Original Delay Differential Equation Research Articles

Short delay limit of the delayed Duffing oscillator.

The delayed Duffing equation, x^{″}+ɛx^{'}+x+x^{3}+cx(t-τ)=0, admits a Hopf bifurcation which becomes singular in the limit ɛ→0 and τ=O(ɛ)→0. To resolve this singularity, we develop an asymptotic theory where x(t-τ) is Taylor expanded in powers of τ. We derive a minimal system of ordinary differential equationsthat captures the Hopf bifurcation branch of the original delay differential equation. An unexpected result of our analysis is the necessity of expanding x(t-τ) up to third order rather than first order. Our work is motivated by laser stability problems exhibiting the same bifurcation problem as the delayed Duffing oscillator [Kovalev etal., Phys. Rev. E 103, 042206 (2021)2470-004510.1103/PhysRevE.103.042206]. Here we substantiate our theory based on the short delay limit by showing the overlap (matching) between our solution and two different asymptotic solutions derived for arbitrary fixed delays.

Linear Quadratic Regulator for Delayed Systems Using the Hamiltonian Approach and Exact Closed-Loop Poles for First-Order Systems

Abstract We consider the linear quadratic regulator (LQR) for linear constant-coefficient delay differential equations (DDEs) with multiple delays. The Hamiltonian approach is used instead of an algebraic Riccati partial differential equation. Two coupled DDEs governing the state and control input are derived using the calculus of variations. This coupled system, with infinitely many roots in both left and right half-planes, defines a boundary value problem. Its left half-plane roots are the exact closed-loop poles of the controlled system. These closed-loop poles have not been used to compute the optimal feedback before. Here, the distributed delay kernel that yields exactly those poles is first computed using an eigenfunction expansion. Increasing the number of terms in the truncated expansion yields a highly oscillatory kernel. However, the oscillatory kernel's antiderivative converges to a piecewise smooth function on the delay interval plus a Dirac delta function at zero. Discontinuities in the kernel coincide with discrete delay values in the original DDE. Using this insight, a fitted piecewise polynomial kernel matches the exact closed-loop poles very well. The twofold contribution of the Hamiltonian approach is thus clarity on the form of the feedback kernel as well as the exact closed-loop poles. Subsequently, the fitted piecewise polynomial kernel can be used for a much simpler control calculation. The polynomial coefficients can be fitted by solving a few simultaneous linear equations. Two detailed numerical examples of the LQR for DDEs, one with two delays and one with three delays, show excellent results.

Exploring complicated behaviors of a delay differential equation

<abstract><p>Complicated behaviors of a delay differential equation are explored through the Euler discretization method. It rigorously shows that the corresponding discrete equation can be chaotic under some conditions, which reflects that there exist complicated behaviors in the original delay differential equation.</p></abstract>

Pole Placement for Delay Differential Equations With Time-Periodic Delays Using Galerkin Approximations

Abstract Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems (TDS). In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

Short-delayed-feedback semiconductor lasers.

We consider the laser rate equations describing the evolution of a semiconductor laser subject to an optoelectronic feedback. We concentrate on the first Hopf bifurcation induced by a short delay and develop an asymptotic theory where the delayed variable is Taylor expanded. We determine a nearly vertical branch of strongly nonlinear oscillations and derive ordinary differential equations that capture the bifurcation properties of the original delay differential equations. An unexpected result is the need for Taylor expanding the delayed variable up to third order rather than first order. We discuss recent laser experiments where sustained oscillations have been clearly observed with a short-delayed feedback.

Fast Generation of Stability Charts for Time-Delay Systems Using Continuation of Characteristic Roots

Abstract Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method can significantly reduce the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.

Spurious roots of delay differential equations using Galerkin approximations

The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerkin approximation method can be applied to a second-order delay differential equation in two ways: either by converting into a second-order partial differential equation and then into a system of second-order ordinary differential equations (the “second-order Galerkin” method) or by first expressing as two first-order delay differential equations and converting into a system of first-order partial differential equations and then into a first-order ordinary differential equation system (the “first-order Galerkin” method). In this paper, we demonstrate that these subtly different formulation procedures lead to different roots of the characteristic polynomial. In particular, the second-order Galerkin method produces spurious roots near the origin, which must then be identified through substitution into the characteristic polynomial of the original delay differential equation. However, spurious roots do not arise if the first-order Galerkin method is used, which can reduce computation time and simplify stability analyses. We describe these two formulation strategies and present numerical examples to highlight their important differences.

On the global attractor of delay differential equations with unimodal feedback not satisfying the negative Schwarzian derivative condition

We study the size of the global attractor for a delay differential equation with unimodal feedback. We are interested in extending and complementing a dichotomy result by Liz and Rost, which assumed that the Schwarzian derivative of the nonlinear feedback is negative in a certain interval. Using recent stability results for difference equations, we obtain a stability dichotomy for the original delay differential equation in the situation wherein the Schwarzian derivative of the nonlinear term may change sign. We illustrate the applicability of our results with several examples.

Reduced ordered modelling of time delay systems using galerkin approximations and eigenvalue decomposition

In this paper, an r-dimensional reduced-order model (ROM) for infinite-dimensional delay differential equations (DDEs) is developed. The eigenvalues of the ROM match the r rightmost characteristic roots of the DDE with a user-specified tolerance of $$\varepsilon $$. Initially, the DDE is approximated by an N-dimensional set of ordinary differential equations using Galerkin approximations. However, only $$N_{c}$$$$(< N)$$ eigenvalues of this N-dimensional model match (with a tolerance of $$\varepsilon $$) the rightmost characteristic roots of the DDEs. By performing numerical simulations, an empirical relationship for $$N_{c}$$ is obtained as a function of N and $$\varepsilon $$ for a scalar DDE with multiple delays. Using eigenvalue decomposition, an r$$(= N_{c})$$ dimensional model is constructed. First, an appropriate r is chosen, and then the minimum value of N at which at least r roots converge is selected. For each of the test cases considered, the time and frequency responses of the original DDE obtained using direct numerical simulations are compared with the corresponding r- and N-dimensional systems. By judiciously selecting r, solutions of the ROM and DDE match closely. Next, an r-dimensional model is developed for an experimental 3D hovercraft in the presence of delay. The time responses of the r-dimensional model compared favorably with the experimental results.

Reduced Ordered Modelling of Time Delay Systems Using Galerkin Approximations and Eigenvalue Decomposition

In this paper, we develop ordinary differential equations (ODEs) based approximations for linear delay differential equations (DDEs). Using the shift of time transformation, the DDE is converted into an equivalent partial differential equation (PDE) with boundary conditions (BCs). The obtained PDE along with BCs is then approximated into a system of linear ODEs using Galerkin approximations. The eigenvalues of these ODEs approximate the characteristic roots of the original DDE. By only retaining the dynamics of these ODEs corresponding to converged eigenvalues of the original DDE, these ODEs are further reduced using eigenvalue decomposition method. The time responses obtained for the Galerkin approximated system models are compared with the same obtained using the direct numerical simulation and the results are found to be satisfactory.

Dynamics of two-cell systems with discrete delays

We consider the system of delay differential equations (DDE) representing the models containing two cells with time-delayed connections. We investigate global, local stability and the bifurcations of the trivial solution under some generic conditions on the Taylor coefficients of the DDE. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including pitchfork, transcritical and Hopf bifurcation) and Takens-Bogdanov bifurcation as a codimension two bifurcation. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. Finally, we show that the analytical results agree with numerical simulations.

Numerical Integration Method for Stability Analysis of Milling With Variable Spindle Speeds

A semi-analytical method is presented in this paper for stability analysis of milling with a variable spindle speed (VSS), periodically modulated around a nominal spindle speed. Taking the regenerative effect into account, the dynamics of the VSS milling is governed by a delay-differential equation (DDE) with time-periodic coefficients and a time-varying delay. By reformulating the original DDE in an integral-equation form, one time period is divided into a series of subintervals. With the aid of numerical integrations, the transition matrix over one time period is then obtained to determine the milling stability by using Floquet theory. On this basis, the stability lobes consisting of critical machining parameters can be calculated. Unlike the constant spindle speed (CSS) milling, the time delay for the VSS is determined by an integral transcendental equation which is accurately calculated with an ordinary differential equation (ODE) based method instead of the formerly adopted approximation expressions. The proposed numerical integration method is verified with high computational efficiency and accuracy by comparing with other methods via a two-degree-of-freedom milling example. With the proposed method, this paper details the influence of modulation parameters on stability diagrams for the VSS milling.

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.

A Legendre Polynomials Based Method for Stability Analysis of Milling Processes

This paper presents a time-domain approach for a semi-analytical prediction of stability in milling using the Legendre polynomials. The governing equation of motion of milling processes is expressed as a delay-differential equation (DDE) with time periodic coefficients. After the DDE being re-expressed in state-space form, the state vector is approximated by a series of Legendre polynomials. With the help of the Legendre–Gauss–Lobatto (LGL) quadrature, a discrete dynamic map is formulated to approximate the original DDE, and utilized to predict the milling stability based on Floquet theory. With numerical examples illustrating the efficiency and accuracy of the proposed approach, an experimental example validates the method.

Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators

The effects of delayed feedback terms on nonlinear oscillators have been extensively studied, and there are important applications in many areas of science and engineering. We study a particular class of second-order delay-differential equations near a point of triple-zero nilpotent bifurcation. Using center manifold and normal form reduction, we show that the three-dimensional nonlinear normal form for the triple-zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay-differential equation.

Nonlinear response of a forced van der Pol–Duffing oscillator at non-resonant bifurcations of codimension two

Non-resonant bifurcations of codimension two may appear in the controlled van der Pol–Duffing oscillator when two critical time delays corresponding to a double Hopf bifurcation have the same value. With the aid of centre manifold theorem and the method of multiple scales, the non-resonant response and two types of primary resonances of the forced van der Pol–Duffing oscillator at non-resonant bifurcations of codimension two are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. It is shown that the non-resonant response of the forced oscillator may exhibit quasi-periodic motions on a two- or three-dimensional (2D or 3D) torus. The primary resonant responses admit single and mixed solutions and may exhibit periodic motions or quasi-periodic motions on a 2D torus. Illustrative examples are presented to interpret the dynamics of the controlled system in terms of two dummy unfolding parameters and exemplify the periodic and quasi-periodic motions. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation.

Nonresonant Hopf bifurcations of a controlled van der Pol–Duffing oscillator

The trivial equilibrium of a van der Pol–Duffing oscillator with a nonlinear feedback control may lose its stability via Hopf bifurcations, when the time delay involved in the feedback control reaches certain values. Nonresonant Hopf–Hopf interactions may occur in the controlled van der Pol–Duffing oscillator when the corresponding characteristic equation has two pairs of purely imaginary roots. With the aid of normal form theory and centre manifold theorem as well as a perturbation method, the dynamic behaviour of the nonresonant co-dimension two bifurcation is investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. In the vicinity of the nonresonant Hopf bifurcation, the oscillator may exhibit the initial equilibrium solution, two periodic solutions as well as a quasi-periodic solution on a two-dimensional torus, depending on the dummy unfolding parameters and nonlinear terms. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation.

Explicit periodic solutions in a model of a relay controller with delay and forcing

In this paper we use a combination of numerical and analytical methods to find and construct solutions of a cameo model of relay control, formulated as a piecewise-constant delay differential equation (DDE). Numerical solutions of a related equation, where the discontinuities of the original DDE are smoothed out, are used to guide the construction of explicit solutions of the original DDE. On the other hand, the construction of explicit solutions provides starting data for numerical continuation of the smoothed equation. The stability of the explicit solutions can also be inferred from the numerical approach.

Delayed feedback control of forced self-sustained oscillations

We consider a weakly nonlinear van der Pol oscillator subjected to a periodic force and delayed feedback control. Without control, the oscillator can be synchronized by the periodic force only in a certain domain of parameters. However, outside of this domain the system possesses unstable periodic orbits that can be stabilized by delayed feedback perturbation. The feedback perturbation vanishes if the stabilization is successful and thus the domain of synchronization can be extended with only small control force. We take advantage of the fact that the system is close to a Hopf bifurcation and derive a simplified averaged equation which we are able to treat analytically even in the presence of the delayed feedback. As a result we obtain simple analytical expressions defining the domain of synchronization of the controlled system as well as an optimal value of the control gain. The analytical theory is supported by numerical simulations of the original delay-differential equations.

Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry

We consider a general class of first-order nonlinear delay-differential equations (DDEs) with reflectional symmetry, and study completely the bifurcations of the trivial equilibrium under some generic conditions on the Taylor coefficients of the DDE. Our analysis reveals a Hopf bifurcation curve terminating on a pitchfork bifurcation line at a codimension two Takens–Bogdanov point in parameter space. We compute the normal form coefficients of the reduced vector field on the centre manifold in terms of the Taylor coefficients of the original DDE, and in contrast to many previous bifurcation analyses of DDEs, we also compute the unfolding parameters in terms of these coefficients. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. We illustrate these results using simple model systems relevant to the areas of neural networks and atmospheric physics, and show that the results agree with numerical simulations.