On the number of limit cycles of a cubic Near-Hamiltonian system
Junmin Yang Maoan Han
Discrete & Continuous Dynamical Systems - A 2009, 24(3): 827-840 doi: 10.3934/dcds.2009.24.827
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.
keywords: homoclinic loop. limit cycle polynomial
Oscillation of second order difference equations with advanced argument
Bi Ping Maoan Han
Conference Publications 2003, 2003(Special): 108-112 doi: 10.3934/proc.2003.2003.108
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.
keywords: oscillation Advanced argument difference equation.
Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems
Jianhe Shen Maoan Han
Discrete & Continuous Dynamical Systems - A 2013, 33(7): 3085-3108 doi: 10.3934/dcds.2013.33.3085
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.
keywords: Polynomial Liénard systems Hopf and jump mechanisms. limit periodic set slow divergence integral bifurcation of limit cycles
Planar quasi-homogeneous polynomial systems with a given weight degree
Yanqin Xiong Maoan Han
Discrete & Continuous Dynamical Systems - A 2016, 36(7): 4015-4025 doi: 10.3934/dcds.2016.36.4015
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.
keywords: Quasi-homogeneous polynomial system weight degree isochronous center.
Some bifurcation methods of finding limit cycles
Maoan Han Tonghua Zhang
Mathematical Biosciences & Engineering 2006, 3(1): 67-77 doi: 10.3934/mbe.2006.3.67
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.
keywords: Melnikov function homoclinic bifurcation stability-changing Hopf bifurcations. Poincare bifurcations limit cycle Hilbert's 16th problem
On some properties and limit cycles of Lienard systems
Maoan Han
Conference Publications 2001, 2001(Special): 426-434 doi: 10.3934/proc.2001.2001.426
Please refer to Full Text.
Theory of rotated equations and applications to a population model
Maoan Han Xiaoyan Hou Lijuan Sheng Chaoyang Wang
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 2171-2185 doi: 10.3934/dcds.2018089

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.

keywords: Periodic solution rotated equation saddle-node bifurcation
Cyclicity of some Liénard Systems
Na Li Maoan Han Valery G. Romanovski
Communications on Pure & Applied Analysis 2015, 14(6): 2127-2150 doi: 10.3934/cpaa.2015.14.2127
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.
keywords: Limit cycle Liénard system. cyclicity
Global stability of a predator-prey system with stage structure and mutual interference
Zhong Li Maoan Han Fengde Chen
Discrete & Continuous Dynamical Systems - B 2014, 19(1): 173-187 doi: 10.3934/dcdsb.2014.19.173
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
Dynamics of an SIS reaction-diffusion epidemic model for disease transmission
Wenzhang Huang Maoan Han Kaiyu Liu
Mathematical Biosciences & Engineering 2010, 7(1): 51-66 doi: 10.3934/mbe.2010.7.51
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.
keywords: stability. epidemic model reaction-diffusion equations

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