Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form

(x(t)+Bx(t−δ)^)′=g₁(t,x(t))+g₂(t,x(t−τ))+p(t).     

Periodic solutions for a neutral nonlinear dynamical equation on a time scale

Let T be a periodic time scale. We use a fixed point theorem due to Krasnosel'ski? to show that the nonlinear neutral dynamic system with delay

     

On the periodic solutions for both nonlinear differential and difference equations: A unified approach

A direct and unifying scheme for disclosure of periodic wave solutions of both nonlinear differential and difference equations is presented. The scheme is based on Hirota's bilinear form and certain Riemann theta function formulae. The relations between the periodic wave solutions and soliton solutions are rigorously established.     

Periodic surface nanostructures on polycrystalline ZnO induced by femtosecond laser pulses

Periodic surface nanostructures induced by femtosecond laser pulses on polycrystalline ZnO are presented. By translating the sample line-by-line under appropriate irradiation conditions, grating-like nanostructures with an average period of 160 nm are fabricated. The dependence of surface morphologies on the processing parameters, such as laser fluence, pulse number and laser polarization, are studied by scanning electronic microscope (SEM). In addition, photoluminescence (PL) analysis at room-temperature indicates that the PL intensity of the irradiated area increases significantly compared with the un-irradiated area. Using femtosecond laser pulses irradiation to fabricate periodic surface nanostructures on polycrystalline ZnO is efficient, simple and low cost, which shows great potential applications in ZnO-based optoelectronic devices.     

Periodic solutions for one dimensional wave equation with bounded nonlinearity

This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient $u(x)$ satisfies $\text{ess inf}{\eta }_u(x)>0$ with ${\eta }_u(x)=\frac{1}{2}\frac{u^{″}}{u}?\frac{1}{4}{\left(\frac{u^{\prime }}{u}\right)}^2$, which actually excludes the classical constant coefficient model. For the case ${\eta }_u(x)=0$, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form $T=\frac{2p?1}{q}$ ($p,q$ are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition $\text{ess inf}{\eta }_u(x)>0$.     

New Peakons and Periodic Peakons of the Modified Camassa-Holm Equation

In this paper, we obtain new peakon and periodic peakon solutions to a modified Camassa-Holm equation. We change the modified Camassa- Holm equation into a planar system. Then the first integral and algebraic curves of this system are obtained. By using the first integral and algebraic curves, a new peakon solution is given by hyperbolic function. Moreover, some new periodic peakons are given by elliptic functions and triangle functions.     

Hopf-zero bifurcation of a delayed predator-prey model with dormancy of predators

In this paper, We investigate Hopf-zero bifurcation with codimension 2 in a delayed predator-prey model with dormancy of predators. First we prove the specific existence condition of the coexistence equilibrium. Then we take the mortality rate and time delay as two bifurcation parameters to find the occurrence condition of Hopf-zero bifurcation in this model. Furthermore, using the Faria and Magalhases normal form method and the center manifold theory, we obtain the third order degenerate normal form with two original parameters. Finally, through theoretical analysis and numerical simulations, we give a bifurcation set and a phase diagram to show the specific relations between the normal form and the original system, and explain the coexistence phenomena of several locally stable states, such as the coexistence of multi-periodic orbits, as well as the coexistence of a locally stable equilibrium and a locally stable periodic orbit.     

Periodic solutions for a type of neutral system with variable parameters

In this paper, we firstly analyze some properties of the linear difference operator $[Ax](t)=x(t)-C(t)x(t-\tau),$ where $C(t)$ is a $n\times n$ matrix function, and then using Mawhin''s continuation theorem, a first-order neutral functional differential system is studied. Some new results on the existence and stability of periodic solutions are obtained. The results are related to the deviating arguments $\tau$ and $\mu$. Meanwhile, the approaches to estimate a $prior$ bounds of periodic solutions are different from the corresponding ones of the known literature.     

Periodic solution of a higher dimensional ecological system

The main purpose of this article is to study the periodicity of a Lotka-Volterra''s competition system with feedback controls. Some new and interesting sufficient conditions are obtained for the global existence of positive periodic solutions. Our method is base on combining matrix''s spectral theory and inequality $|x(t)|\leq x(t_{0})+\int_{0}^{\omega }|\dot{x}(t)|{\rm d}t$. Some examples and their simulations show the feasibility of our main result.