Nonlinear Delay Differential Equations Research Articles

New criteria for mean square exponential stability of stochastic delay differential equations

By a novel approach, we present several new criteria for the mean square exponential stability of general non-linear stochastic delay differential equations. Both delay-independent and delay-dependent stability criteria are provided. A discussion of the obtained results is given.

Oscillation criteria for higher-order neutral type difference equations

In this paper, oscillation criteria are obtained for higher-order neutral-type nonlinear delay difference equations of the form ∆ rn ∆k−1 yn + pnyτn + qnf yσn = 0, n ≥ n0, 0.1 where rn, pn, qn ∈ [n0, ∞ , rn > 0, qn > 0; 0 ≤ pn ≤ p0 0; τσ = στ ; f u u ≥ m > 0 for u = 0. Moreover, we provide some examples to illustrate our main results.

A Study of Nonlinear Behavior of Flexible Tool and Workpiece in Turning Operation With Regenerative Effect Under Internal and Primary Resonance Conditions

Abstract In this work, the nonlinear dynamic behavior of turning operation has been studied considering flexible tool and thin cylindrical workpiece. The system is taken as a two-degree-of-freedom system with nonlinear stiffness and is subjected to self-excited vibration because of the regenerative effect. The cutting force is considered to be a combination of a constant and periodic force in addition to the force due to the regenerative effect. The regenerative effect during turning operation is included in the mathematical model, resulting in a nonlinear delay differential equation. Here, the natural frequency of the workpiece is assumed to be closed to that of the tool, leading to 1:1 internal resonance. Further, the frequency of the time-varying cutting force is assumed to be closed to the natural frequency of the workpiece giving rise to primary resonance condition. The nonlinear responses and the stability of the tool and the workpiece have been determined using a higher-order method of multiple scales (MMS) under internal and primary resonance conditions. The solution of the equation of motion using the MMS is validated by comparing the solution obtained using the numerical method. The effect of the tool and workpiece stiffness nonlinearities on steady-state frequency responses and stability is investigated, and system parameters are also identified to have stable turning operation. This work will find applications in estimating the system parameters for chatter-free turning operation with a flexible tool and workpiece when their dynamic compliances are comparable.

Delay in ATP-dependent calcium inflow may affect insulin secretion from pancreatic beta-cell

Any dysfunction in the secretion of insulin and/or in its use is critical in the development of type 2 diabetes. Calcium plays a very important role in the insulin secretion mechanism from beta-cell. In this paper, we studied the dynamic interaction of glucose-induced insulin secretion mechanism through glucose metabolism and ATP-dependent calcium influx. We proposed and analyzed a four dimensional system of nonlinear delay differential equations to give insights into different possible mechanisms for maintaining plasma glucose homeostasis through calcium-induced insulin secretion. The time duration between ATP formation and subsequent calcium influx through cell membrane has been found to be critical in maintaining insulin homeostasis. How variational effects of system parameters lead to various glycemic states are demonstrated. Possibility of new potential drug therapies is also discussed through parameter recalibration.

Oscillatory behavior of $n$-th order nonlinear delay differential equations with a nonpositive neutral term

We study the oscillation problem for solutions of a class of $n$-th order nonlinear delay differential equations with nonpositive neutral terms. The obtained results improve and correlate many of the known oscillation criteria in the literature for neutral and non-neutral equations.

Steady-state periodic solutions of the nonlinear wave propagation problem of a one-dimensional lattice using a new methodology with an incremental harmonic balance method that handles time delays

A new methodology is developed in this work to solve a one-dimensional (1D) nonlinear wave propagation problem. In its response, the space variable is converted to time functions at different locations, and the resultant time functions are merged with the time variable. Since the merged variables are essentially time-delay variables, the main point of the methodology is that the governing equation of the nonlinear wave propagation problem can be converted to a corresponding nonlinear delay differential equation (DDE) with multiple delays. The new methodology is shown how to formulate the converted DDE, and a modified incremental harmonic balance method is used to solve the 1D nonlinear DDE by introducing a delay matrix operator, where a formula of the Jacobian matrix is derived and can be efficiently and automatically used in the Newton method. This new methodology is demonstrated by solving examples of 1D monatomic chains under the nonlinear Hertzian contact law and with cubic springs. Results well match those in previous works, but derivation effort of solutions and calculation time can be significantly reduced here. When the algorithm of the methodology is compiled as a computer program, there is no additional derivation required to solve wave propagation problems associated with different governing equations.

Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations

In this paper, the authors studied oscillatory behavior of solutions of fourth-order delay difference equation Δc3nΔc2nΔc1nΔun+pnfun−k=0 under the conditions ∑n=n0∞cin<∞, i=1, 2, 3. New oscillation criteria have been obtained which greatly reduce the number of conditions required for the studied equation. Some examples are presented to show the strength and applicability of the main results.

New oscillation criteria for nonlinear delay differential equations of fourth-order

By using a generalized Riccati substitution, we present a new criterion for oscillation of solutions of fourth-order quasi-linear differential equations (r(t)(x‴(t))α)′ + f(t,x(σ(t)))=0, in noncanonical case ∫∞r−1/α(s)ds < ∞. We utilize an approach that gives rise to two or three independent conditions, eliminating non-oscillatory solutions. Moreover, we establish conditions in a non-traditional form (limsup(·)>1), while condition (limsup(·)=∞) cannot be applied. Our results essentially improve and complement a number of related ones. Examples are provided to illustrate the results.

Approximate analytical solution of the linear and nonlinear multi-pantograph delay differential equations

In this paper, a combination of Laplace transform and variational iteration method are applied to get an approximate analytic solution for the multi-pantograph delay with higher order differential equations. Lagrange multiplier technique is constructed a correction functional which obtained by using Laplace transform with the variational theory. Numerical studies for the application of the present method for the considered problems are given and graphically illustrated, our proposed method is compared favorably with other methods. The simplicity and efficiency of the method.

On a Predator–Prey System with Digestion Delay and Anti-predation Strategy

Predator–prey interactions are among the most complicated interactions between biological species, in which there may be both direct effect (through predation) and indirect effect (e.g., fear effect). In the literature, the indirect effect has been largely missing in predator–prey models, until some recent works. Based on the recent work (Wang et al. in J Math Biol 73:1179–1204, 2016) where a fear effect is considered in an ODE model as a cost, in this paper, we also consider a benefit from the anti-predation response in addition to the cost, as well as a time delay in the transfer of biomass from the prey to the predator after predation. This results in a system of delay differential equations (DDEs). By analyzing this nonlinear DDE system, we obtain some insights on how the anti-predation response level (indirect effect) and the biomass transfer delay jointly affect the population dynamics; particularly we show how the nonlinearity in the predation term mediated by the fear effect affects the long term dynamics of the model system. We also perform some numerical computations and simulations to demonstrate our results. These results seem to suggest a need to revisit existing predator–prey models in the literature by incorporating the indirect effect and biomass transfer delay.

Effect of two forms of feedback on the performance of the Rate Control Protocol (RCP)

The Rate Control Protocol (RCP) uses explicit feedback from routers to control network congestion. RCP estimates it’s fair rate from two forms of feedback: rate mismatch and queue size. An important design question that remains open in RCP is whether the presence of queue size feedback is helpful, given the presence of feedback from rate mismatch. The feedback from routers to end-systems is time-delayed, and may introduce instabilities and complex nonlinear dynamics. Delay dynamical systems are often modeled using delay differential equations to facilitate a mathematical analysis of their performance and dynamics. The RCP models with and without queue size feedback give rise to two distinct nonlinear delay differential equations. Earlier work on this design question was based on methods of linear systems theory. For further progress, it is quite natural to employ nonlinear techniques. In this study, we approach this design question using tools from control and bifurcation theory. The analytical results reveal that the removal of queue feedback could enhance both stability and convergence properties. Further, using Poincaré normal forms and center manifold theory, we investigate two nonlinear properties, namely, the type of the Hopf bifurcation and the asymptotic stability of the bifurcating limit cycles. We show that the presence of queue feedback in the RCP can lead to a sub-critical Hopf bifurcation, which would give rise either to the onset of large amplitude limit cycles or to unstable limit cycles. Whereas, in the absence of queue feedback, the Hopf bifurcation is always super-critical, and the bifurcating limit cycles are stable. The analysis is complemented with computations and some packet-level simulations as well. In terms of design, our study suggests that the presence of both forms of feedback may be detrimental to the performance of RCP.

A New Convergent Pseudospectral Method for Delay Differential Equations

Delay differential equations have a wide range of applications in science and engineering. When these equations are nonlinear, we cannot usually obtain an exact solution. Hence, we must utilize an efficient numerical method with high convergence rate and low error to approximate the solution. In this paper, we propose a new shifted pseudospectral method to solve nonlinear delay differential equations. First, we convert the problem into an equivalent continuous-time optimization problem and then use a pseudospectral method to discretize the problem. By solving the obtained discrete-time problem, we achieve an approximate solution for the main delay differential equation. Here, we analyze the convergence of the method and solve some practical delay differential equations to show the efficiency of the method.

A new stable collocation method for solving a class of nonlinear fractional delay differential equations

In this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal bases of W^{1}_{2,0}. Error estimations of approximate solutions are given and the highest convergence order can reach four in the sense of the norm of W_{2,0}^{1}. To overcome the nonlinear condition, we make use of Newton’s method to transform the nonlinear equation into a sequence of linear equations. For the linear equations, a rigorous theory is given for obtaining their ε-approximate solutions by solving a system of equations or searching the minimum value. Stability analysis is also obtained. Some examples are discussed to illustrate the efficiency of the proposed method.

Effect of nonlinear cladding stiffness on the stability and Hopf bifurcation of a heat-exchanger tube subject to cross-flow

The linear stability of a heat-exchanger tube modeled as a single-span cantilever beam subjected to cross-flow has been studied with two parameters: (1) varying stiffness of the baffle-cladding at the free end and (2) varying flow velocity. A mathematical model incorporating the motion-dependent fluid forces acting on the beam is developed using the Euler–Bernoulli beam theory, under the inextensible condition. The partial delay differential equation governing the dynamics of the continuous system is discretized to a set of finite, nonlinear delay differential equations through a Galerkin method in which a single mode is considered. Unstable regions in the parametric space of dimensionless cladding stiffness and flow velocity are identified, along with the magnitude of damping in the stable region. This information can be used to determine the cladding stiffness at which the system should be operated to achieve maximum damping at a known operational flow velocity. Furthermore, the system is found to lose stability by Hopf bifurcation and the method of multiple scales is used to analyze its post-instability behavior. Stable and unstable limit cycles are observed for different values of the linear component of the dimensionless cladding stiffness. A global bifurcation analysis indicates that the number of limit cycles decreases with increasing linear cladding stiffness. An optimal range for the linear cladding stiffness is recommended where tube vibrations would either diminish to zero or assume a relatively low amplitude associated with a stable limit cycle.

Oscillatory behavior for nonlinear delay differential equation with several non-monotone arguments

This paper is devoted to obtaining some new sufficient conditions for the oscillation of all solutions of first order nonlinear differential equations with several deviating arguments. Finally, an illustrative example related to our results is given.

Oscillatory behavior of second order nonlinear delay differential equations with positive and negative neutral terms

Oscillatory behavior of second order nonlinear delay differential equations with positive and negative neutral terms

Supercritical and Subcritical Hopf Bifurcations in a Delay Differential Equation Model of a Heat-Exchanger Tube Under Cross-Flow

Abstract Nonlinear vibrations of a heat-exchanger tube modeled as a simply supported Euler–Bernoulli beam under axial load and cross-flow have been studied. The compressive axial loads are a consequence of thermal expansion, and tensile axial loads can be induced by design (prestress). The fluid forces are represented using an added mass, damping, and a time-delayed displacement term. Due to the presence of the time-delayed term, the equation governing the dynamics of the tube becomes a partial delay differential equation (PDDE). Using the modal-expansion procedure, the PDDE is converted into a nonlinear delay differential equation (DDE). The fixed points (zero and buckled equilibria) of the nonlinear DDE are found, and their linear stability is analyzed. It is found that stability can be lost via either supercritical or subcritical Hopf bifurcation. Using Galerkin approximations, the characteristic roots (spectrum) of the DDE are found and reported in the parametric space of fluid velocity and axial load. Furthermore, the stability chart obtained from the Galerkin approximations is compared with the critical curves obtained from analytical calculations. Next, the method of multiple scales (MMS) is used to derive the normal-form equations near the supercritical and subcritical Hopf bifurcation points for both zero and buckled equilibrium configurations. The steady-state amplitude response equation, obtained from the MMS, at Hopf bifurcation points is compared with the numerical solution. The coexistence of multiple limit cycles in the parametric space is found, and has implications in the fatigue life calculations of the heat-exchanger tubes.

On oscillatory second-order nonlinear delay differential equations of neutral type

On oscillatory second-order nonlinear delay differential equations of neutral type

Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay

This paper generalizes a previously conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipliers. As a consequence, it is shown how critical points on the constraint manifold can be found through several stages of continuation along a sequence of connected one-dimensional manifolds of solutions to increasing subsets of the necessary optimality conditions. Due to the presence of delayed and advanced arguments in the original and adjoint differential equations, care must be taken to determine the degree of smoothness of the Lagrange multipliers with respect to time. Such considerations naturally lead to a formulation in terms of multi-segment boundary-value problems (BVPs), including the possibility that the number of segments may change, or that their order may permute, during continuation. The methodology is illustrated using the software package coco on periodic orbits of both linear and nonlinear delay-differential equations, keeping in mind that closed-form solutions are not typically available even in the linear case. Finally, we demonstrate optimization on a family of quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with delay and parametric forcing. The quasiperiodic case is a further original contribution to the literature on optimization constrained by partial differential BVPs.

Some results of certain nonlinear delay differential equation

In this paper, we investigate the non-vanishing entire solutions with finite order of nonlinear delay differential equation where , are polynomials, are meromorphic functions with growth and . Particularly, we show that the exponential polynomial solutions of equation must reduce to rather specific forms under certain proper conditions.