Stability of a second-order differential equation with retarded argument

Effective criteria for exponential stability of second-order linear equations based on the positivity of the Cauchy function are proposed. Bounds of the 'smallness' of the deviation of the argument are established under which the equation with retarded argument is still stable if the ordinary differential equation with the same coefficients is stable. Based on the criteria obtained, an applied problem on the regulation of concentration is analyzed     

一类非线性时滞双曲型偏微分方程的振动性

研究一类非线性时滞双曲型偏泛函微分方程解的振动性，利用微分不等式方法和广义Riccati变换，获得了该类方程在第一类边值条件下振动的新的充分条件，所得结果通过实例加以阐明．     

Periodic solutions in systems of piecewise- linear differential equations

Motivated by the periodic behaviour of regulatory networks within cell biology and neurology, we have studied the periodic solutions of piecewise-linear, first- order differential equations with identical relative decay rates. The flow of the solution trajectories can be represented qualitatively by a directed graph. By examining the cycles in this graph and solving the eigenvalue problem for corresponding mapping matrices, all closed, period-1 orbits can be found by analytical means. Theorems about their exist- ence, stabiiiiy and uniqueness are derived. For three-dimensional systems, the basins of attraction of the limit cycles can be explicitly determined and it is shown that higher periodic and chaotic solutions do not exist.     

Exact solutions of the Boeder differential equation for macromolecular orientations in a flowing liquid

The Boeder differential equation is solved in this work over a wide range of , yielding the probability density functions (PDF), that describe the average orientations of rod-like macromolecules in a flowing liquid. The quantity is the ratio of the hydrodynamic shear rate to the rotational diffusion coefficient. It characterises the coupling of the motion of the macromolecules in the hydrodynamic flow to their thermal diffusion. Previous analytical work is limited to approximate solutions for small values of . Special analytical as well as numerical methods are developed in the present work in order to calculate accurately the PDF for a range of covering several orders of magnitude, 10⁻⁶10⁸. The mathematical nature of the differential equation is revealed as a singular perturbation problem when becomes large. Scaling results are obtained over the differential equation for 10³. Monte Carlo Brownian simulations are also constructed and shown to agree with the numerical solutions of the differential equation in the bulk of the flowing liquid, for an extensive range of . This confirms the solidity of the developed analytical and numerical methods.     

A comparison of integral and differential equation solutions for field problems

Field computation in three dimensions is necessarily expensive. To minimise the cost and increase the reliability of the results it is important to use well behaved formulations. Several integral equation formulations are shown and their conditioning is examined, point matching and weighted residual solutions are compared. Finally the integral methods are compared with finite element solutions of partial differential equations. The applicability of the two approaches is discussed in relation to the development of new computer hardware and costs.     

Subcritical and supercritical bifurcations of the first- and second-order Benney equations

The problem of the stability threshold of thin-film dynamics as described by the Benney equation of both first and second orders is revisited. The main result is that the primary Hopf bifurcation of the Benney equation of first order is supercritical for smaller values of Reynolds number and subcritical for its larger values. This result is numerically validated and further investigated analytically to reveal coexisting stable and unstable traveling waves. However, the primary bifurcation of the second-order Benney equation is supercritical for any Reynolds numbers. Sideband instability of traveling-wave regimes whose amplitude and frequency arise from the corresponding complex Ginzburg-Landau equation (CGLE) is found for the Benney equation of both first and second orders.     

The optimization of a functional within the class of solutions of a partial differential equation

Summary This paper deals with a class of variational problems involving multiple integrals with one unknown function. Else than in the classical calculus of variations where the unknown function e.g. must be continuous and must take fixed values on the boundary, the unknown function must be a solution of a partial differential equation.Physically one could imagine a process, described by a partial differential equation and controlled by the boundary conditions, while this process must be optimized in some sense by choosing the best boundary values.     

余维4的Duffing-Van der Pol方程全局分岔分析

本文采用级数展开形式的Melnikov函数解决高余维分岔问题。通过研究一类5次项和3次项共存,具有异宿轨的Duffing-Van der Pol方程的余维4全局分岔问题,得到了该系统的分岔方程及全局拓扑结构,说明了该方法的可行性。研究结果表明,该系统有单个极限环、单个异宿轨、异宿轨和极限环共存、两个极限环共存等情况。最后通过数值模拟验正了理论分析结果的正确性。     

Two results on stable rapidly oscillating periodic solutions of delay differential equations

In 1999 Ivanov and Losson [A.F. Ivanov and J. Losson, Stable rapidly oscillating solutions in delay differential equations with negative feedback, Differ. Int. Eqns 12 (1999), pp. 811–832] presented a computer assisted proof that a particular delay differential equation (with negative feedback) admits a stable rapidly oscillating periodic solution (ROPS). In this article the delay equation of Ivanov and Losson is embedded in a five-parametric class of differential equations. Conditions on the parameters are given such that the delay equation admits a stable ROPS. Moreover, it is shown that for odd n?>?1 the delay equation admits a stable ROPS with n humps per unit time if the parameters satisfy some explicitly given conditions. The delay equation of Ivanov and Losson satisfies all conditions on the five parameters. This gives an analytic proof and a considerable generalization of the result of Ivanov and Losson. The conditions on the parameters are believed to be sharp in a certain sense. The second result proves part of a conjecture in Stoffer [D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Differ. Eqns 20(1) (2008), pp. 201–238]. For a class of stiff delay differential equations with piecewise constant nonlinearity (positive or negative feedback) and for every n the following holds: if the stiffness parameter is sufficiently large then there are 2a(n) essentially different stable ROPSs with n humps per time unit. a(n) is the number of essentially different binary n-stage shift register sequences.     

Asymptotic stability of a stochastic delay equation

The investigation of a nonlinear stochastic delay equation with structural tool and regenerative time delays is presented. The conditions of Hopf bifurcation are computed in order to describe the regions of stability and instability. Explicit expressions characterizing the influence of nonlinear and stochastic perturbations, valid in the first order centre manifold approximation, are derived. In addition to this, we describe the underlying mathematical ideas of the centre manifold reduction of delay differential equations to ordinary differential equations for fixed time delays.     

Oscillatory and asymptotic behaviour of a multiplicative delay logistic equation

Sufficient conditions are obtained for the oscillation of all positive solutions of about the positive equilibrium K and for the global attractivity of the positive equilibrium     

电动力绳索系统的周期解求解及其稳定性分析

摘 要：对运行在倾斜圆轨道上的电动力绳索系统的动力学特性进行了分析研究。首先建立了系统的动力学模型,分别采用摄动法及推广后的数值算法求得系统的基本周期解,并运用所给数值算法中的稳定性判据分析了周期解的稳定性,得出该系统周期运动不稳定的结论。最后进行仿真验证,结果表明在摄动量较小时,两种求解算法得到的周期解基本相同,但当摄动量较大时,摄动法求得的周期解发生了畸变,不理想此时通常借助数值算法加以求解;仿真结果同样证实了所得周期解的不稳定特性。     

Periodic oscillations in a model of the glucose-insulin interaction with delay and periodic forcing

Using degree theoretic methods, the existence of periodic solutions is established for a two-dimensional delay model representing the glucose-insulin interaction in a person with reduced insulin secretion and receiving oscillatory infusions of both insulin and glucose. For the case when the parameters are time-independent, the possibility of a periodic solution is investigated by a Hopf bifurcation analysis. The results of some numerical simulations are also presented and discussed.     

Smoothness loss of periodic solutions of a neutral functional differential equation: on a bifurcation of the essential spectrum

The linearized Poincare operator of a periodic solution of a neutral functional differential equation is, unlike the situation for retarded functional differential equations, no longer a compact operator. It has both a point and an essential spectrum. In the existing theory one commonly requires that the essential spectrum should be inside the unit circle and bounded away from it. However, during continuation the essential spectrum may move and approach the unit circle, causing a bifurcation that is inherently infinite-dimensional in nature since it involves an infinite number of eigenmodes. In this paper we analyse a specific system with such a bifurcation. We prove its existence and show that the smoothness of the corresponding branch of periodic solutions is lost beyond the bifurcation point.     

Approximate solutions to a nonlinear diffusion equation

Approximate similarity solutions to the porous-medium equation, c ₁ = · (c ^m c), are obtained in one and two dimensions. The problems considered arise in the modelling of dopant diffusion in semiconductors, the two-dimentional problems corresponding to diffusion under a mask edge.     

New X-wave solutions of free-space scalar wave equation and their finite size realization

Based on the method proposed by Donnelly and Ziolkowski [1], [2], a new general solution has been obtained for the isotropic/homogeneous scalar wave equation in cylindrical coordinates. It is shown that well-known limited diffraction beams such as Durnin's Bessel beams [4], Lu and Greenleaf's X-wave [15], localized waves of Donnelly and Ziolkowski [1], [2], and limited-diffraction, band-limited waves of Li and Bharath [19], [20] can be obtained from this generic solution as particular cases. In addition, we have obtained new X-wave solutions and have calculated the field characteristics for one of them using a finite aperture realization. It is shown that with a proper choice of the free parameter values, well-behaved X-waves with narrow beamwidths and large depths of field can be achieved. For similar source spectra, the results are compared with Lu and Greenleaf's zeroth-order X-wave, and it is shown that the depth of field and beamwidth are very comparable.     

Linear stability criteria in a reaction-diffusion equation with spatially inhomogeneous delay

Time delays play an important role in many biological and ecological systems. However, they are usually incorporated into mathematical models in a way not explicitly dependent on space, even though the delay may be modelling some environmental aspect of the system. In this paper we study a scalar reaction-diffusion equation with a spatially inhomogeneous delay, taken for simplicity in the form of a step function of the spatial coordinate. We derive the dispersion relation from which analytical results on the stability or instability of the uniform steady states can be determined. We confirm and extended these results by numerical simulations which confirm the possibility of qualitatively different types of behaviour on different parts of the spatial domain.