State-dependent Time Delay Research Articles

Bifurcation analysis of thin-walled structures trimming process with state-dependent time delay

Our previous study [1] revealed that trimming of thin-walled structures has a strongly nonlinear nature, hence this investigation delves into nonlinear dynamic analysis. Specifically, a perturbation method was applied to analyze the linear stability, whilst a direct numerical integration approach was employed to construct bifurcation diagrams, where unsafe cutting caused by multiple stabilities had been found. Basins of attraction were computed to evaluate the cutting safety. Finally, the stability charts for both linearized and fully nonlinear models were constructed showing a good degree of similarity but the linear one has a larger area of stable trimming. The data from our focused experimental studies was then superimposed on both diagrams and it is clear that more refinements in models’ calibration are needed. This work has laid firm foundations for further advanced modelling investigations and experimental studies to gain more insights into nonlinear dynamics of the machining process.

Perimeter Control With State-Dependent Delays: Optimal Control Model and Computational Method

Perimeter control is to manipulate traffic flows in different regions through adjusting traffic signals at the border of the regions. Traditionally, the complete trips in a region are assumed to be dependent on the current accumulated vehicles in this region. This assumption is invalid because the vehicle needs time to complete its trip in a region. In order to solve this shortcoming, some recent studies have introduced a time-delay dynamical system to describe the dynamical behaviour of the accumulated vehicles in a region. However, in these studies, the perimeter control problem is formulated as a tracking problem with a given optimal reference point. In reality, such an optimal reference point is unavailable in advance. This paper will fill this gap through formulating perimeter control as an optimal control problem governed by a state-dependent time delay system. The control parametrization technique and an exact penalty method are introduced to solve such a challenging optimal control problem. Model predictive control is applied to obtain close-loop solution through solving a series of online optimal control problems. Some experiments are performed to demonstrate the effectiveness of our method.

Effect of state-dependent time delay on dynamics of trimming of thin-walled structures

Trimming of cantilever thin-walled structures is commonly seen in aerospace industry, including trimming of blades. Trimming with helix angle tools can cause the vibration of the thin-walled workpiece along the tool-axis, which may disturb the time delay between cutting by the current and previous teeth. The time delay dependent on the vibration state makes the stability analysis of trimming process challenging. This paper is the first attempt to uncover the effect of state-dependent time delay of the trimming process caused by workpiece vibration on chatter stability. Modeling of the cutter-workpiece interactions, state-dependent time delay and the dynamic chip generation mechanism are presented. A time-domain numerical algorithm with an improved stability metrics is constructed to analyze the trimming stability behaviors. We found that the two dominant states can occur, namely, period-n instabilities with time-varying time delay and stability with constant time delay. A focused experimental study was carried out to calibrate this new finding. This study reveals the way the workpiece vibration affects the time delay and stability in the trimming process.

An Improved Model to Characterize Drill-String Vibrations in Rotary Drilling Applications

A specific model is elaborated for stick-slip and bit-bounce vibrations, which are dangerous dynamic phenomena typically encountered in the context of rotary drilling applications. Such a model takes into account two coupled degrees of freedom of drill-string vibrations. Moreover, it assumes a state-dependent time delay and a viscous damping for both the axial and torsional vibrations and relies on a sawtooth function to account for the cutting force fluctuation. In the frame of this theoretical approach, the influence of rock brittleness on the stability of the drill string is calculated via direct integration of the model equations. The results show that the rock brittleness has a great influence on the rotational speed and bit depth.

State dependent regenerative stability and surface location error in peripheral milling of thin-walled parts

Chatter avoidance and error control are the core issues accompanying milling processes of thin-walled parts. Owing to the high flexibility of workpiece, the cutting vibrations can easily reach dozens of micrometers and become comparable to the nominal chip thickness. However, the classic dynamic models usually neglect the influence of system state on regenerative stability and surface location error (SLE), which consequently reduces the prediction accuracy. In this paper, the coupling relationships between cutter-workpiece engagement, time delay and system state are modeled by means of analyzing the true teeth trajectories that are composed of tool rotation, feed movement and cutting vibrations. The resultant dynamic model is a state dependent delay differential equation, which is capable of accounting for the fly-over and multi-regeneration nonlinear phenomena too. Afterwards, an efficient numerical algorithm is proposed to accurately compute the dynamic responses. By contrast to classic time domain simulation techniques, the proposed algorithm uses discrete root finding scheme to determine the state dependent time delay and the state dependent cutter-workpiece engagement, which ensures its high computational efficiency. Based on the proposed dynamic model and numerical algorithm, the stability limits and SLE are finally obtained, and experimentally validated on two thin-walled plates using two variable pitch tools. Theoretical comparison and experimental results reveal that the proposed state dependent dynamic model successfully predicts the milling stability and SLE while the classic model fails. The state dependency of regenerative stability and SLE is further investigated and explained in detail.

Complex Dynamics and Periodic Oscillation Mechanism in Two Novel Gene Expression Models with State-Dependent Delays

The reaction time delay in the transcription process depends on the concentration of the protein because the transportation of mRNA from the nucleus to the cytoplasm becomes saturated. Thus the gene regulatory network is a state-dependent delayed model. This study aims to provide some mathematical explanations for the dynamics of the system, such as the linear stability and periodic oscillation, using mathematical techniques, such as formal linearization, linear stability analysis, the method of multiple scale (MMS), and the normal form. First, Hopf bifurcation of the state-dependent delayed gene regulatory networks model in the gene expression is analyzed by the method of multiple scales (MMS). Mechanism of periodic oscillations is obtained by Hopf bifurcation. The findings show that when degradation effects of the mRNA and protein are very strong, the oscillatory gene expression disappears. Then, a more realistic version of the aforementioned model with both constant and state-dependent time delays is established due to the existence of the constant time delay in the protein degradation process. Its nonresonant double Hopf bifurcation is found and analyzed using MMS. Interesting complex dynamic phenomena, such as periodic, quasi-periodic, and global period-[Formula: see text] solutions, are also discovered. These observations indicate that both state-dependent delay and constant delay could induce richer dynamics of the system, and the modified model may potentially describe the real dynamical mechanism (both the transcription process and the degradation process) more accurately in the gene expression. The findings may provide important guidance or hints to understand the real dynamic mechanism of the gene expression process.

Dynamics and control of the active control system with the state-dependent actuation time delay

In this paper, a model of active control system with state-dependent actuation time delay is investigated. Galerkin projection scheme is used to obtain its low dimensional approximation system. State-dependency and effect on dynamics and performance of actuation delay in this active control system are considered. The following results are obtained. Firstly, a state-dependent actuation delay is proposed, which has never been reported in the previous works. It displays that actuation delay is state-dependent. Secondly, employing Galerkin projection scheme, low dimensional approximation system of this delayed active control system is obtained. At last, its stability and Hopf bifurcation are investigated by Routh-Hurwitz Criterion and Hopf bifurcation theory. Using software WinPP, numerical simulation is executed and supports the theoretical results in this paper.

Periodic solutions and complicated dynamics in network congestion control systems with state-dependent round-trip delays and non-smooth dropping probability.

In this paper, the harmonic balance method with the alternating frequency/time (HB-AFT) domain technique is extended to the dynamical systems with state-dependent delays and non-smooth right-hand side for the first time. Two types of network congestion control models [the modified transmission control protocol-random early detection (TCP-RED) model and the fluid-flow TCP-additive increase multiplicative decease (AIMD)/RED model] with state-dependent round-trip time delays and non-smooth right-hand side are considered in detail. First, their dynamics and bifurcation are analyzed by the numerical analysis method. Then, the analytical approximate expressions of the periodic solutions are obtained by employing the semi-analytical method named as HB-AFT. The results of the numerical simulation and HB-AFT agree with each other very well. It indicates that the HB-AFT technique is simple, valid, effective, and accurate for the non-smooth dynamical systems with state-dependent time delays. Besides, more complicated and rich dynamical phenomena, such as period-2 solution, period-3 solution, chaos and the period-doubling bifurcation, and the window of period-3 solution, which lead to chaos, are explored in the TCP-AIMD/RED model. Moreover, two different kinds of multi-stabilities, such as the coexistence of a stable period-1 solution, a stable period-3 solution, and an asymptotically stable equilibrium, and that of a stable period-1 solution, a stable period-2 solution, chaos, and an asymptotically stable equilibrium, are also discovered in the TCP-AIMD/RED congestion control model. It discloses that the interaction of the state-dependent delay and non-smooth function can easily induce the richer and more complicated dynamics. It can explain the complicated dynamical phenomena in the Internet network. Thus, the control and parameters tuning based on the above results could be used to avoid the congestion and optimize the performance. It can give the theoretical guidance to the application and research in network congestion control, which is very important in practical application.

Influence of bit design on the stability of a rotary drilling system

This paper extends the RGD model originally proposed by Richard et al. (J Sound Vib 305(3):432–456, 2007, https://doi.org/10.1016/j.jsv.2007.04.015) to investigate drilling-induced vibrations, by considering idealized drag bits characterized either by multiple angular offsets between the blades or by a combination of full and partial blades, a departure from the symmetric bit of the RGD model consisting of multiple continuous blades. The RGD model is built on a two-degrees of freedom discrete representation of the drillstring combined with a rate-independent bit/rock interaction law. By relaxing the original constraints on the bit geometry, the coupled state-dependent delay differential equations governing the angular and axial motions of the bit now contain multiple state-dependent time delays. A linear stability analysis leads to explicit expressions for the critical rotational speed(s) defining two different regimes of stability of the coupled dynamics. Stability maps indicate that both groups of bits have a better axial stability than their RGD counterparts. It is also found that multiple critical rotation speeds exist for certain bit geometries, in contrast to the single axial stability boundary of the original RGD model. Time simulations confirm the predictions of the linear stability analysis and show evidence of the existence of different types of quasi-periodic responses before the establishment of a limit cycle characterized by both axial and torsional stick-slip events.

Nonlinear dynamics of lump mass model of drill-string in horizontal well

We develop a dynamic model of a drill-string in a horizontal well having six Degrees-Of-Freedom, which accounts for longitudinal, lateral and torsional motions. In this model, nonlinearities that arise due to intermittent contacts of a drill-pipe with a borehole wall and complex interactions between a drill-bit and a rock formation are all considered. The lateral motions of the drill-bit are restricted and its interactions with a borehole generate an interlaced force being result of a friction and a cutting that has a regenerative effect. In the cutting process, a state-dependent time delay is introduced to couple the axial and torsional motions of the drill-bit. The dynamic model established in this article is tested where the friction and cutting effects are gradually switched on, which shows that the model is robust. Subsequently, the complex whirling of a horizontal drill-string is analyzed and a particular attention is given to the influence of driving rotation speed and dynamic friction coefficient. The study should help us to better understand nonlinear dynamic effects of drill-strings in horizontal wells, which will lay a foundation for optimizing drilling parameters.

Wavefronts of a Nonlocal State-dependent Time Delay Model

This paper is concerned with the travelling wavefronts of a nonlocal dispersal cooperation model with harvesting and state-dependent delay, which is assumed to be an increasing function of the population density with lower and upper bound. Especially, state-dependent delay is introduced into a nonlocal reaction-diffusion model. The conditions of Schauder’s fixed point theorem are proved by constructing a reasonable set of functions Γ (see Section 2) and a pair of upper-lower solutions, so the existence of traveling wavefronts is established. The present study is continuation of a previous work that highlights the Laplacian diffusion.

Dynamics of four-neuron recurrent inhibitory loop with state-dependent time delays

Abstract In previous works, recurrent inhibitory loops with state-dependent propagation delays, firing process and absolute refractory period have been successfully employed to solve capacity-simplicity dilemma in associative memory attractors networks. But in realistic networks, usually there are more than two or three neurons. In order to disclose their natures, in this paper, we consider the dynamics of periodic patterns mainly in a four-neuron recurrent inhibitory loop. We explicitly address how to give rise to its enormous periodic patterns and obtain their existence conditions. At last, we further execute numerical simulations to address the difference from the five-neuron loop and demonstrate similar periodic patterns. Even for smaller τ, the coexistence of three periodic patterns is found in four-neuron loop and that of four periodic patterns is discovered in five-neuron loop. New periodic patterns are also generated and the maximum values of τ for the existence of possible periodic patterns also decrease with the increment of the number of neurons in loops. They will improve loops performance greatly. Our research shows that delays are significant and remarkable for the dynamics of recurrent inhibitory loops, since loops with more neurons have more complicated dynamical behavior and loops performance are also enhanced due to the the increase of synaptic delay and propagation delay.

Intra-specific competition and insect larval development: a model with time-dependent delay

We derive a stage-structured model for an insect population in which a larva matures on reaching a certain size, and in which there is intra-specific competition among larvae that hinders their development, thereby prolonging the larval phase. The model, a system of delay differential equations for the total numbers of adults and larvae, assumes two forms. One of these is a system with a variable state-dependent time delay determined by a threshold condition, the other has constant and distributed delays, a size-like independent variable replacing time t, and no threshold condition. We prove theorems on boundedness and on the linear stability of equilibria.

State Dependent Delayed Drill-string Vibration: Theory, Experiments and New Model

We have reviewed the development and the analysis of the coupled two degrees-of-freedom model of drill-string vibration, which assumes a state-dependent time delay and a viscous damping for both the axial and torsional vibration. This model corrects the instability deficiencies of the previous models but it is still based on a uniform angular distribution of blades on a drill-bit, which can lead to unrealistic regenerative types of instability. However, the model sheds some lights on the origin and the interplay between stick-slip and bit-bounce, the dangerous dynamic phenomena encountered during rotary drilling. The experimental rig developed by the Centre of Applied Dynamics Research at the University of Aberdeen is used to show some shortcomings of this model and to develop a new more realistic one. The new model assuming a non-uniform distribution of blades for drag type drill-bits is proposed.

Pauses of Larval Development and Their Consequences for Stage-Structured Populations

This paper considers some novel predictions of a mathematical model for a stage-structured insect species that undergoes diapause if faced with strong intraspecific competition among larvae. The model consists of a system of two delay differential equations with a state-dependent time delay of threshold type. When the model has an Allee effect, we show that diapause may cause extinction in some parameter regimes even where the initial population is high. We also demonstrate that the model can have diapause-induced periodic solutions that can arise even if the birth function is strictly increasing, a situation in which solutions for the constant delay case always converge to an equilibrium.

Stochastic modeling of biochemical systems with multistep reactions using state-dependent time delay

To deal with the growing scale of molecular systems, sophisticated modelling techniques have been designed in recent years to reduce the complexity of mathematical models. Among them, a widely used approach is delayed reaction for simplifying multistep reactions. However, recent research results suggest that a delayed reaction with constant time delay is unable to describe multistep reactions accurately. To address this issue, we propose a novel approach using state-dependent time delay to approximate multistep reactions. We first use stochastic simulations to calculate time delay arising from multistep reactions exactly. Then we design algorithms to calculate time delay based on system dynamics precisely. To demonstrate the power of proposed method, two processes of mRNA degradation are used to investigate the function of time delay in determining system dynamics. In addition, a multistep pathway of metabolic synthesis is used to explore the potential of the proposed method to simplify multistep reactions with nonlinear reaction rates. Simulation results suggest that the state-dependent time delay is a promising and accurate approach to reduce model complexity and decrease the number of unknown parameters in the models.

Regenerative and frictional chatter in plunge grinding

This paper studies self-excited vibrations, regenerative and frictional chatters, in a plunge grinding process. In consideration of lateral and torsional workpiece movements, a dynamic model with four degrees of freedom is proposed, which involves state-dependent time delays and Stribeck effect for regenerative and frictional effects, respectively. With this model, by linearising contact angle, regenerative grinding depth, frictional velocity and the state-dependent delays, eigenvalue analysis yields stability diagrams for the grinding, where boundaries for both regenerative and frictional instabilities are determined. Then, near the boundaries, simulations and bifurcation analyses are performed to reveal patterns of chatter onset. Bifurcation diagrams show coexistence of stable grinding and the regenerative chatter near the regenerative boundary, and a sudden switch between the stable and the frictionally unstable on the frictional boundary. Moreover, simulation results also show various dynamical properties in the grinding chatters, such as effect of losing contact and stick–slip motion.

Dynamical Analysis of the Hindmarsh-Rose Neuron With Time Delays.

This brief is mainly concerned with a series of dynamical analyses of the Hindmarsh-Rose (HR) neuron with state-dependent time delays. The dynamical analyses focus on stability, Hopf bifurcation, as well as chaos and chaos control. Through the stability and bifurcation analysis, we determine that increasing the external current causes the excitable HR neuron to exhibit periodic or chaotic bursting/spiking behaviors and emit subcritical Hopf bifurcation. Furthermore, by choosing a fixed external current and varying the time delay, the stability of the HR neuron is affected. We analyze the chaotic behaviors of the HR neuron under a fixed external current through time series, bifurcation diagram, Lyapunov exponents, and Lyapunov dimension. We also analyze the synchronization of the chaotic time-delayed HR neuron through nonlinear control. Based on an appropriate Lyapunov-Krasovskii functional with triple integral terms, a nonlinear feedback control scheme is designed to achieve synchronization between the uncontrolled and controlled models. The proposed synchronization criteria are derived in terms of linear matrix inequalities to achieve the global asymptotical stability of the considered error model under the designed control scheme. Finally, numerical simulations pertaining to stability, Hopf bifurcation, periodic, chaotic, and synchronized models are provided to demonstrate the effectiveness of the derived theoretical results.

The effect of state dependent delay and harvesting on a stage-structured predator–prey model

We propose and analyze a stage-structured predator–prey model in which the time from birth to maturity is directly related to the number of individuals present. Individuals mature more quickly when there are fewer of them around. The state dependent time delay is taken to be an increasing differentiable bounded function. In this research, we study the dynamics of our model analytically. We present results on positivity and boundedness of all populations. Criteria for the existence of all equilibria and uniqueness of a positive equilibrium are given. In order to observe the effect of state-dependent maturation delay, local stability analysis around all equilibria of the proposed model is discussed due to variation of maturation delay. Also, global stability of trivial and the boundary equilibria is investigated, using Liapunov functional and LaSalle invariant principle. To investigate the effect of state-dependent maturation delay and the harvesting effort of all species we carried out numerical simulations.

State-Dependent Delay Influenced Drill-String Oscillations and Stability Analysis

In this paper, the authors present a discrete system model to study the coupled axial–torsional dynamics of a drill string. The model is developed taking into account state-dependent time delay and nonlinearities due to dry friction and loss of contact. Simulations are carried out by using a 32-segment model with 128 states. Bit bounce is observed through time histories of axial vibrations, while stick-slip phenomenon is noted in the torsion response. The normal strain contours of this spatial–temporal system demonstrate the existence of strain wave propagation along the drill string. The shear strain wave exhibits features of wave nodes and wave loops along the drill string, which indicate that the torsional motion has the properties of a standing wave. When the penetration rate is varied, qualitative changes are observed in the system response. The observed behavior includes chaotic and hyperchaotic dynamics. Stability analysis reveals a stable region for the degenerate one-segment model. This stable region becomes infinitesimally small, as the resolution of spatial discretization is increased. This finding suggests that drill-string motions have a high likelihood of being self-exited in practical drilling operations.