First-order Nonlinear Delay Differential Equation Research Articles

Oscillation tests for nonlinear differential equations with nonmonotone delays

In this paper, our aim is to investigate a class of first-order nonlinear delay differential equations with several deviating arguments. In addition, we present some sufficient conditions for the oscillatory solutions of these equations. Differing from other studies in the literature, delay terms are not necessarily monotone. Finally, we give examples to demonstrate the results.

Hyers-Ulam Stability of First-Order Non-Linear Delay Differential Equations with Fractional Integrable Impulses

This paper proves the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of first-order non-linear delay differential equations with fractional integrable impulses. Our approach uses abstract Gronwall lemma together with integral inequality of Gronwall type for piecewise continuous functions

Oscillatory and monotone solutions of first-order nonlinear delay differential equations

Abstract For first order nonlinear delay differential equations, necessary and sufficient conditions are established for the oscillation of all proper solutions as well as for the existence of at least one vanishing at infinity proper Kneser solution.

Numerical oscillations for first-order nonlinear delay differential equations in a hematopoiesis model

In this paper, we consider the oscillations of numerical solutions for the nonlinear delay differential equations in a hematopoiesis model. Using two θ-methods, namely the linear θ-method and the one-leg θ-method, several conditions, under which the numerical solutions oscillate, are obtained. Moreover, it is proved that every non-oscillatory numerical solution tends to an equilibrium point of the original system. Some numerical experiments are provided to support the theoretical results. MSC:65L05, 65L20.

Chaotic dynamics of a frequency-modulated microwave oscillator with time-delayed feedback

We present a chaotic frequency-modulated microwave source that is governed by a simple, first-order nonlinear delay differential equation. When a sinusoidal nonlinearity is incorporated, the dynamical behaviors range from fixed-point to periodic to chaotic, depending on the feedback strength. When the sinusoidal nonlinearity is replaced by a binary nonlinearity, the system exhibits a complex periodic attractor with no fixed-point solution.

Bifurcating periodic solution for a class of first-order nonlinear delay differential equation after Hopf bifurcation

In this paper, we predict the accurate bifurcating periodic solution for a general class of first-order nonlinear delay differential equation with reflectional symmetry by constructing an approximate technique, named residue harmonic balance. This technique combines the features of the homotopy concept with harmonic balance which leads to easy computation and gives accurate prediction on the periodic solution to the desired accuracy. The zeroth-order solution using just one Fourier term is applied by solving a set of nonlinear algebraic equations containing the delay term. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. It is shown that the solutions are valid for a wide range of variation of the parameters by two examples. The second-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration. Moreover, the residue harmonic balance method works not only in determining the amplitude but also the frequency of the bifurcating periodic solution. The method can be easily extended to other delay differential equations.

Local Hopf bifurcation and global existence of periodic solutions in TCP system

This paper investigates the dynamics of a TCP system described by a first-order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the positive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifurcating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu (Wu, J. H. Symmetric functional differential equations and neural networks with memory.

Local Hopf bifurcation and global existence of periodic solutions in a kind of physiological system

The dynamics of a physiological control systems described by a first-order nonlinear delay differential equations are investigated. we proved that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838].

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.

Oscillation for First-Order Nonlinear Delay Differential Equations

Almost sharp oscillation and nonoscillation criteria are obtained for the first-order nonlinear delay differential equationx′(t)+p(t)∏j=1m|x(t−τj)|αjsign[x(t−τ1)]=0,t≥t0,where ∑j=1mαj>1. Some applications are given.