Search

We present a new method for obtaining stability conditions for certain classes of delay differential equations. The method is based on the transition from an individual equation to a family of equations, and next the selection of a representative of this family, the test equation, asymptotic properties of which determine those of all equations in the family. This approach allows us to obtain the conditions that are the criteria for the stability of all equations of a given family. These conditions are formulated in terms of the parameters of the class of equations being studied, and are effectively verifiable. The main difference of the proposed method from the known general methods (using Lyapunov–Krasovsky functionals, Razumikhin functions, and Azbelev W-substitution) is the emphasis on the exactness of the result; the difference from the known exact methods is a significant expansion of the range of applicability. The method provides an algorithm for checking stability conditions, which is carried out in a finite number of operations and allows the use of numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, the stability of an equilibrium of a feedback nonlinear system with time delay and structural switching is studied. The critical case of a zero root of the characteristic equations of the linearised systems is treated by applying a Malkin-type theorem using a complete Lyapunov–Krasovskii functional. [ABSTRACT FROM AUTHOR]

In this paper the stability of the zero equilibrium of a system with time delay is studied. The critical case of a multiple zero root of the characteristic equation of the linearized system is treated by applying a Malkin type theorem and using a complete Lyapunov-Krasovskii functional. An application to a model for malaria under treatment considering the action of the immune system is presented. [ABSTRACT FROM AUTHOR]

• Stability properties of DEs with a general distributed delay are investigated. • Saddle-node and Hopf bifurcation curves are determined in a general framework. • Stability region of the equilibrium is fully characterized in the parameter plane. • Criticality of Hopf bifurcation is analyzed by the multiple times scales method. • Theoretical results are showcased in the framework of a simple neural model. For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural model. [ABSTRACT FROM AUTHOR]

Based on the idea of collocation methods, a spline-based integration method for the stability prediction of systems governed by time-periodic delay differential equations (DDEs) is presented. A B-spline curve is adopted to approximate the solution of the DDEs using the direct integration technique. The stability of the dynamic system is then predicted using the Floquet theory based on the established state transition matrix. The proposed method can apply to time-periodic DDEs where the delay and the period are incommensurate, and the proposed method is extended to study the stability of milling processes as well. For stability analysis of variable pitch cutters, the proposed method use one or multiple B-spline curves to approximate the solution of the DDEs according to the relationship between the cutter angles and the radial immersion, which can describe exactly the free vibration of the cutter. Compared with the numerical integration method and the Chebyshev collocation method, the simulation results demonstrate that the proposed technique is more efficient and accurate for stability prediction at low radial immersion cases in milling processes with variable pitch cutters. [ABSTRACT FROM AUTHOR]

The paper discusses stability and asymptotic properties of a fractional-order differential equation involving both delayed as well as non-delayed terms. As the main results, explicit necessary and sufficient conditions guaranteeing asymptotic stability of the zero solution are presented, including asymptotic formulae for all solutions. The studied equation represents a basic test equation for numerical analysis of delay differential equations of fractional type. Therefore, the knowledge of optimal stability conditions is crucial, among others, for numerical stability investigations of such equations. Theoretical conclusions are supported by comments and comparisons distinguishing behaviour of a fractional-order delay equation from its integer-order pattern. [ABSTRACT FROM AUTHOR]

The main aim of this paper is the investigation of the stability problem for ordinary delay differential equations. More precisely, we would like to study the following problem. Assume that for a continuous function a given delay differential equation is fulfilled only approximately. Is it true that in this case this function is close to an exact solution of this delay differential equation?. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: It is acknowledged that coral reefs are globally threatened. P.J. Mumby et al. [10] constructed a mathematical model with ordinary differential equations to investigate the dynamics of coral reefs. In this paper, we first provide a detailed global analysis of the coral reef ODE model in [10]. Next we incorporate the inherent time delay to obtain a mathematical model with delay differential equations. We consider the grazing intensity and the time delay as focused parameters and perform local stability analysis for the coral reef DDE model. If the time delay is sufficiently small, the stability results remain the same. However, if the time delay is large enough, macroalgae only state and coral only state are both unstable, while they are both stable in the ODE model. Meanwhile, if the grazing intensity and the time delay are endowed some suitable values, the DDE model possesses a nontrivial periodic solution, whereas the ODE model has no nontrivial periodic solutions for any grazing rate. We study the existence and property of the Hopf bifurcation points and the corresponding stability switching directions. [Copyright &y& Elsevier]

Energy efficient contact less suspension of high-speed rotor shafts in application areas such as power-plants and Flywheel Energy Storage Systems (FESS), can be achieved through Active Magnetic Bearings (AMBs). Inevitably, there exists time delay in sensing the rotor displacement and actual application of control force by electromagnets, thus leading to system instability. This paper therefore analyses stability of AMBs, with constant time delay. FESS is modelled as a rigid rotor-AMB system. Performance of two different control laws, (i) PID and (ii) a recently introduced Four-Element (FE) law, are compared. The governing equations are transformed to frequency domain, to exploit Pade's approximation in modeling time delayed value of feedback signal and stability analysis is carried out by analyzing the system poles. This methodology has been verified using experimental results from the existing literature. Numerical simulation of delay differential equations is also used to validate analytical results. The FE control law shows promise in tolerating delay, as compared to conventional PID controller thus leading to AMB system with higher possible spin speed and relaxed sampling constraints on AMB system instrumentation. One of the prospective applications of this work is in design of flywheel energy storage systems for an electric vehicle. • Time delay in Active Magnetic Bearing (AMB) system is inevitable. • Delayed control force from AMB on rotor may cause instability. • Stability analysis of 4-DOF rotor AMB system with constant time delay is reported. • For the first time, such analysis is done with AMB using integral control action. • Verification with experimental results and numerical simulation is carried out. [ABSTRACT FROM AUTHOR]

• Stability analyses of nuclear-coupled thermal hydraulics for a natural circulation heavy metal (lead-cooled) fast reactor. • Modal expansion method for neutronics and reduced order model for thermal hydraulic part. • Delay differential equations are solved for the closed system. • Different scenarios initiated by reactivity and thermal hydraulics conditions are simulated using the coupled models. Stability analyses of nuclear-coupled thermal hydraulics are carried out for SUPERSTAR – a natural circulation heavy metal (lead-cooled) fast reactor. Attributed to its natural circulation characteristic, flow stability as well as nuclear-coupled thermal hydraulics stability are of some concern. Modal expansion method as an extension for point reactor kinetics is used for neutronics and reduced order model for thermal hydraulic part. These two separate models are coupled through feedback effects. Different scenarios initiated by reactivity and thermal hydraulics conditions are simulated using the coupled models in time domain. It is shown that SUPERSTAR design has considerable stability margin at the nominal operating point. Design becomes less stable as transit time between core exit and core inlet is increased. [ABSTRACT FROM AUTHOR]

We study first order linear impulsive delay differential equations with periodic coefficients and constant delays. This study presents some new results on the asymptotic behavior and stability. Thus, a proper real root was used for a representative characteristic equation. Applications to special cases, such as linear impulsive delay differential equations with constant coefficients, were also presented. In this study, we gave three different cases (stable, asymptotic stable and unstable) in one example. The findings suggest that an equation that is in a way that characteristic equation plays a crucial role in establishing the results in this study. [ABSTRACT FROM AUTHOR]

Alcoholism is a social phenomenon that affects all social classes and is a chronic disorder that causes the person to drink uncontrollably, which can bring a series of social problems. With this motivation, a delayed drinking model including five subclasses is proposed in this paper. By employing the method of characteristic eigenvalue and taking the temporary immunity delay for alcoholics under treatment as a bifurcation parameter, a threshold value of the time delay for the local stability of drinking-present equilibrium and the existence of Hopf bifurcation are found. Then the length of delay has been estimated to preserve stability using the Nyquist criterion. Moreover, optimal strategies to lower down the number of drinkers are proposed. Numerical simulations are presented to examine the correctness of the obtained results and the effects of some parameters on dynamics of the drinking model. [ABSTRACT FROM AUTHOR]