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### Open Access Journals

DCDS

For the near-Hamiltonian system $\dot{x}=y+\varepsilon
P(x,y),\dot{y}=x-x^2+\varepsilon Q(x,y)$, where $P$ and $Q$ are
polynomials of $x,y$ having degree 3 with varying coefficients we
obtain 5 limit cycles.

PROC

In this paper, we are mainly concerned with the second order difference equations with advanced argument and give sufficient conditions for their solutions to be oscillatory.

DCDS

This paper is concerned with bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems.
By analyzing the multiplicities of the zeroes of the slow divergence integrals and their complete unfolding, the
upper bounds of canard limit cycles bifurcating from the suitable limit periodic sets through respectively
the generic Hopf breaking mechanism, the generic jump breaking mechanism and a succession of the Hopf and jump mechanisms
in these polynomial Liénard systems are obtained.

DCDS

In this paper, we investigate a class of quasi-homogeneous polynomial systems with a given weight degree. Firstly, by some analytical skills, several properties about this kind of systems are derived and an algorithm can be established to obtain all possible explicit systems for a given weight degree. Then, we focus on center problems for such systems and provide some necessary conditions for the existence of centers. Finally, for a specific quasi-homogeneous polynomial system, we characterize its center and prove that the center is not isochronous.

MBE

In this paper we outline some methods of finding limit cycles for planar autonomous systems with small parameter perturbations. Three ways of studying Hopf bifurcations and the method of Melnikov functions in studying Poincaré bifurcations are introduced briefly. A new method of stability-changing in studying homoclinic bifurcation is described along with some interesting applications to polynomial systems.

DCDS

We consider a family of scalar periodic equations with a parameter and establish theory of rotated equations, studying the behavior of periodic solutions with the change of the parameter. It is shown that a stable (completely unstable) periodic solution of a rotated equation varies monotonically with respect to the parameter and a semi-stable periodic solution splits into two periodic solutions or disappears as the parameter changes in one direction or another. As an application of the obtained results, we give a further study of a piecewise smooth population model verifying the existence of saddle-node bifurcation.

CPAA

The Liénard system and its generalizations are important models of nonlinear oscillators. We study small-amplitude limit cycles of two families of
Liénard systems and find exact number of such limit cycles bifurcating from
a center or focus at the origin for these families, thus obtaining the
precise bound for cyclicity of the families.

DCDS-B

In this paper, we consider a predator-prey system with stage structure
and mutual interference. By analyzing the characteristic equations,
we study the local stability of the interior equilibrium of the system. Using an iterative method,
we investigate the global stability of this equilibrium.

keywords:
global attractivity.
,
mutual interference
,
local stability
,
Predator-prey
,
stage structure

MBE

Recently an SIS epidemic reaction-diffusion model with Neumann (or no-flux) boundary condition has been proposed and studied by several authors to understand the dynamics of disease transmission in a spatially heterogeneous environment in which the individuals are subject to a random movement. Many important and interesting properties have been obtained: such as the role of diffusion coefficients in defining the reproductive number; the global stability of disease-free equilibrium; the existence and uniqueness of a positive endemic steady; global stability of endemic steady for some particular cases; and the asymptotical profiles of the endemic steady states as the diffusion coefficient for susceptible individuals is sufficiently small. In this research we will study two modified SIS diffusion models with the Dirichlet boundary condition that reflects a hostile environment in the boundary. The reproductive number is defined which plays an essential role in determining whether the disease will extinct or persist. We have showed that the disease will die out when the reproductive number is less than one and that the endemic equilibrium occurs when the reproductive number is exceeds one. Partial result on the global stability of the endemic equilibrium is also obtained.

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