State-dependent Delay Differential Equation Research Articles

Existence of a periodic solution for a state-dependent delay differential equation with a singular perturbation

The idea of this article came from the works dones by Khung and Smith on the one hand in [2] for and Arino and Al in [1] for on the other hand, to proves the existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard. I proposed the following question: Do the result remain valid if 1 ? by the same method with using the homotopy well defined link between the application of Khung and Smith and the application of Arino and and indeed, I came to prove existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard, therefore the result obtained in this work is generalization of two special cases. A State-dependent Delay Differential equation play important role in recent years to modeling of several phenomenon, in dynamics of populations, in biology and even in the social life for example: When we take our driving test, we learn that the reaction time of our nervous system when driving is of the order of a few seconds, and that we must take care to maintain a sufficient distance between two cars following each other, from this example, we notice that the delay can be useful. Although this work has added a new addition to the scientific field which is a generalization and linking between two special cases, it remains somewhat limited because it takes special case of Arino and Al. Therefore in the futur, we are thinking about a new work that takes into account the general case of Arino and .

Exploiting delay differential equations solved by Eta functions as suitable mathematical tools for the investigation of thickness controlling in rolling mill

This work is dedicated to introducing the properties and application of Eta functions. We derive the properties of the Eta function, such as the generating function, integral representation, and the Laplace transform. Also, some properties of the Eta-based functions are introduced. To show the advantages of the Eta-based functions in the computational method, we develop a new numerical method to solve the state-dependent and time-dependent neutral delay differential equation based on the Eta-based function. We introduce the operational matrix of derivative for the Eta-base functions to develop the new numerical method. This method uses the operational matrix of derivative and collocation method to convert the delay differential equation to a system of nonlinear algebraic equations. We derive the technique’s error bound and establish the method’s accuracy by solving some examples, which are state-dependent and time-dependent delay differential equations. In the end, we study the model of the metal forming process by rolling the mill using the new numerical method to show the advantages of using the Eta-based function for solving a more practical problem.

Global Stability and Hopf Bifurcation for a Stage Structured Model with Competition for Food

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.

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.

A discussion on the decoupling assumption of axial and torsional dynamics in bit-rock models

A common strategy for analyzing the stability of coupled axial and torsional dynamics in drilling is by using a two-degree-of-freedom mass-spring-damper model derived as a second-order ordinary differential equation. This implies using a single point mass to represent the inertia of the bottom-hole assembly and thus the characteristics of an otherwise distributed system. For the bit-rock model developed in Richard et al. (2007), the stability characteristics of the drilling system are in many previous works derived by assuming that the axial dynamics can be decoupled from the slower torsional dynamics. Using this bit-rock model, the friction forces and torques in the system are dependent on a time-delay term, dictating the stability of the system. Consequently, a set of critical drilling parameters, i.e., rotation speed, drill string stiffness, and fluid damping, can be investigated to identify the properties of the equilibrium for the nominal solution.In this paper, we have generalized the drill string dynamics by modeling the drill string as a series of alternating springs and point masses. By making the same assumptions as in previous works, the dynamics are derived as a lumped-multi-element second-order model including a time-delay dependent on the system states. The model is then defined as a state-dependent delay differential equation. The stability analysis of the decoupled generalized system indicates that the stable region for the chosen drilling parameters reduces and is trending towards zero when the number of model elements increases. The implication of this is that the nonlinearity of the coupled axial and torsional dynamics is required for a thorough understanding of the stability of the drilling system. Furthermore, we have performed a numerical analysis of the coupled dynamics by simulating relevant drilling scenarios.

Existence of global attractor for a nonautonomous state-dependent delay differential equation of neuronal type

The analysis of the long-term behavior of the mathematical model of a neural network constitutes a suitable framework to develop new tools for the dynamical description of nonautonomous state-dependent delay equations (SDDEs). The concept of global attractor is given, and some results which establish properties ensuring its existence and providing a description of its shape, are proved. Conditions for the exponential stability of the global attractor are also studied. Some properties of comparison of solutions constitute a key in the proof of the main results, introducing methods of monotonicity in the dynamical analysis of nonautonomous SDDEs. Numerical simulations of some illustrative models show the applicability of the theory.

Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods

We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.

Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations

We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.

Equivalences between age structured models and state dependent distributed delay differential equations

We use the McKendrick equation with variable ageing rate and randomly distributed maturation time to derive a state dependent distributed delay differential equation. We show that the resulting delay differential equation preserves non-negativity of initial conditions and we characterise local stability of equilibria. By specifying the distribution of maturation age, we recover state dependent discrete, uniform and gamma distributed delay differential equations. We show how to reduce the uniform case to a system of state dependent discrete delay equations and the gamma distributed case to a system of ordinary differential equations. To illustrate the benefits of these reductions, we convert previously published transit compartment models into equivalent distributed delay differential equations.

A system of state-dependent delay differential equation modeling forest growth I: semiflow properties

In this article, we investigate the semiflow properties of a class of state-dependent delay differential equations which is motivated by some models describing the dynamics of the number of adult trees in forests. We investigate the existence and uniqueness of a semiflow in the space of Lipschitz and $$C^1$$ weighted functions. We obtain a blow-up result when the time approaches the maximal time of existence. We conclude the paper with an application of a spatially structured forest model.

A system of state-dependent delay differential equation modelling forest growth II: Boundedness of solutions

In this article we consider a class of state-dependent delay differential equations which models the dynamics of the number of adult trees in forests. We prove the boundedness and the dissipativity of the solutions for a single species model and an n-species model.

Dynamical properties of nonautonomous functional differential
 equations with state-dependent delay

A type of nonautonomous n-dimensional state-dependent delay differential equation (SDDE) is studied. The evolution law is supposed to satisfy standard conditions ensuring that it can be imbedded, via the Bebutov hull construction, in a new map which determines a family of SDDEs when it is evaluated along the orbits of a flow on a compact metric space. Additional conditions on the initial equation, inherited by those of the family, ensure the existence and uniqueness of the maximal solution of each initial value problem. The solutions give rise to a skew-product semiflow which may be noncontinuous, but which satisfies strong continuity properties. In addition, the solutions of the variational equation associated to the SDDE determine the Frechet differential with respect to the initial state of the orbits of the semiflow at the compatibility points. These results are key points to start using topological tools in the analysis of the long-term behavior of the solution of this type of nonautonomous SDDEs.

Local bifurcations analysis of a state-dependent delay differential equation

In this paper, a first-order equation with state-dependent delay and a nonlinear right-hand side is considered. The conditions of the existence and uniqueness of the solution of the initial value problem are supposed to be executed.The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. The local dynamics depends on real parameters, which are coefficients of the right-hand side decomposition in a Taylor series.The parameter, which is a coefficient at the linear part of this decomposition, has two critical values that determine the stability domain of the zero equilibrium. We introduce a small positive parameter and use the asymptotic method of normal forms in order to investigate local dynamics modifications of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the first of these critical values, and the supercritical Andronov–Hopf bifurcation occurs near the second of these values (provided the sufficient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered to be an example. The bifurcation parameter of this equation has the only critical value. A simple sufficient condition of the occurrence of the supercritical Andronov–Hopf bifurcation in the considered equation near a critical value has been obtained as a result of applying the method of normal forms.

Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method II: Analytic case

We construct analytic quasi-periodic solutions of a state-dependent delay differential equation with quasi-periodically forcing. We show that if we consider a family of problems that depends on one dimensional parameters (with some non-degeneracy conditions), there is a positive measure set Π of parameters for which the system admits analytic quasi-periodic solutions.The main difficulty to be overcome is the appearance of small divisors and this is the reason why we need to exclude parameters. Our main result is proved by a Nash–Moser fast convergent method and is formulated in the a-posteriori format of numerical analysis. That is, given an approximate solution of a functional equation which satisfies some non-degeneracy conditions, we can find a true solution close to it.This is in sharp contrast with the finite regularity theory developed in [18]. We conjecture that the exclusion of parameters is a real phenomenon and not a technical difficulty. More precisely, for generic families of perturbations, the quasi-periodic solutions are only finitely differentiable in open sets in the complement of parameters set Π.

An analysis on the stability of a state dependent delay differential equation

Abstract In this paper, we present an analysis for the stability of a differential equation with state-dependent delay. We establish existence and uniqueness of solutions of differential equation with delay term τ ( u ( t ) ) = a + b u ( t ) c + b u ( t ) . $\tau (u(t)) = \frac{{a + bu(t)}}{{c + bu(t)}}.$ Moreover, we put the some restrictions for the positivity of delay term τ(u(t)) Based on the boundedness of delay term, we obtain stability criterion in terms of the parameters of the equation.

Periodic Solutions of a Singularly Perturbed Delay Differential Equation with Two State-Dependent Delays

Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of singular solution is introduced which is based on a continuous parametrisation of the possibly discontinuous limiting solution. This reduces the construction of the limiting profiles to an algebraic problem. A model two state-dependent delay differential equation is studied in detail and periodic singular solutions are constructed with one and two local maxima per period. A complete characterisation of the conditions on the parameters for these singular solutions to exist facilitates an investigation of bifurcation structures in the singular case revealing folds and possible cusp bifurcations. Sophisticated boundary value techniques are used to numerically compute the bifurcation diagram of the state-dependent DDE when the perturbation parameter is close to zero. This confirms that the solutions and bifurcations constructed in the singular case persist when the perturbation parameter is nonzero, and hence demonstrates that the solutions constructed using our singular solution definition are useful and relevant to the singularly perturbed problem. Fold and cusp bifurcations are found very close to the parameter values predicted by the singular solution theory, and we also find period-doubling bifurcations as well as periodic orbits with more than two local maxima per period, and explain the alignment between the folds on different bifurcation branches.

Local Bifurcations Analysis of a State-Dependent Delay Differential Equation

In this paper, a first-order equation with state-dependent delay and with a nonlinear right-hand side is considered. Conditions of existence and uniqueness of the solution of initial value problem are supposed to be executed. The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. Local dynamics depends on real parameters which are coefficients of equation right-hand side decomposition in a Taylor series. The parameter which is a coefficient at the linear part of this decomposition has two critical values which determine a stability domain of zero equilibrium. We introduce a small positive parameter and use the asymtotic method of normal forms in order to investigate local dynamics modifications of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the first of these critical values, and the supercritical Andronov – Hopf bifurcation occurs near the second of them (if the sufficient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered as an example. The bifurcation parameter of this equation has one critical value. A simple sufficient condition of Andronov – Hopf bifurcation occurence in the considered equation near a critical value is obtained as a result of applying the method of normal forms.

Using AQM for performance improvement of networked control systems

This paper presents a new scheme to jointly control the plant and the network to improve the overall performance of the networked control system (NCS). Using an analytical transmission control protocol (TCP) model for the network, an active queue management (AQM) is designed to achieve desired queue length. This leads to keeping network induced delay and its variation to be small enough to improve the overall performance of the NCSs. By augmenting the TCP communication model with the control system equations, the closed loop system is transformed to a state-dependent delay differential equation (SDDDE). Then using the Lyapunov-Krasovskii method, a sufficient condition for the stability of this class of systems is presented in terms of linear matrix inequality (LMI). Simulation results are presented and compared to some other methods, the superiority of the proposed approach is shown.

Observer Based Temperature Control of a Tank: From Constant to Nonlinear State-Dependent Delay

The problem of temperature control of a fluid in a tank is expounded in this paper. In our problem, the delay is a an explicit nonlinear function of the state (temperature) of the system and the resulting dynamics are governed by a nonlinear State-Dependent Delay differential Equation (SD-DDE). We perform the stability analysis using Lyapunov-Razumikhin based approach. Using the new concept of delay injection with inversion, an infinite dimensional observer is designed to estimate the state of the system. We also shed some light on the state space required for characterizing the Cauchy problem and the evolution of the SD-DDE based system. A juxtaposition of fixed delay and state-dependent delay is given for a better illustration, comparison and contrast of results. Simulation results are depicted at the end which portray the effectiveness, validity and usefulness of the proposed observer scheme.

Singular Hopf bifurcation in a differential equation with large state-dependent delay

We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol's equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.