On the existence of periodic solutions for nonlinear system with multiple delays

IntroductionInthispaper,westudyT_periodicsolutionsofthefollowingnonlinearsystemwithmultipledelays x(t) =f(t,x(t) ,x(t-τ1(t) ) ,… ,x(t -τm(t) ) ) ,(1 )wherex(t) ∈C(R ,R) ,fiscontinuous,f(t+T ,·) =f(t,·) ,τi(t) (i=1 ,2 ,… ,m)arecontinuousperiodicfunctionsofperiodT .AlemmaisintroducedfordiscussingtheexistenceofT_periodicsolutionofsystem (1 ) .LetXbeaBanachSpace ,considerthefollowingoperatorequation :Lx =λNx　　 (λ∈ [0 ,1 ] ) ,whereL :DomL∩X→Xisalinearoperator,λ∈ [0 ,1 ]isapa…     

On the existence of periodic solutions to higher dimensional periodic system with delay

Ｃｏｎｓｉｄｅｒｉｎｇｔｈｅｈｉｇｈｅｒｄｉｍｅｎｓｉｏｎａｌｐｅｒｉｏｄｉｃｓｙｓｔｅｍｓｗｉｔｈｄｅｌａｙｏｆｔｈｅｆｏｒｍｘ′（ｔ）＝Ａ（ｔ,ｘ（ｔ））ｘ（ｔ）＋ｆ（ｔ,ｘ（ｔ－τ））,（１）ｘ′（ｔ）＝ｇｒａｄＧ（ｘ（ｔ））＋ｆ（ｔ,ｘ（ｔ－...     

Problem of periodic solutions for neutral functional differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＣ(ｋ- 1)2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓ (ｋ -1 )＿ｔｈｏｒｄｅｒｃｏｎｔｉｎｕｏｕｓｄｉｆｆｅｒｅｎｔｉａｂｌｅａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　Ｃ2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓｃｏｎｔｉｎｕｏｕｓａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　‖ｈ(ｔ)‖ =ｓｕｐｔ∈ [0 ,2π] ｜ｈ(ｔ) ｜ ,　　‖ｈ(ｔ)‖Ｐｋ- 1 =ｍａｘ‖ｈ(ｔ)‖ ,‖ｈ′(ｔ)‖ ,… ,‖ｈ(ｋ- 1) (ｔ)‖ ,　　ｘ(ｍ) (ｔ+ ·) (θ) =ｘ(ｍ) (ｔ+θ)　　θ∈Ｒ (ｍ =0 ,1 ,2 ,… ,ｋ-1 ) .Ｃｌｅａｒｌｙ ,ｘ(ｍ) (ｔ + ·) ∈Ｃ2π, …     

Existence and uniqueness of almost periodic solution to a class of nonautonomous system

Ｈｉｇｈ＿ｔｅｎｓｉｏｎｅｌｅｃｔｒｉｃｉｔｙｎｅｔｗｏｒｋ,ｗｉｔｈｏｕｔｗｈｉｃｈｏｕｒｍｏｄｅｒｎｓｏｃｉｅｔｙｃａｎｎｅｖｅｒｌａｓｔｆｏｒａｄａｙ,ｍａｙｓｏｍｅｔｉｍｅｓｂｅｒｅａｌｌｙｕｎｓｔａｂｌｅ,ｄｕｅｔｏｖａｒｉｏｕｓｋｉｎｄｏｆｏｓｃｉｌｌａｔｉｏｎｓｏｆｃｕｒｒｅｎｔｉｎｔｅｎｓｉｔｙ．Ｉｎｒｅｃｅｎｔｙｅａｒｓ,ｐｅｒｉｏｄｉｃｏｓｃｉｌｌａｔｉｏｎｓｏｒａｌｍｏｓｔｐｅｒｉｏｄｉｃｏｓｃｉｌｌａｔｉｏｎｓｈａｖｅａｐｐｅａｒｅｄｉｎＣｅｎｔｒａｌｇｒｉｄａｎｄＮｏｒ…     

Periodic solutions of a class of higher order neutral type equations

In this paper, by ssing the theory of Fourier series, some necessary and sufficient conditions of existence and uniqueness of periodic solutions of a class of higher order neutral type equations are obtained. The main results by Shi Jianguo in “Discussion on the periodic solutions for linear equation of neutral type with constant coefficients” are improved, i. e., the condition |b₀|≠1 instead of the condtion |b₀|<1/2 of Theorem 1 by Shi Jianguo is given. Other theorems by Shi are rebuilt and improved according to the new assumption. Foundation item: the Natural Science Foundation of Yunnan Province, China (97A012G)     

一类非自治动力系统超次谐周期解的不存在性

在非线性动力系统的研究中， Melnikov函数被广泛地用来作为微扰哈密顿系统是否发生次谐或超次谐分岔乃至混沌的判 据. 但是在大多数情况下，经典的Melnikov方法往往只给出存在次谐周期解的结论. 产生 该结果的原因被归之为在经典的Melnikov方法中只采取了一阶近似，因而高阶Melnikov方 法被发展用来判断超次谐周期解的存在性. 本文对一类非自治微分动力系统进行了研究，证 明了在这样一类系统中如果存在周期解则只可能是次谐周期解，超次谐周期解不可能存在， 并进一步证明了在一类平面问题中所定义的旋转(R)型超次谐周期解同样不可能存在.作为 该结论的一个应用，文中考察了几个典型的算例，结果表明现有的二阶Melnikov方法判断 平面扰动系统是否存在超次谐周期解的结论是不恰当的，并提供了一个简单的几何上的解释.     

RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

ＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦ　ＮＯＮＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＤＩＦＦＥ...     

非线性电报方程组的3个双周期正解问题

This paper studies existence of at least three positive doubly periodic solutions of a coupled nonlinear telegraph system with doubly periodic boundary conditions. First, by using the Green function and maximum principle, existence of solutions of a nonlinear telegraph system is equivalent to existence of fixed points of an operator. By imposing growth conditions on the nonlinearities, existence of at least three fixed points in cone is obtained by using the Leggett-Williams fixed point theorem to cones in ordered Banach spaces. In other words, there exist at least three positive doubly periodic solutions of nonlinear telegraph system.     

Robust stability of uncertain neutral linear stochastic differential delay system

The LaSalle-type theorem for the neutral stochastic differential equations with delay is established for the first time and then applied to propose algebraic criteria of the stochastically asymptotic stability and almost exponential stability for the uncertain neutral stochastic differential systems with delay. An example is given to verify the effectiveness of obtained results.     

A new method for the periodic solution of strongly nonlinear systems

A new method for the periodic solution of strongly nonlinear system is given. By using this method, the existance and stability of the periodic solution can be decided, and the approximate expression of the periodic solution can also be found. The project supported by the National Natural Science Foundation of China     

On the existence and stability of periodic solutions for hopfield neural network equations with delay

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｒｅｃｅｎｔｙｅａｒｓ,ｗｉｔｈａｌｏｔｏｆａｐｐｌｉｃａｔｉｏｎｓｏｆｎｅｕｒａｌｎｅｔｗｏｒｋｍｏｄｅｌｓ,ｍａｎｙａｕｔｈｏｒｓ［１～３］ａｒｅｉｎｔｅｒｅｓｔｅｄｉｎｔｈｅｒｅｓｅａｒｃｈｏｆｔｈｅｓｔｒｕｃｔｕｒｅａｎｄｐｅｒｆｏｒｍａｎｃｅｆｏｒｔｈｅｓｅｎｅｔｗｏｒｋｓ．ＢｅｃａｕｓｅＨｏｐｆｉｅｌｄｎｅｕｒａｌｎｅｔｗｏｒｋｗｅｌｌｓｉｍｕｌａｔｅｔｈｅｅｃｏｌｏｇｉｃａｌｓｙｓｔｅｍ,ｍａｎｙｓｔｕｄｉｅｓａｒｅｃｏｎｃｅｎｔｒａｔｅｄｏｎｔｈ…     

非自治系统用瞬态响应求周期解稳定度的方法

根据Floquet理论关于线性周期系数系统解的性质及稳定性条件 ,定义了衡量非线性非自治系统周期解受扰后的衰减速率指标—稳定度。从动力系统流的概念出发 ,给出了利用非线性非自治系统稳态周期解受扰后的瞬态响应信息计算周期解稳定度的方法。以不平衡滑动轴承 弹性转子系统为例 ,说明了该方法的有效性。将稳定度等于零作为临界判据 ,该方法不仅解决了工频周期解失稳边界的确定问题 ,而且解决了渐进稳定域的估计和抗冲击扰动裕度的计算问题     

On static output-feedback stabilization of a periodic system

Relations are derived that allow standard MATLAB routines to be used to solve the static output-feedback control problem for a periodic discrete-time system. The efficiency of the approach proposed to design optimal static output-feedback controllers is demonstrated by examples __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 127–134, March 2006.     

DISCUSSION ON THE PERIODIC SOLUTIONS FOR HIGHER-ORDER LINEAR EQUATION OF NEU

ＳｉＪｉａｎｇｕｏ（司建国）（ＲｅｃｅｉｖｅｄＭａｙ３０，１９９４，ＣｏｍｍｕｎｉｃａｔｅｄｂｙＬｉｎＺｏｎｇｃｈｉ）ＤＩＳＣＵＳＳＩＯＮＯＮＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＦＯＲＨＩＧＨＥＲ－ＯＲＤＥＲＬＩＮＥＡＲＥＱＵＡＴＩＯＮＯＦＮＥＵ...     

一种非线性系统周期解的延拓算法

提出了一种非线性系统周期解的延拓算法。指出了非线性系统周期解在分岔点处由于雅可比矩阵奇异而导致一般延拓方法延拓失败问题;然后基于推广的打靶法的思想,将普通延拓算法推广,提出了一种用于周期解延拓的算法。对于非线性动力系统,该算法可以在已知某一参数下的周期解的基础上,求解出在一定参数范围内非线性动力系统的解随参数的连续变化情况。应用该方法对非线性柔性转子-轴承系统的周期解与参数的依赖关系进行了求解,验证了方法的有效性。     

Local Hopf bifurcation and global existence of periodic solutions in TCP system

This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.     

A periodic solution to a nonlinear diffusion equation

An analytic solution to the one dimensional heat diffusion equation is presented where the diffusion coefficient varies as a power of temperature. The discussion is motivated by the transmission of heat through the strongly nonlinear medium of soil. Under boundary conditions representing the daily, or seasonal, sinusoidal fluctuation in temperature it is seen that, despite the nonlinearity, the period of the oscillation is preserved on passage through the medium. The nonlinearity acts to accelerate the heating phase and retard the cooling phase within a period which itself remains stable. These effects are calculable from a second harmonic arising in the analysis.     

STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

ＳＴＡＢＩＬＩＴＹＡＮＡＬＹＳＩＳＯＦＬＩＮＥＡＲＡＮＤＮＯＮＬＩＮＥＡＲＰＥＲＩＯＤＩＣＣＯＮＶＥＣＴＩＯＮＩＮＴＨＥＲＭＯＨＡＬＩＮＥＤＯＵＢＬＥ－ＤＩＦＦＵＳＩＶＥＳＹＳＴＥＭＳＺｈａｎｇＤｉｍｉｎｇ（张涤明）;ＬｉＬｉｎ（李琳）;ＨｕａｎｇＨ...     

Development of a doubly periodic system of cavities in viscous bodies

The plane unknown-boundary problem of the development of a doubly periodic system of cavities in viscous media in the presence of finite strains is considered. Under conditions of timedependent slow Newtonian viscous fluid flow the solution of the doubly periodic problem of the development of a system of identical cavities whose centers are arranged in square and triangular grids is obtained.     

The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｏｂｌｅｍｉｎｔｈｅＲｅｓｅａｒｃｈｏｆＴｏｒｏｉｄＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｅｘｉｓｔｅｎｃｅｏｆ2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｔｏｔｈｅｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｏｆｆｉｒｓｔ＿ｏｒｄｅｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈａｄｅｖｉａｔｉｎｇａｒｇｕｍｅｎｔ ｘ(ｔ) =Ｂｘ(ｔ) Ｆ(ｘ(ｔ-τ) ) ｐ(ｔ) ,( 1 )ｗｈｅｒｅｘ(ｔ)∈Ｒ2 , ｘ(ｔ) =ｄｄｔｘ(ｔ) ,τ∈Ｒ ,Ｂ∈Ｒ2×2 ,Ｆ :Ｒ2 →Ｒ2 ｉｓｂｏｕｎｄｅｄａｎｄｐ∈Ｃ(…