受迫广义KdV-Burgers方程行波解的存在唯一性

本文研究具有受迫性的广义二维KdV-Burgers方程的周期行波解,为了获得周期行波解的存在唯一性定理,使唤用特定系数法和Schauder不动点定理获得了受迫广义KdV-Burgers方程周期行波解存在唯一性的条件.并获得了周期行波解的一些先验估计式.     

参数振动受迫响应的三角级数解

基于调制反馈方法,对参数周期与激励力周期不相同情况下,研究其参数系统受迫振动响应三角级数解．采用谐波的线性组合形式从数学上表达受迫振动响应解,然后通过运用谐波平衡,将参数振动方程转化成无限阶线性代数方程组,解出其谐波的系数．上述方法的特点在于:1） 用三角级数来表达振动受迫响应,十分便于参数振动的频域分析,剖析受迫响应性质;2） 从解的表达可直接推出组合谐波共振条件; 3） 采用标准的Runge Kutta算法得到的相图证实上述方法结果的精确性．研究结果表明:该方法适用于参数振动完整受迫响应解的数学表达与分析．     

含时受迫谐振子的相干态与AA相

给出了构造含时受迫谐振子的相干态的新方法 ,讨论了与经典受迫谐振子的关系 ,最小测不准关系和AA (Aharonov Anandan)相 .外加周期作用力 (非共振情况 )一般将使AA相减小     

非线性项依赖于时间和空间变量的梁方程拟周期解的存在性

本文研究带有空间周期和时间拟周期非线性项的常数势能梁方程,证明了对于大多数频率向量和大多数势能常数,方程存在小振幅、线性稳定的时间拟周期解.通过对本质上无穷多个小除数的测度估计,本文构建了一个实解析的辛坐标变换,将Hamilton函数化为其Birkhoff标准型.利用一个无穷维Kolmogorov-Arnold-Moser(KAM)定理,本文证明了拟周期解的存在性.     

受迫耗散Boussinesq方程的可积性质和孤波解的探讨

考虑到耗散效应和地形外力,Rossby波的振幅可由受迫耗散Boussinesq方程来描述.当包含这两项时,模型比较复杂,不具有Painleve性质.通过将模型双线性化,双线性方法是一个可寻找孤波解和B(a|¨)cklund变换的方法.通过截断的Painleve展开式,得到了将方程双线性化的合适的因变量变换.然后得到了受迫耗散Boussinesq方程的单孤波解和B(a|¨)cklund变换.     

有限个渐近伪压缩映射近迫点序列的收敛性

研究了Banach空间中有限个渐近伪压缩映射近迫点序列的收敛性问题,此结果推广了以前的结论.     

有限个李普希兹伪压缩映射近迫点序列的收敛性

研究了Banach空间中有限个李普希兹伪压缩映射近迫点序列的收敛性问题,此结果推广了以前的结论.     

添加超实数而不加实数的力迫

本文研究了加强型Mathias力迫及其在不可数情形下的推广.通过力迫法,证明了Mathias力迫添加支配性实数,而加强型Mathias力迫添加的是无界、非支配性的实数.还证明了ω₁上的Mathias型力迫添加的是无界、非支配性的ω₁类实数且不添加新的实数.这些结论可应用于对实数上的基数不变量的研究.     

负阻尼周期运动的经过时间

初始时作周期运动的系统被负阻尼作用逐渐托出势能井,其周期运动的经过时间由多重变量展开解得.一强非线性系统的算例表明其结果近似性好且计算简便.     

带有振动系数的一类高阶中立型非线性受迫微分方程的振动准则

在本文中,我们获得形如[y(t) l∑j=1pj(t)y(σj(t))](n) ∫0q(t,s)f(y(t s))dσ(s)=h(t)的带有振动系数的一类高阶中立型非线性受迫微分方程的振动准则.     

Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction

A well known theorem says that the forced pendulum equation has periodic solutions if there is no friction and the external force has mean value zero. In this paper we show that this result cannot be extended to the case of linear friction.

     

A variational approach to chaotic dynamics in periodically forced nonlinear oscillators

We prove that a class of problems containing the classical periodically forced pendulum equation displays the main features of chaotic dynamics. The approach is based on the construction of multibump type heteroclinic solutions to periodic orbits by the use of global variational methods.     

Periodic Liénard-type delay equations with state-dependent impulses

Conditions are obtained for Liénard-type equations with delay and state-dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation (and consequently with Lebesgue integrable second derivative). The results are applied to Josephson's equation and the nonconservative forced pendulum equation.     

Teaching the computer how to discover(!) and then prove(!!) (all by itself(!!!)) analogues of Collatz's notorious 3x + 1 conjecture

In this paper, we study the global attractivity for a class of periodic difference equation with delay which has a generalized form of Pielou's difference equation. The global dynamics of the equation is characterized by using a relation between the upper limit and lower limit of the solution. There are two possible global dynamics: zero solution is globally attractive or there exists a periodic solution which is globally attractive. Recent results by Camouzis and Ladas [Periodically forced Pielou's equation, J. Math. Anal. Appl. 333 (1) (2007) 117–127] are generalized. Two examples are given to illustrate our results.     

Chaotic behavior in differential equations driven by a Brownian motion

In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.     

Almost periodically forced circle flows

We study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li-Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2,R)-valued, continuous or discrete cocycle.Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multi-frequency-driven dynamical irregularities and complexities in non-monotone dynamical systems.     

A global solution curve for a class of periodic problems,including the relativistic pendulum

Using continuation methods, we study the global solution structure of periodic solutions for a class of periodically forced equations, generalizing the case of relativistic pendulum. We obtain results on the existence and multiplicity of periodic solutions. Our approach is suitable for numerical computations, and in fact we present some numerically computed bifurcation diagrams illustrating our results.     

不动点理论与里卡提方程两个正周期解的存在性

利用压缩映射原理,得到里卡提方程一个正周期解的存在性;利用变量变换方法,将里卡提方程转化为伯努利方程.根据伯努利方程的周期解和变量变换,得到里卡提方程的另一个周期解.并讨论了两个正周期解的稳定性,一个周期解在某个区间上是吸引的,另一个周期解在R上是不稳定的.     

Limit-periodic solutions of integro-differential equations in a critical case

We consider equations with nonlinear terms representable by power series in the variable and functionals in integral form. The equation depends on a small exponentially limitperiodic perturbation, i.e., on a function that exponentially tends to a periodic function as the independent variable increases. In the Lyapunov critical case of one zero root, we prove the existence of a family of exponentially limit-periodic solutions of the equation in the form of power series in the small parameter and arbitrary initial values of the noncritical variables.     

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 …