Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system
Zhirong He Weinian Zhang
Discrete & Continuous Dynamical Systems - A 2009, 24(3): 841-854 doi: 10.3934/dcds.2009.24.841
In this paper we investigate critical periods for a planar cubic differential system with a periodic annulus linking to equilibria at infinity. The monotonicity of the period function is decided by the sign of the second order derivative of a Abelian integral. We derive a Picard-Fuchs equation from a system of Abelian integrals and further give an induced Riccati equation for a ratio of derivatives of Abelian integrals. The number of critical points of the period function for periodic annulus is determined by discussing an planar autonomous system, the orbits of which describe solutions of the Riccati equation.
keywords: critical period Riccati equation. Abelian integral cubic system Hamiltonian system Picard-Fuchs equation
Dynamics of epidemic models with asymptomatic infection and seasonal succession
Yilei Tang Dongmei Xiao Weinian Zhang Di Zhu
Mathematical Biosciences & Engineering 2017, 14(5&6): 1407-1424 doi: 10.3934/mbe.2017073

In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and evaluated directly for this model, and uniform persistence of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0≤ 1$, and as $\mathcal{R}_0>1$ the disease-free equilibrium is unstable and there is an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infectives and symptomatic infectives are close. These theoretical results provide an intuitive basis for understanding that the asymptomatically infective individuals and the seasonal disease transmission promote the evolution of the epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.

keywords: Epidemic model asymptomatic infection seasonal succession basic reproduction number threshold dynamics
Bifurcations of a discrete prey-predator model with Holling type II functional response
Shuping Li Weinian Zhang
Discrete & Continuous Dynamical Systems - B 2010, 14(1): 159-176 doi: 10.3934/dcdsb.2010.14.159
Recently a discrete-time prey-predator model with Holling type II was discussed for its bifurcations so as to show its complicated dynamical properties. Simulation illustrated the occurrence of invariant cycles. In this paper we first clarify the parametric conditions of non-hyperbolicity, correcting a known result. Then we apply the center manifold reduction and the method of normal forms to completely discuss bifurcations of codimension 1. We give bifurcation curves analytically for transcritical bifurcation, flip bifurcation and Neimark-Sacker bifurcation separately, showing bifurcation phenomena not indicated in the previous work for the system.
keywords: discrete prey-predator model; invariant cycle; transcritical bifurcation; flip bifurcation; Neimark-Sacker bifurcation.
Identification of focus and center in a 3-dimensional system
Lingling Liu Bo Gao Dongmei Xiao Weinian Zhang
Discrete & Continuous Dynamical Systems - B 2014, 19(2): 485-522 doi: 10.3934/dcdsb.2014.19.485
In this paper we identify focus and center for a generalized Lorenz system, a 3-dimensional quadratic polynomial differential system with four parameters $a$, $b$, $c$, $\sigma$. The known work computes the first order Lyapunov quantity on a center manifold and shows the appearance of a limit cycle for $a\neq b$, but the order of weak foci was not determined yet. Moreover, the case that $a=b$ was not discussed. In this paper, for $a\neq b$ we use resultants to decompose the algebraic varieties of Lyapunov quantities so as to prove that the order is at most 3. Further, we apply Sturm's theorem to determine real zeros of the first order Lyapunov quantity over an extension field so that we give branches of parameter curves for each order of weak foci. For $a=b$ we prove its Darboux integrability by finding an invariant surface, showing that the equilibrium of center-focus type is actually a rough center on a center manifold.
keywords: center-focus Darboux integrability. algebraic variety Generalized Lorenz system extension field
Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems
Xingwu Chen Jaume Llibre Weinian Zhang
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3953-3965 doi: 10.3934/dcdsb.2017203

It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in $\mathbb{R}^2$ is $$. In contrast here we consider discontinuous differential systems in $\mathbb{R}^2$ defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of $\mathbb{R}^2$, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of $\mathbb{R}^2$. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.

keywords: Hopf bifurcation cyclicity discontinuous differential system limit cycle
Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity
Hailong Zhu Jifeng Chu Weinian Zhang
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 1935-1953 doi: 10.3934/dcds.2018078

In the setting of mean-square exponential dichotomies, we study the existence and uniqueness of mean-square almost automorphic solutions of non-autonomous linear and nonlinear stochastic differential equations.

keywords: Mean-square almost automorphic solutions mean-square exponential dichotomy stochastic differential equations
Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network
Qiuyan Zhang Lingling Liu Weinian Zhang
Mathematical Biosciences & Engineering 2017, 14(5&6): 1499-1514 doi: 10.3934/mbe.2017078

There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.

keywords: Enzyme-catalyzed reaction cusp Bogdanov-Takens bifurcation normal form

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