State-dependent Delay Differential Equations Research Articles

Global Hopf Bifurcation of State-Dependent Delay Differential Equations

We apply the [Formula: see text]-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 [Formula: see text]-equivariant degree. As a result, the local Hopf bifurcation is detected and the global continuation of periodic solutions is described.

Dynamic Analysis of a Downhole Regulator for Drilling in Interbedded Formations

Summary In this paper, we investigate the effect of a downhole passive regulator [anti-stall tool (AST)] on the dynamics of rotary drilling systems in interbedded formations. Drilling in interbedded formations can significantly affect the vibrational signature of these systems and the associated drilling performance. Hence, models to assess the impact of drilling in such formations are needed. Here, we construct a dynamic model of the drillstring system which incorporates the bit/rock interface laws for the transitional phase of bit motion between rock layers with distinct mechanical properties. In the model, the axial and torsional dynamics of the drillstring systems are coupled by these interface laws and cast in the form of discontinuous (state-dependent) delay differential equations. Next, we include the AST in the dynamic model to enable the assessment of the influence of this downhole tool on drilling performance, in particular in terms of rate-of-penetration (ROP) and drilling efficiency. Furthermore, we also investigate the mechanical specific energy (MSE) and steady-state power losses at the bit (due to frictional torque) for different operational conditions and rock layer thicknesses. The analysis reveals that an increased drilling efficiency and lower MSE are obtained by incorporating the downhole tool in the drillstring, resulting in a higher ROP and a lower frictional contact between the bit and rock in interbedded formations.

Almost Periodicity in Time-Dependent and State-Dependent Delay Differential Equations

We study the response of various linear and nonlinear differential equations to almost periodic forced oscillations. A special attention is given to differential equations with state-dependent and almost periodic type delays. To the best of our knowledge, there are no results in the literature that address this problem.

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.

An Alternative Formulation for Modeling Self-Excited Vibrations of Drillstring With Polycrystalline Diamond Compact Bits

Abstract This work describes an alternative formulation of a system of nonlinear state-dependent delay differential equations (SDDDEs), which governs the coupled axial-torsional vibrations of a 2 DOF drillstring model with a realistic representation of polycrystalline diamond compact (PDC) bits. The regenerative effect associated with the complex cutter layout for such bits can introduce up to 100 state-dependent delays in the equations of motion. This large number of state-dependent delays renders the computational efficiency of conventional solution strategies unacceptable. The regeneration of the bottom-hole surface can alternatively be described by the bit trajectory function, whose evolution is governed by a partial differential equation (PDE). Thus the original system of SDDDEs can be replaced by a nonlinear coupled system of a PDE and ordinary differential equations (ODEs). Via the application of the Galerkin Method, this system of PDE-ODEs is transformed into a system of coupled ODEs, which can readily be solved. The algorithm is further extended to perform a linear stability analysis of the bit motion. The resulting stability boundaries are verified with time-domain simulations. The reported algorithm could, in principle, be applied to a more realistic drillstring model, which may lead to an in-depth understanding of the mitigation of self-excited vibrations through PDC bit designs.

Lyapunov-Razumikhin techniques for state-dependent delay differential equations

We present Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin methods. Our results apply to DDEs with multiple discrete state-dependent delays, which may be nonautonomous for the Lyapunov stability result, but autonomous (or periodically forced) for the asymptotic stability result. Our main technique is to replace the DDE by a nonautonomous ordinary differential equation (ODE) where the delayed terms become source terms in the ODE. The asymptotic stability result is based on a contradiction argument together with the Arzelà-Ascoli theorem. This approach alleviates the need to construct auxiliary functions to ensure the asymptotic contraction, which is a feature of other Lyapunov-Razumikhin asymptotic stability results of which we are aware. We apply our results to a state-dependent model equation which includes Hayes equation as a special case, to directly establish asymptotic stability in parts of the stability domain together with lower bounds on the size of the basin of attraction.

Expansions in the delay of quasi-periodic solutions for state dependent delay equations* * Supported in part by NSF Grant DMS 1800241.

We consider several models of state dependent delay differential equations (SDDEs), in which the delay is affected by a small parameter. This is a very singular perturbation since the nature of the equation changes. Under some conditions, we construct formal power series, which solve the SDDEs order by order. These series are quasi-periodic functions of time. This is very similar to the Lindstedt procedure in celestial mechanics.Truncations of these power series can be taken as input for a posteriori theorems, that show that near the approximate solutions there are true solutions. In this way, we hope that one can construct a catalogue of solutions for SDDEs, bypassing the need of a systematic theory of existence and uniqueness for all initial conditions.

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.

An alternative formulation for modeling self-excited oscillations of rotary drilling systems

This paper describes an alternative formulation of the nonlinear equations governing the coupled axial and torsional dynamics of a discrete model of a rotary drilling system equipped with a drag bit. This model assumes a rate-independent bit/rock interface law that accounts for both cutting and frictional contact processes. The regenerative effect associated with the cutting introduces a term with delay, and thus a feedback, in the equations of motion, while the frictional contact results in set-valued contact forces and discontinuities in the boundary conditions, which are responsible for the occurrence of axial and torsional stick-slip oscillations. Inspired by P. Wahi and A. Chatterjee (Self-interrupted regenerative metal cutting in turning, International Journal of Non-Linear Mechanics 43:2, 111–123, 2008), the regenerative effect is captured by a bit trajectory function, whose evolution is governed by a first order partial differential equation. With this approach, the original state-dependent delay differential equations (SDDDEs) governing the dynamics of the discrete model are replaced by the partial differential equation of the bit trajectory function together with the axial and angular equations of motion of the two degrees-of-freedom model. The equation governing the evolution of the bit trajectory function is further approximated by a system of first order ordinary differential equations through application of the Galerkin method. The introduction of the bit trajectory function makes it possible to consider a bit with two state-dependent delays (and in principle multiple delays) without reducing the computational efficiency. The accuracy of this approximation is validated using published results, and the convergence and robustness properties of the proposed method are analyzed using three cases corresponding to different regimes of instability. Finally, we extend the discrete model to a more generic lumped-parameter model with multiple degrees-of-freedom, which captures the propagation of axial and torsional vibrations along the drillstring.

Numerical solution of stochastic state-dependent delay differential equations: convergence and stability

Numerical analysis of stochastic delay differential equations has been widely developed but frequently for the cases where the delay term has a simple feature. In this paper, we aim to study a more general case of delay term which has not been much discussed so far. We mean the case where the delay term takes random values. For this purpose, a new continuous split-step scheme is introduced to approximate the solution and then convergence in the mean-square sense is investigated. Moreover, given a test equation, the mean-square asymptotic stability of the scheme is presented. Numerical examples are provided to further illustrate the obtained theoretical results.

An [formula omitted]-version of the discontinuous Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays

We present and analyze an hp-version of the discontinuous Galerkin (DG) time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bounds in the L2- and L∞-norm that are fully explicit with respect to the local time steps, the local approximation orders, and the local regularity of the exact solution. We further prove that the hp-DG scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start-up singularities. Moreover, we also propose an hp-DG scheme for the state dependent delay differential equations. We illustrate the theoretical results with a series of numerical experiments.

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.

Superconvergence of discontinuous Galerkin methods for nonlinear delay differential equations with vanishing delay

In this paper, we investigate the local superconvergence of the discontinuous Galerkin (DG) solutions on quasi-graded meshes for nonlinear delay differential equations with vanishing delay. It is shown that the optimal order of the DG solution at the mesh points is O ( h 2 m + 1 ) . By analyzing the supercloseness between the DG solution and the interpolation Π h u of the exact solution, we get the optimal order O ( h m + 2 ) of the DG solution at characteristic points. We then extend the convergence results of DG solutions to state dependent delay differential equations. Numerical examples are provided to illustrate the theoretical results.

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.

Non-linear vibration analysis of hard rock TBMs with regenerative and stick–slip effects

The significant vibration caused by hard rock tunnel boring machines (TBMs) increases structural damage and reduces excavation efficiency and therefore has become a subject of major study within the tunnelling process. A two degrees-of-freedom model that accounts for axial and torsional vibrations that are coupled through the disc cutter–rock interaction is established in order to qualitatively analyse the complicated dynamic behaviours of the TBM system. Introducing a regenerative cutting effect into the cutter–rock interaction laws results in three interaction modes which are formulated based on the Colorado School of Mines (CSM) model. The regenerative effects and the friction caused by axial stick–slip motion are the sources responsible for the TBM self-excited vibration, of which the dynamics are described by state-dependent delay differential equations (DDEs). The DDEs are linearised and discretised into piecewise ordinary differential equations (ODEs) by using the semi-discretization method. Based on the Floquet theory, the stability limits of a TBM system are obtained, showing that the stable lobes are intermittently distributed in the operational parameter space. The parameter regime corresponding to the practical tunnelling is unstable, indicating that the vibrations of TBMs are easily excited by disturbances and never vanish in practical tunnelling. The axial vibration of TBMs with regenerative and stick–slip effects experiences a subcritical Hopf bifurcation, when the TBM goes from unstable to stable parameter regimes. The stick–slip effect suppresses the vibrations, which would otherwise increase without limit until cutter bounce appears if only the regenerative effect is considered. It is found that both the amplitude and the frequency of the axial stick–slip vibration increase with the increase in the steady advance velocity. The predicted axial stick–slip vibration and its amplitude in this investigation are both in good agreement with the observations obtained from a practical tunnelling project. This study provides reasonable insight into the vibration mechanism of TBMs and serves as a basis for choosing proper operational parameters in a practical project to achieve smooth and efficient tunnelling.

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.

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.

Construction of Quasi-periodic Solutions of State-Dependent Delay Differential Equations by the Parameterization Method I: Finitely Differentiable, Hyperbolic Case

In this paper, we use the parameterization method to construct quasi-periodic solutions of state-dependent delay differential equations. For example $$\begin{aligned} \left\{ \begin{aligned} \dot{x}(t)&=f(\theta ,x(t),\epsilon x(t-\tau (x(t))))\\ \dot{\theta }(t)&=\omega . \end{aligned} \right. \end{aligned}$$ Under the assumption of exponential dichotomies for the \(\epsilon =0\) case, we use a contraction mapping argument to prove the existence and smoothness of the quasi-periodic solution. Furthermore, the result is given in an a posteriori format. The method is very general and applies also to equations with several delays, distributed delays etc.

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.