Hopf Bifurcation and Stability of Periodic Solutions of Differential-difference and Integro-differential Equations

Periodic solutions bifurcating from a steady state of differential-differenceand integrodifferential equations are studied. An algorithmfor determining the asymptotic orbital stability, directionof bifurcation, period, and asymptotic form of these solutionsis presented. This algorithm is applied to the Hutchinson-Wrightequation and to an equation with two delays. Finally some integro-differentialequations, modelling two and three trophic level competition,introduced by May, are discussed. Some numerical work supportingthe theory is described.     

二阶脉冲积微分方程周期过值问题的解(英)

利用上下解方法和不动点定理，给出了二阶混合型脉冲积微分方程周期边值问题解的存在性．     

带有无界时滞的积分微分方程的正解

我们得到一类带有无界时滞的积分微分方程存在正解的必要充分条件.     

On the Existence and Global Bifurcation of Periodic Solutions to Planar Differential Delay Equations

This paper is concerned with periodic solutions to one-parameter families of planar differential delay equations. The concept of slowly oscillating periodic solution is extended to this setting and we state the existence of an unbounded continuum of such solutions.     

一类半线性积分微分方程初边值问题的爆破解和全局解

本文研究初边值问题的爆破解和全局解，证明了在ｆ的凸性假设和一定的增长性假定下解在有限时刻爆破，而在ｆ的其他假设下证明了全局解的存在性。     

Stability of Periodic Solutions of Ordinary Differential Equations in Banach Spaces

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On Stability of Solutions to Quasilinear Periodic Systems of Differential Equations

We consider a quasilinear system of differential equations with periodic coefficients in the linear terms. We obtain estimates for the attraction domain of the zero solution and establish estimates for the decay rate of solutions at infinity. The results are stated in terms of the integrals of the norm of a periodic solution to the Lyapunov differential equation.     

Periodic Solutions For a Kind of Neutral Rayleigh Equations With Variable Parameter

Employing the coincidence degree theory of Mawhin we obtain some existence results of periodic solutions for a type of neutral Rayleigh equation with variable parameter $$((x(t) - c(t)x(t - \tau))'' + f(x'(t)) + g(x(t - \gamma(t))) = e(t).$$ It is worth noting that c(t) is no longer a constant which is different from the corresponding ones of past work. Furthermore, our results generalize corresponding work in the past.     

Uniformly Asymptotic Stability and Existence of Almost Periodic Solutions for Functional Differential Equations

Ｉｔｉｓｗｅｌｋｎｏｗｎｔｈａｔｆｏｒａｎａｌｍｏｓｔｐｅｒｉｏｄｉｃｏｒｄｉｎａｒｙｄｉｆｅｒｅｎｔｉａｌｓｙｓｔｅｍｔｈｅｅｘｉｓｔｅｎｃｅｏｆａｕｎｉｆｏｒｍｌｙａｓｙｍｐｔｏｔｉｃａｌｙｓｔａｂｌｅｂｏｕｎｄｅｄｓｏｌｕｔｉｏｎｉｍｐｌｉｅｓ...     

Asymptotic Stability of Solutions to Perturbed Linear Difference Equations with Periodic Coefficients

We consider perturbed linear systems of difference equations with periodic coefficients. The zero solution of a nonperturbed system is assumed asymptotically stable, i.e., all eigenvalues of the monodromy matrix belong to the unit disk {||<1}. We obtain conditions on the perturbation of this system under which the zero solution of the system is asymptotically stable and also establish continuous dependence of one class of numeric characteristics of asymptotic stability of solutions on the coefficients of the system.     

Lienard方程的周期解

主要利用Leray-Schauder不动点理论研究Lienard方程周期边值问题x f(x)x g(t,x)=e(t)x(0)=x(T),x(0)=x(T)的正解及多个正解的存在性.     

一类微分差分方程的周期解

本文研究微分差分方程ｘ＇（ｔ）＝－ｆ（ｘ（ｔ），ｘ（ｔ－τ１））－ｆ（ｘ（ｔ），ｘ（ｔ－τ２））－．．．－ｆ（ｘ（ｔ），ｘ（ｔ－τｎ））非平凡周期解的存在性问题，得到了一些判别准则，推广和改进了文［１－４］的工作。     

Stable Periodic Solutions of Periodic Systems of Differential Equations

An infinitely differentiable periodic two-dimensional system of differential equations is considered. It is assumed that there is a hyperbolic periodic solution and there exists a homoclinic solution to the periodic solution. It is shown that, for a certain type of tangency of the stable manifold and unstable manifold, any neighborhood of the nontransversal homoclinic solution contains a countable set of stable periodic solutions such that their characteristic exponents are separated from zero.     

Periodic Solutions of Linear Integro-Differential Equations

Using a degree-theoretic result of Granas, a homotopy is constructed enabling us to show that if there is an a priori bound on all possible T-periodic solutions of a Volterra equation, then there is a T-periodic solution. The a priori bound is established by means of a Liapunov functional. The latter result is unusual in that no bounds on the Liapunov functional are required. Thus, in addition to the periodic solution, the equation may have both bounded and unbounded Solutions.     

The Asymptotic Property of Solutions to a Class of Integrodifferential Equations

The asymptotic property of solutions to a class of integrodifferential systems containing a small parameter singularly is studied.     

Periodic Solutions for Nonlinear Diferential Equations

ＰｅｒｉｏｄｉｃＳｏｌｕｔｉｏｎｓｆｏｒＮｏｎｌｉｎｅａｒＤｉｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎｓＣｏｎｇＦｕｚｈｏｎｇ（从福仲）（ＯｆｆｉｃｅｏｆＭａｔｈｅｍａｔｉｃｓ，８６００３Ｕｎｉｔ，Ｃｈａｎｇｃｈｕｎ，１３００２２）ＭａｏＤｏｎｇｍｉｎｇ（...     

Solitary and Periodic Solutions of Nonlinear Nonintegrable Equations

The singular manifold method and partial fraction decomposition allow one to find some special solutions of nonintegrable partial differential equations (PDE) in the form of solitary waves, traveling wave fronts, and periodic pulse trains. The truncated Painlevé expansion is used to reduce a nonlinear PDE to a multilinear form. Some special solutions of the latter equation represent solitary waves and traveling wave fronts of the original PDE. The partial fraction decomposition is used to obtain a periodic wave train solution as an infinite superposition of the "corrected" solitary waves.