一类非线性系统的周期扰动Hopf分支

研究了小周期扰动对一类存在Hopf分支的非线性系统的影响.特别是应用平均法讨论了扰动频率与Hopf分支固有频率在共振及二阶次调和共振的情形周期解分支的存在性.表明了在某些参数区域内,系统存在调和解分支和次调和解分支,并进一步讨论了二阶次调和分支周期解的稳定性.     

周期扰动下的尖分支

本文研究周期扰动对一个发生余维2尖分支的系统的影响,应用Lyapunov-Schmidt方法,将周期扰动系统的周期解问题化为一个分支函数的零点问题.研究发现,在扰动幅度很小时,系统产生周期解的折分支和尖分支,与无扰动系统的平衡点分支相对应;并且新的分支偏离于原分支的程度与扰动幅度和扰动函数紧密相关.同时,为了进一步说明这些理论结果,本文还利用数值分支方法给出几个范例系统的分支分析.     

一类摄动系统的Poincare分叉与极限环

本文利用临界情形的隐函数存在定理讨论了一类摄动系统分支周期解的存在性与稳定性，利用后继函数法讨论了该系统极限环的存在性、唯一性和稳定性。     

具有共振特征值的轨道翻转双同宿环分支

该文研究了具有轨道翻转的双同宿环四维系统,在主特征值共振和沿轨道奇点处切方向共振下的两种分支．我们分别在系统奇点小邻域内利用规范型的解构造一个奇异映射,再在双同宿环的管状邻域内引起局部活动坐标架,利用系统线性变分方程的解定义了一个正则映射,通过复合两个映射而得到分支研究中一类重要的Poincaré映射,经过简单的计算最终得到后继函数的精确表达式．对分支方程细致地研究,我们给出了原双同宿环的保存性条件,并证明了“大” 1-同宿环分支曲面,2-重“大”1-周期轨分支曲面,“大”2-同宿环分支曲面的存在性、存在区域和近似表达式,及其分支出的“大”周期轨和“大”同宿轨的存在性区域和数量．     

强共振情况下不变环面的分支

本文利用周期变换和积分流形理论研究平面周期扰动系统，在强共振情况下获得了从一阶细焦点分支出不变环面的简洁条件，本文中的非共振条件不同于［５］中所给出的非共振条件．     

一般自治系统周期解的局部分支

本文研究高维自治系统周期解的Hopf分支，作为应用，对非半单特征值的情况给出了周期解的分支条件     

一类摄动系统的Poincaré分叉与极限环

本文利用临界情形的隐函数存在定理讨论了一类摄动系统分支周期解的存在性与稳定性,利用后继函数法讨论了该系统极限环的存在性、唯一性和稳定性。     

高维同宿环扭曲分支与稳定性

利用沿同宿环的线性变分方程的线性独立解作为在同宿环的小管状邻域内的局部坐标系来建立Poincaré映射,研究了高维系统扭曲同宿环的分支问题．在非共振条件和共振条件下,获得了1-同宿环、 1-周期轨道、 2-同宿环、 2-周期轨道和两重2-同期轨道的存在性、 存在个数和存在区域．给出了相关的分支曲面的近似表示．同时,研究了高维系统同宿环和平面系统非扭曲同宿环的稳定性．     

具有脉冲效应和综合害虫控制的捕食系统

本文通过生物控制和化学控制提出了具有周期脉冲效应与害虫控制的捕食系统. 系统保护天敌避免灭绝,在一些条件下可以使害虫灭绝.就是说当脉冲周期小于某一临界值时,存在全局稳定害虫灭绝周期解.脉冲周期增大大于临界值时,平凡害虫灭绝周期解失去稳定性并产生正周期解,利用分支理论来研究正周期解的存在性.进而,利用李雅普诺夫函数和比较定理确定了持续生存的条件.     

Poincaré非线性振动理论在连续介质力学中的推广(Ⅱ)——若干应用

文本是文[1]的继续.文[1]中,提出和建议使用非线性偏微分方程直接摄动与加权积分方程法,计算连续介质系统的共振与非共振周期解.本文中,应用该方法计算了定跨度弹性梁在各种常见边界条件下强迫振动的共振与非共振周期解,方板在集中周期荷载作用下的共振周期解.指出了,非主振型对非线性振动周期解的影响及静荷载对幅频特性曲线的影响.     

Lazer–Leach type conditions on periodic solutions of semilinear resonant Duffing equations with singularities

In this paper, we study the existence of positive periodic solutions of resonant Duffing equations with singularities. Some Lazer–Leach type conditions are given to ensure the existence of positive periodic solutions of singular resonant Duffing equations.     

EXISTENCE OF PERIODIC SOLUTIONS OF A CLASS OF SECOND-ORDER NON-AUTONOMOUS SYSTEMS

In this paper, we are concerned with the existence of periodic solutions of second-order non-autonomous systems. By applying the Schauder''s fixed point theorem and Miranda''s theorem, a new existence result of periodic solutions is established.     

Periodic solutions of a periodically forced and undamped beam resting on weakly elastic bearings

We study the problem of existence of periodic solutions to a partial differential equation modelling the behavior of an undamped beam subject to an external periodic force. We assume that the ordinary differential equation associated to the first two modes of vibration of the beam has a symmetric homoclinic solution. By using methods borrowed by dynamical systems theory we prove that, if the period is non resonant with the (infinitely many) internal periods of the PDE, the equation has a weak periodic solution of the same period as the external force. In particular we obtain continua of periodic solutions for the undamped beam in absence of external forces. This result may be considered as an infinite dimensional analogue of a result obtained in [16] concerning accumulation of periodic solutions to homoclinic orbits in finite dimensional reversible systems. Matteo Franca: Partially supported by G.N.A.M.P.A. – INdAM (Italy).     

Solutions of periodic boundary value problems for second-order nonlinear differential equations via variational methods

By introducing a variational framework for a class of second order nonlinear differential equations with non-separated periodic boundary value conditions, some results on the existence of non-trivial, positive and negative solutions of the problems are obtained. Some results by Atici-Guseinov, Graef-Kong, etc. obtained by topological degree methods are extended. The resonant case of the problems where the nonlinearities are unbounded and satisfy Ahmad-Lazer-Paul type conditions is also considered.     

Nontrivial periodic solutions to a type of delayed resonant differential equations

In this paper, we consider a type of delayed resonant differential equations. We focus on the existence of periodic solutions. Employing the Clark dual, we provide two sets of criteria on the existence of at least one periodic solution. In fact, the periodic solutions are critical points minimizing the dual functional of the coupled Hamiltonian system on certain subspaces of a Banach space.     

含时滞的反应扩散方程组周期解的存在与稳定性

本文研究了一类反应项非单调的时滞反应扩散方程组.利用上、下解方法及不动点理论获得了此系统边值问题周期解存在性的充分条件,给出了证明其周期解稳定性的方法.最后通过化学中的一个典型模型说明了所得结果的意义.     

Dynamics of a congestion control model in a wireless access network

In this paper, a congestion control algorithm with heterogeneous delays in a wireless access network is considered. We regard the communication time delay as a bifurcating parameter to study the dynamical behaviors, i.e., local asymptotical stability, Hopf bifurcation and resonant codimension-two bifurcation. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the delay passes through a sequence of critical value. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. In the meantime, the resonant codimension-two bifurcation is also found in this model. Some numerical examples are finally performed to verify the theoretical results.     

Periodic solutions of a resonant third-order equation

We study the existence of periodic solutions for a third-order equation of resonant type. Under suitable conditions we prove the existence of at least one periodic solution of the problem applying Mawhin coincidence degree theory.     

EXISTENCE AND UNIQUENESS OF ALMOST PERIODIC SOLUTIONS FOR FORCED RAYLEIGH EQUATIONS

1 IntroductionEquationor its nonhomogeneous formwhere p, q) r and w are constants, originated with the study of aerodinamica.And then as its generalization, forced Rayleigh equationespecially its periodic solution has been studied by many authors. For example,R.Reissig [1], proved equation (1.1) has at least one w-periodic solution underthe following conditionsi) F(v), g(x) and e(t) are continuous functions;n) Suppose that m 5 0 5 M, and that there exists a positive number V,such that when …