopf bifurcation induced by small time delay are first investigated by analyzing the related characteristic equation. Then, the direction of Hopf bifurcations and stability of Hopf-bifurcating periodic oscillations are determined by calculating the normal form on the center manifold. Next, spatiotemporal patterns of these Hopf-bifurcating periodic oscillations are analyzed by using the symmetric bifurcation theory of delay differential equations. Finally, it is found that numerical simulations agree with the associated analytic results. These phenomena could be induced although time delay is very small. Therefore, it is shown that time delay is an important factor which influences the sensitivity and accuracy of the gyroscope system and cannot be neglected during the design and manufacture. [ABSTRACT FROM AUTHOR]

In this paper, we study a fractional-order model with time-delay to describe the dynamics of Ebola virus infection with cytotoxic T-lymphocyte (CTL) response in vivo. The time-delay is introduced in the CTL response term to represent time required to stimulate the immune system. Based on fractional Laplace transform, some conditions on stability and Hopf bifurcation are derived for the model. The analysis shows that the fractional-order with time-delay can effectively enrich the dynamics and strengthen the stability condition of fractional-order infection model. Finally, the derived theoretical results are justified by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study how to capture smooth oscillations arising from delay-differential equations with distributed delays. For this purpose, we introduce a modified version of the frequency-domain method based on the Graphical Hopf Bifurcation Theorem. Our approach takes advantage of a simple interpretation of the distributed delay effect by means of some Laplace-transformed properties. Our theoretical results are illustrated through an example of two coupled neurons with distributed delay in their communication channel. For this system, we compute several bifurcation diagrams and approximations of the amplitudes of periodic solutions. In addition, we establish analytical conditions for the appearance of a double zero bifurcation and investigate the unfolding by the proposed methodology. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the codimension-two double Hopf bifurcation in delay-coupled van der Pol-Duffing oscillators. By using normal form theory of delay differential equations, the normal form associated with the codimension-two double Hopf bifurcation is calculated. Choosing appropriate values of the coupling strength and the delay can result in nonresonance and weak resonance double Hopf bifurcations. The dynamical classification near these bifurcation points can be explicitly determined by the corresponding normal form. Periodic, quasi-periodic solutions and torus are found near the bifurcation point. The numerical simulations are employed to support the theoretical results. [ABSTRACT FROM AUTHOR]

In a previous paper, we developed a new way to apply the Homotopy Analysis Method (HAM) in the search for periodic orbits in dynamical systems modeled by ordinary differential equations. This method differs from the original in the heuristic search of the frequencies of the cycles. In this paper, we show that the method can be extended to the search for periodic orbits in delay differential equations. Herein, this methodology is applied twice, firstly in an equation of van der Pol type and secondly in an anharmonic oscillator, both systems with a delayed feedback. [ABSTRACT FROM AUTHOR]

In the present paper, we study the effect of time delays in evolutionary games with one population of users and two strategies. The case where the delays, corresponding to different strategies, are not the same is considered. The local stability of the stationary state for the replicator dynamics is analyzed. We show that there is Hopf bifurcation. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and normal form theory. The stochastic evolutionary game with delay is taken into consideration. We also study the behavior of the first and second solution moments for linear stochastic differential delay equation in the presence of white and colored noise. The last part of the paper includes numerical simulations and conclusions. [ABSTRACT FROM AUTHOR]

In this paper, employing the normal form theory of delay differential equations due to Faria and Magalhães, we present explicit formulas of the coefficients of a normal form associated with the flow on a center manifold with the unfolding for general delay differential equations under the cases of steady-state, Hopf and steady-state-Hopf singularities. The explicit conditions determining the transcritical and pitchfork bifurcations for steady-state singularity, determining the direction and stability of Hopf bifurcations, and determining the coefficients of a normal form with universal unfolding for steady-state-Hopf singularity up to third order are obtained. Using the obtained results, we give a complete description of bifurcation scenario of the damped harmonic oscillator with delay feedback near the zero equilibrium. Finally, numerical simulations are given to illustrate our theoretical results and some numerical extensions are obtained as a supplement to our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper we apply the homotopy analysis method (HAM) to study the van der Pol equation with a linear delayed feedback. The paper focuses on the calculation of periodic solutions and associated bifurcations, Hopf, double Hopf, Neimark-Sacker, etc. In particular, we discuss the behavior of the systems in the neighborhoods of double Hopf points. We demonstrate the applicability of HAM to the analysis of bifurcation and stability. [ABSTRACT FROM AUTHOR]