The Hopf Bifurcation in a Parabolic Free Boundary Problem with Double Layers

We consider a parabolic free boundary problem which has a bifurcation parameter and double interfaces. We investigate the sign change in a real part of eigenvalues and the transversality condition as a bifurcation parameter cross the critical value in order to examine the stability of the stationary solutions. The occurence of a Hopf bifurcation will be shown at a critical value.     

A Hopf Bifurcation in a Generalized Mckean Type of Free Boundary Problem Satisfying the Dirichlet Boundary Condition

In this paper, we consider the free boundary problem satisfying the Dirichlet boundary condition. This problem is derived from the reaction diffusion equations with the generalized McKean reaction dynamics. We shall show a Hopf bifurcation occurs at some critical point τ when the stationary solution (v^⋅ (x), s^⋅) satisfies 1/3 < s^⋅ < 1.     

时滞微分方程广义Hopf 分岔

本文运用Lyapunov-Schmidt 约化方法研究了一般时滞微分方程的分岔情况, 具体分析了当参数达到一个临界值时, 系统的无穷小生成元具有一对k 重非半单纯虚特征值的情形, 得到了判定分岔周期解存在性和分岔方向的判据, 而且该判据明显依赖于系统参数, 并通过对van der Pol 方程的详细分析进一步验证了我们的结果.     

具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

研究了具有时滞耦合的n个van derPol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2:5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2:5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.     

Stability and bifurcation analysis in the delay-coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol equations with a finite delay are investigated. We show that there exist the stability switches and a sequence of Hopf bifurcations occur at the zero equilibrium when the delay varies. Using the theory of normal form and the center manifold theorem, the explicit expression for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.     

Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay

A system of three coupled van der Pol oscillators with delay is considered. Hopf bifurcations at the zero equilibrium as the delay increases are exhibited. The existence and stability of multiple periodic solutions are established using a symmetric Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799-4838).     

Euler-Bernoulli梁在自激励边界反馈下的动力行为

考虑一端固定 ,一端在 van der Pol自激励边界反馈下 Euler-Bernoulli梁的动力行为 .首先设计梁方程的有限元离散格式 ,然后对选定的初始条件计算振动能量的终极变化 .数值结果表明 ,当反馈增益常数由小到大变化时 ,振动能量的变化经历慢周期 ,快周期 ,混沌 ,然后再回到周期振动的过程 .为进一步的理论研究提供了直观的数值表示 .     

The Well Posedness in a Parabolic Multiple Free Boundary Problem

We consider a free boundary problem in a parabolic partial differential equation with multiple interfacial curves which is reduced to a reaction-diffusion equation. The forcing term of this problem is not continuously differentiable and thus we use Green's function to make a regular one. The existence, uniqueness and dependence. on initial conditions will be shown in this paper.     

椭圆外区域上Helmholtz问题的耦合法

以椭圆外区域上Helmholtz方程为例,研究一种带有椭圆人工边界的自然边界元与有限元耦合法,给出了耦合变分问题的适定性及误差分析并给出数值例子.理论分析及数值结果表明,用方法求解椭圆外问题是十分有效的.为求解具有长条型内边界外Helmholtz问题提供了一种很好的数值方法.     

The Perturbation of the Interface of the Two-dimensional Diffraction Problem and an Approximating Muskat Modal

ln this paper, a new transformation is found out to straighten the interface Γ_2 x = f(y), f ∈ C^{2+a}([0, a]), f_y|_y =0, δ < f < l-δ,δ > 0,δ,l=constants and a perturbation of the interface is considered for a two dimensional diffraction problem. And the existence, uniqueness and regularity of an appeoximating Muskat model are proved.     

Period-doubling bifurcation in an extended van der Pol system with bounded random parameter

An extended van der Pol system with bounded random parameter subjected to harmonic excitation is investigated by Chebyshev polynomial approximation. Firstly the stochastic extended van der Pol system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then we explored nonlinear dynamical behavior about period-doubling bifurcation in stochastic system. Numerical simulations show that similar to the conventional period-doubling phenomenon in deterministic extended van der Pol system, stochastic period-doubling bifurcation may also occur in the stochastic extended van der Pol system. Besides, different from the deterministic case, in addition to the conventional bifurcation parameters, i.e. the amplitude and frequency of harmonic excitation, in the stochastic case the intensity of random parameter should also be taken as a new bifurcation parameter.     

Global stability and the Hopf bifurcation for some class of delay differential equation

In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right‐hand side depending only on the past. We extend the results from paper by U. Fory? (Appl. Math. Lett. 2004; 17 (5):581–584), where the right‐hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd.     

The Uniqueness of Steady-state Solution for Two-phase Continuous Casting Problem

Concerning steady-state continuous casting problem, we know that if the number of phases is one, both existence and uniqueness had been solved ([1], [2], [3]), if the number of phases is two, the existence had been proved ([4]), but the uniqueness of weak solution is an open problem all the time. This paper is devoted to solving this problem.     

Stochastic Hopf bifurcation and chaos of stochastic Bonhoeffer–van der Pol system via Chebyshev polynomial approximation

In this paper, we analyzed stochastic chaos and Hopf bifurcation of stochastic Bonhoeffer–van der Pol (SBVP for short) system with bounded random parameter of an arch-like probability density function. The modifier ‘stochastic’ here implies dependent on some random parameter. In order to study the dynamical behavior of the SBVP system, Chebyshev polynomial approximation is applied to transform the SBVP system into its equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. Thus, we can further explore the nonlinear phenomena in SBVP system. Stochastic chaos and Hopf bifurcation analyzed here are by and large similar to those in the deterministic mean-parameter Bonhoeffer–van der Pol system (DM–BVP for short) but there are also some featuring differences between them shown by numerical results. For example, in the SBVP system the parameter interval matching chaotic responses diffuses into a wider one, which further grows wider with increasing of intensity of the random variable. The shapes of limit cycles in the SBVP system are some different from that in the DM–BVP system, and the sizes of limit cycles become smaller with the increasing of intensity of the random variable. And some biological explanations are given.     

Bifurcation Analysis for a Free Boundary Problem Mo deling Growth of Solid Tumor with Inhibitors

This paper is concerned with the bifurcation analysis for a free boundary problem modeling the growth of solid tumor with inhibitors.In this problem,surface tension coefficient plays the role of bifurcation parameter,it is proved that there exists a sequence of the nonradially stationary solutions bifurcate from the radially symmetric stationary solutions.Our results indicate that the tumor grown in vivo may have various shapes.In particular,a tumor with an inhibitor is associated with the growth of protrusions.     

Global Existence and Asymptotic Behavior of Solutions to a Free Boundary Problem for the 1D Viscous Radiative and Reactive Gas

In this paper, we study a free boundary problem for the 1D viscous radiative and reactive gas. We prove that for any large initial data, the problem admits a unique global generalized solution. Meanwhile, we obtain the time-asymptotic behavior of the global solutions. Our results improve and generalize the previous work.     

Numerical methods for weak solution of wave equation with van der Pol type nonlinear boundary conditions

We develop computational methods for solving wave equation with van der Pol type nonlinear boundary conditions under the framework of weak solutions. Based on the wave reflection on the boundaries, we first solve the Riemann invariants by constructing two iteration mappings, and then show that the weak solution can be obtained by the integration of the Riemann invariants on the boundaries. If the compatible conditions are not satisfied or only hold with a low degree, a high‐order integration method is developed for the numerical solution. When the initial condition is sufficiently smooth and compatible conditions hold with a sufficient degree, we establish a sixth‐order finite difference scheme, which only needs to solve a linear system at any given time instance. Numerical experiments are provided to demonstrate the effectiveness of the proposed approaches. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 373–398, 2016     

求解互补问题的Gauss-Newton方法的全局收敛性

求解互补问题的Gauss-Newton方法是由Subramamian提出的．本文研究了此方法的收敛性质，在较弱的情况下，建立了一个全局收敛结果，此结果是相关文献中的结果的推广．     

Global Weak Solutions of Initial Boundary Value Problem for Boltzmann－Poisson System with Absorbing Boundary

This paper deals with the initial boundary value problem for the Boltzmann-Poisson system, which arises in semiconductor physics, with absorbing boundary. The global existence of weak solutions is proved by using the stability of velocity averages and the compactness results on L1-theory under weaker conditons on initial boundary values.