Form Of Delay Differential Equations Research Articles

Stability and Bifurcation Analysis in Turning of Flexible Parts with Spindle Speed Variation Using FEM Simulation Data

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.

Analyzing population dynamics models via Sumudu transform

This study demonstrates how to construct the solutions of a more general form of population dynamics models via a blend of variational iterative method with Sumudu transform. Evolution of population growth models are presented and new models which are more general, are proposed in form of delay differential equations of pantograph type. This study presents suitable reformulation and reconstruction for some existing population growth models in terms of delay differential equations of pantograph type. Also, presentation is given on innovative ways to obtain the solutions of population growth models where other analytic methods fail. Stimulating procedures for finding patterns and regularities in seemingly chaotic processes are elucidated in this paper. Some single and interacting species population models are illustrated graphically and analyzed. How, when and why the changes in population sizes occur can be deduced through this study.

Bifurcation Analysis of a Duopoly Game with R&D Spillover, Price Competition and Time Delays

The aim of this study is to analyse a discrete-time two-stage game with R&D competition by considering a continuous-time set-up with fixed delays. The model is represented in the form of delay differential equations. The stability of all the equilibrium points is studied. It is found that the model exhibits very complex dynamical behaviours, and its Nash equilibrium is destabilised via Hopf bifurcations.

Modelling and stability analysis of complex balanced kinetic systems with distributed time delays

Kinetic systems are positive polynomial systems that can be equipped with discrete and/or distributed time delays for describing process models often appearing in complex reaction kinetic systems. Mathematically these models are in the form of delay-differential equations with discrete and/or distributed time delays.The origin of the delays may be an inherent phenomena or some approximation of the biochemical system model. It is shown in the paper that even the simplest transport mechanism, the spatially distributed convection combined with chemical reaction is resulted in a delayed chemical reaction network model, where the parameters of the transport are analytically related to the parameters of the induced delay – discrete for the plug flow and distributed for the laminar flow – in the model.We proved that – similarly to the case of kinetic systems without delay – every positive complex balanced equilibrium of a time delayed kinetic system with arbitrary nondecreasing cumulative delay distribution function is locally asymptotically stable relative to its positive stoichiometric compatibility class.The results and notations were illustrated on a simple example using simulation investigations.

Dynamics of Drilling Systems With an Antistall Tool: Effect on Rate of Penetration and Mechanical Specific Energy

Summary This paper considers the effect of an antistall tool on the dynamics of deep drilling systems. Field results show that the antistall tool increased the rate of penetration (ROP) of drilling systems when compared with ROP in offset wells drilled without this tool. We developed a model–based approach to investigate the effect of this downhole tool on the ROP and on the mechanical specific energy. Toward this, a drillstring model including the antistall tool was constructed; it describes the coupled axial/torsional dynamics in the form of delay differential equations. Simulation results and a dynamic analysis based on averaging the obtained steady–state response show that an increased drilling efficiency was obtained using the antistall tool, resulting in a higher ROP.

Modified wavelets–based algorithm for nonlinear delay differential equations of fractional order

Most of the physical phenomena located around us are nonlinear in nature and their solutions are of great significance for scientists and engineers. In order to have a better representation of these physical phenomena, fractional calculus is developed. Some of these nonlinear physical models can be represented in the form of delay differential equations of fractional order. In this article, a new method named Gegenbauer Wavelets Steps Method is proposed using Gegenbauer polynomials and method of steps for solving nonlinear fractional delay differential equations. Method of steps is used to convert the fractional nonlinear fractional delay differential equation into a fractional nonlinear differential equation and then Gegenbauer wavelet method is applied at each iteration of fractional differential equation to find the solution. To check the accuracy and efficiency of the proposed method, the proposed method is implemented on different nonlinear fractional delay differential equations including singular-type problems also.

Age-dependent intra-specific competition in pre-adult life stages and its effects on adult population dynamics

Intra-specific competition in insect and amphibian species is often experienced in completely different ways in their distinct life stages. Competition among larvae is important because it can impact on adult traits that affect disease transmission, yet mathematical models often ignore larval competition. We present two models of larval competition in the form of delay differential equations for the adult population derived from age-structured models that include larval competition. We present a simple prototype equation that models larval competition in a simplistic way. Recognising that individual larvae experience competition from other larvae at various stages of development, we then derive a more complex equation containing an integral with a kernel that quantifies the competitive effect of larvae of ageāon larvae of agea. In some parameter regimes, this model and the famous spruce budworm model have similar dynamics, with the possibility of multiple co-existing equilibria. Results on boundedness and persistence are also proved.

Regenerative delay, parametric forcing and machine tool chatter: A review

Two intrinsic component of machine tool chatter modeling is regenerative time delay and parametric forcing. The corresponding governing equations are therefore given in the form of delay-differential equations with time-periodic coeficients. In this paper, a brief review is given on the mechanism of these two effects, and recent numerical techniques from the literature are categorized and discussed.

Modelling the dynamics of bluetongue disease and the effect of seasonality.

We present mathematical models for the midge-borne disease bluetongue, with cattle and sheep as hosts. The models take the form of delay differential equations and incorporate the incubation time of bluetongue in cattle, sheep and midges, and also the larval developmental time of midges. Recovery in cattle and sheep is also included. Both an autonomous and a periodic model are considered, to take account of seasonality. For both models we present conditions for the disease-free state to be linearly stable, and a detailed interpretation of those conditions. The results of simulations are also presented. Important findings include the need for prompt diagnosis of latent infection and appropriate action before the animal turns infectious, and the need for measures that reduce insect bites.

Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory

In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.

Closed-form solution to the problem of reaction with fixed bed adsorption using delay-differential equations

Delay-differential equations (DDEs) can describe many chemical engineering models. However, the formalism of DDEs appears to be underutilized in chemical engineering. We have recast the canonical chemical engineering problem of batch reaction with fixed bed sorption into the form of a delay-differential equation, obtaining a more intuitive model and a simpler closed form solution than those previously reported. Considerable model reduction is possible through the use of DDE formalism when one considers that chemical processes can be partially represented by networks of transportation and state delays. Analytical and numerical methods for solution, as well as controllability and stability theory for systems of DDEs, are nearly as rich and developed as those for ordinary differential equations. Significant progress thus may be possible in areas such as the modeling, synthesis, and control of chemical processes, if the governing equations can be expressed in the form of delay-differential equations.

Modelling periodic oscillations during somitogenesis

We consider a model of genetic network that has been previously presented by J. Lewis. This model takes the form of delay differential equations with two delays. We give conditions for the local stability of the non-trivial steady state. We investigate the condition under which stability is lost and oscillations occur. In particular, we show that when the ratio of the time delays passes a threshold, sustained oscillations occur through a Hopf bifurcation. Through numerical simulations, we further investigate the ways in which various parameters influence the period and the amplitude of the oscillations. In conclusion, we discuss the implications of our results.

Learning dynamics from nonstationary time series: analysis of electroencephalograms.

We propose an empirical modeling technique for a nonstationary time series analysis. Proposed methods include a high-dimensional (N>3) dynamical model construction in the form of delay differential equations, a nonparametric method of respective time delay calculation, the detection of quasistationary regions of the process by recurrence analysis in the space of model coefficients, and final fitting of the model to quasistationary segments of observed time series. We also demonstrate the effectiveness of our approach for nonstationary signal classification in the space of model coefficients. Applying the empirical modeling technique to electroencephalogram (EEG) records analysis, we find evidence of high-dimensional nonlinear dynamics in quasistationary EEG segments. Recurrence analysis of model parameters reveals long-term correlations in nonstationary EEG records. Using the dynamical model as a nonlinear filter, we find that different emotional states of subjects can be clearly distinguished in the space of model coefficients.

The Portevin-Le Ch�telier Effect: A Description Based on Time Delay

A mathematical description of the Portevin-Le Châtelier (PLC) effect is presented in the form of delay differential equations. Inclusion of a delay time in the dislocation evolution equations has been found to be helpful in explaining a number of experimentally observed results. The results of the simulation reproduce some basic features of the PLC effect such as the strain-rate window within which the effect occurs, increase in stress drop and period of stress oscillation with change in strain rate, and the occurrence of chaotic stress oscillations under certain conditions.