This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

We investigate the following differential equations: $-(y^{[1]}(x))\text{'}+q(x)y(x)=\lambda f(x,y(x))$, with impulsive and integral boundary conditions $-\mathrm{\Delta }(y^{[1]}(x_i))=I_i(y(x_i))$, $i=\mathrm{1,2},\dots ,m$, $y(0)-a{y}^{[1]}(0)={\int }_0^{\omega }{g}_0(s)y(s)ds$, $y(\omega )-b{y}^{[1]}(\omega )={\int }_0^{\omega }{g}_1(s)y(s)ds$, where $y^{[1]}(x)=p(x)y\text{'}(x)$. The expression of Green's function and the existence of positive solution for the system are obtained. Upper and lower bounds for positive solutions are also given. When $p(t)$, $I(·)$, $g_0(s)$, and $g_1(s)$ take different values, the system can be simplified to some forms which has been studied in the works by Guo and LakshmiKantham (1988), Guo et al. (1995), Boucherif (2009), He et al. (2011), and Atici and Guseinov (2001). Our discussion is based on the fixed point index theory in cones.

By using the Riccati transformation technique and constructing a class of Philos-type functions on time scales, we establish some new interval oscillation criteria for the second-order damped nonlinear dynamic equations with forced term of the form $(r(t)x^{\mathrm{\Delta }}(t){)}^{\mathrm{\Delta }}+p(t)x^{\mathrm{\Delta }\sigma }(t)+q(t)(x^{\sigma }(t){)}^{\alpha }=F(t,x^{\sigma }(t))$ on a time scale $𝕋$ which is unbounded, where $\alpha$ is a quotient of odd positive integer. Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.

The dynamics of a coupled optoelectronic feedback loops are investigated. Depending on the coupling parameters and the feedback strength, the system exhibits synchronized asymptotically stable equilibrium and Hopf bifurcation. Employing the center manifold theorem and normal form method introduced by Hassard et al. (1981), we give an algorithm for determining the Hopf bifurcation properties.

By means of novel analytical techniques, we have established several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale $𝕋$, that is, $(r(t)|Z^{\mathrm{\Delta }}(t){|}^{\alpha -1}{Z}^{\mathrm{\Delta }}(t){)}^{\mathrm{\Delta }}+f(t,x(\delta (t)))=0$, with respect to the case ${\int }_{t_0}^{\infty }{r}^{-1/\alpha }(s)\mathrm{\Delta }s=\infty$ and the case , where $Z(t)=x(t)+p(t)x(\tau (t))$,$\alpha$ is a constant, $|f(t,u)|\mathrm{⩾q}(t)|u^{\beta }|$, $\beta$ is a constant satisfying $\alpha \mathrm{⩾\beta }>0$, and $r$, $p$, and $q$ are real valued right-dense continuous nonnegative functions defined on $𝕋$. Noting the parameter value $\alpha$ probably unequal to $\beta$, our equation factually includes the existing models as special cases; our results are more general and have wider adaptive range than others' work in the literature.

A chaotic system arising from double-diffusive convection in a fluid layer is investigated in this paper based on the theory of dynamical systems. A five-dimensional model of chaotic system is obtained using the Galerkin truncated approximation. The results showed that the transition from steady convection to chaos via a Hopf bifurcation produced a limit cycle which may be associated with a homoclinic explosion at a slightly subcritical value of the Rayleigh number.

This paper is concerned with the higher order nonlinear neutral delay differential equation $[a(t)(x(t)+b(t)x(t-\tau ){)}^{(m)}{]}^{(n-m)}+[h(t,x(h_1(t)),\dots ,x(h_l(t))){]}^{(i)}+f(t,x(f_1(t)),\dots ,x(f_l(t)))=g(t),$ for all $t\ge t_0$. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

This paper concerns limit cycle bifurcations by perturbing a piecewise linear Hamiltonian system. We first obtain all phase portraits of the unperturbed system having at least one family of periodic orbits. By using the first-order Melnikov function of the piecewise near-Hamiltonian system, we investigate the maximal number of limit cycles that bifurcate from a global center up to first order of $\epsilon$.

This paper deals with the existence and iteration of positive solutions for nonlinear second-order impulsive integral boundary value problems with $p$-Laplacian on infinite intervals. Our approach is based on the monotone iterative technique.

In this paper, a two-species nonautonomous stochastic mutualism system is investigated. The intrinsic growth rates of the two species at time $t$ are estimated by $r_i\left(t\right)+{\sigma }_i\left(t\right){\dot{B}}_i(t),\hspace{0.17em}\hspace{0.17em}i=\mathrm{1,2,}$ respectively. Viewing the different intensities of the noises ${\sigma }_i(t)$, $i=1,2$ as two parameters at time $t$, we conclude that there exists a global positive solution and the $p$th moment of the solution is bounded. We also show that the system is permanent, including stochastic permanence, persistence in mean, and asymptotic boundedness in time average. Besides, we show that the large white noise will make the system nonpersistent. Finally, we establish sufficient criteria for the global attractivity of the system.

We establish a new existence result on homoclinic solutions for a second-order nonperiodic Hamiltonian systems. This homoclinic solution is obtained as a limit of solutions of a certain sequence of nil-boundary value problems which are obtained by the minimax methods. Some recent results in the literature are generalized and extended.

We study the existence of a positive periodic solution for second-order singular semipositone differential equation by a nonlinear alternative principle of Leray-Schauder. Truncation plays an important role in the analysis of the uniform positive lower bound for all the solutions of the equation. Recent results in the literature (Chu et al., 2010) are generalized.