一类带约束的二维弱奇异积分方程的解*

本文找出二维弱奇异第一类积分方程作用着约束方程的解p.p=p(r,θ)={2/[π2k(φ₀]}√F(r,θ)-c_*(0≤r≤r_*)其中是(s,φ)原点在M(r,θ)的局部极坐标,(r,θ)是原点在O(0,0)的总体极坐标:k和F是给出的连续函数;φ₀是一常数;F(_*,θ)=c_*(常数)是研究域Q的边界围线。所用方法可推广到三维情形。     

广义神经传播型非线性拟双曲方程解的爆破

本文讨论了广义神经传播型非线性拟双曲方程u_tt-Δu_t=F(x,t,u,?u,u_t,?u_t)分别具Neumann边界和Dirichlet边界的两类混合问题.在非线性部分F(x,t,u,?u,u₁,?u₁)和初值满足某些条件时,我们得到了解的爆破性质.     

非自治时滞微分方程的扰动全局吸引性*

考虑具有扰动项的非自治时滞微分方程x>(t)=-a(t)x(t-τ)+F(t,x_t),t≥0(*)其中F:[0,∞)×C[-δ,0]→R且连续,C[-δ,0]表示将[-δ,0]映射到R的所有连续函数集合.F(t,0)≡0,a(t)∈C((0,∞),(0,∞)),τ≥0.通常文献对a(t)不依赖于t即a(t)为自治情形,研究了方程(*)零解的局部或全局渐近性质[1～5,7].本文对a(t)为非自治即依赖于t之情形,获得了方程(*)零解全局吸引的充分条件,所得结论在某种意义上说是不可改进的.本文改进和推广了已有文献的相应结果,同时本文采用的方法可应用到非自治非线性扰动方程.     

正则变换的一种群表示

经典力学中的哈密顿正则变换所涉及的4个母函数F₁(q,Q),F₂(q,P),F₃(p,P),F₄(p,Q)和4种正则变量q,p,Q,P之间所有的关系,可以由7个基本关系式经线性变换而得到,这些变换是勒让德变换,变换是由32个8×8的变换矩阵来实现的,而这32个矩阵以4:1的关系与具有8个群元的D₄点群同态。热力学中的4个状态函数G(P,T),H(P,S),U(V,S),F(V,T)和4个热力学变量P,V,T,S之间的变换关系恰好与正则变换关系相同。热力学状态方程是源于宏观测量的实验结果的概括,而哈密顿正则变换是经典力学的理论性总结,它们的群表示是相同的,即它们的数学结构是相同的, 这种共性表明热力学变换是一维哈密顿正则变换的实例。     

轴对称载荷下旋转壳弹性小应变的轴向任意大挠度问题

本文建议以径向位移u,轴向位移w,子午线切线转角X,径向内力H和经向弯矩M_φ作为描述轴对称载荷下旋转壳弹性小应变的轴向任意大挠度问题的状态变量.在此基础上,本文建立了整体坐标下的一阶非线性微分方方程.进而用权余法得到该问题的最小位能原理.又用引入拉格朗日乘子并加以识别的方法,得到这一问题的广义变分原理. 本文还提出了以载荷参数为尺度的变特征无量纲化方法,可以有效地提高非线性计算的成功率.所得到的无量纲微分方程和无量纲位能原理,可以作为轴对称壳任意大挠度问题数值计算的理论基础.     

Liénard方程极限环的存在性

本文在没有常设条件G(±∞)=+∞的情况下,证明了Liénard方程存在极限环的几个充分性定理,推广了文[3～6]的某些结果.这些定理给出的条件均可估计极限环的存在区域.至少在n个极限环的充分性定理3、4的条件既不要求F(x)是奇函数,也不要求F(x)"n重互相相容"或"n重互相包含".     

关于圆柱壳定解方程的通解问题

本文证明了Vlasov[5]所提出的以应变一应力函数F(ξ,η)所表达的解,为圆柱壳的定解偏微分方程组的通解。     

渐近法在一类强非线性系统中的应用

本文采用文[1、2]的渐近解形式,将渐近法推广到如下较为广泛一类的强非线性振动系统式中g和f为x,的非线性解析函数,ε>0为小参数,并假设对应于ε=0的派生系统有周期解.本文推得系统(0.1)的渐近解递推方法,并应用于实例.     

粘弹性与弹性平面问题间的某些恒等关系*

本文以应力函数F(x_a,t)为基本未知量,首先建立了各向同性、线性粘弹性平面问题(开孔与不开孔)的基本边值问题;其次,详细讨论了粘弹性与弹性平面问题之间位移和应力的某些恒等关系,得到了若干重要和有意义的结论.作为应用,研究了具有中心微孔的粘弹性平板在单向拉伸时微孔的变形响应.     

Maxwell方程的多余阶次与恰当解

设pu=f在Ω内对任意f有解,其中p是任意线性常系数偏微分算子.在本文中我们证明Maxwell方程相当于四阶方程,齐次Maxwell方程的一般解为 其中φ_i满足□φ_i=0 i=1.2.     

Nonlinear open-plus-closed-loop (NOPCL) control of dynamic systems

A nonlinear open-plus-closed-loop (NOPCL) control strategy is proposed in this paper to entrain complex dynamic systems to arbitrarily given goal dynamics. It contains nonlinear closed-loop control actions as well as nonlinear open-loop control actions. Compared to the existing open-plus-closed-loop (OPCL) control, the NOPCL control usually results in much simpler error equations that determine the basins of entrainment of the controlled dynamic systems. For a large number of dynamic systems such as the Lorenz, Rossler, Duffing, Van der Pol, and Chua systems, the NOPCL control is shown to have global basins of entrainment for arbitrarily given goal dynamics.     

Chaos synchronization of a fractional‐order modified Van der Pol–Duffing system via new linear control,backstepping control and Takagi–Sugeno fuzzy approaches

In this article, based on the stability theory of fractional‐order systems, chaos synchronization is achieved in the fractional‐order modified Van der Pol–Duffing system via a new linear control approach. A fractional backstepping controller is also designed to achieve chaos synchronization in the proposed system. Takagi‐Sugeno fuzzy models‐based are also presented to achieve chaos synchronization in the fractional‐order modified Van der Pol–Duffing system via linear control technique. Numerical simulations are used to verify the effectiveness of the synchronization schemes. © 2015 Wiley Periodicals, Inc. Complexity 21: 116–124, 2016     

Synchronization of a chaotic electronic circuit system with cubic term via adaptive feedback control

This paper presents an adaptive feedback control scheme for the synchronization of the chaotic system consisting of Van der Pol oscillators coupled to linear oscillators with cubic term when the parameters of the master system are unknown and different with the those of the slave system. Based on the Lyapunov stability theory, an adaptive control law is derived to make the states of two slightly mismatched chaotic systems asymptotically synchronized. This method is efficient and easy to implement. Numerical simulations results confirming the analytical predictions are shown and pspice simulations are also performed to confirm the efficiency of the proposed control scheme.     

参外激励下强非线性广义Van der Pol方程的亚谐共振解（英文）

In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.     

非线性动力系统的自适应显式Magnus数值方法

基于最近发展的矩阵李群上非线性微分方程的显式Magnus展式,给出了非线性动力系统的有效的数值算法,并且在数值求解过程中具有自适应的步长控制特点,可以显著地提高计算效率．最后,通过非线性动力系统典型问题Duffing方程和强刚性的Van derPol方程以及非线性振子的Hamilton方程的数值实验来说明方法的有效性．     

Chaos generalized synchronization of new Mathieu–Van der Pol systems with new Duffing–Van der Pol systems as functional system by GYC partial region stability theory

In this paper, a new strategy by using GYC partial region stability theory is proposed to achieve generalized chaos synchronization. via using the GYC partial region stability theory, the new Lyapunov function used is a simple linear homogeneous function of states and the lower order controllers are much more simple and introduce less simulation error. Numerical simulations are given for new Mathieu–Van der Pol system and new Duffing–Van der Pol system to show the effectiveness of this strategy.     

Mode decomposition for a synchronous state and its applications

Synchronization of coupled dynamical systems including periodic and chaotic systems is investigated both anlaytically and numerically. A novel method, mode decomposition, of treating the stability of a synchronous state is proposed based on the Floquet theory. A rigorous criterion is then derived, which can be applied to arbitrary coupled systems. Two typical numerical examples: coupled Van der Pol systems (corresponding to the case of coupled periodic oscillators) and coupled Lorenz systems (corresponding to the case of chaotic systems) are used to demonstrate the theoretical analysis.     

Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.     

A Hilbert-Huang transform based identification method for general linear time-varying systems and weakly nonlinear systems

This paper proposes an identification method for general linear time-varying (LTV) MDOF systems and weakly nonlinear systems based on the Hilbert-Huang Transform (HHT)[1]. The proposed method uses Empirical mode decomposition (EMD) to decompose the response signals of systems into intrinsic mode functions (IMFs) and residues, and then analyzes the IMFs and the residues by Hilbert transform (HT) to obtain the analytical IMFs and analytical residues. After that, the above signals are synthesized to form new response signals and new analytical response signals. Finally, the new synthesized signals are used to identify the stiffness and damping coefficients of the systems. Three types of variation: smooth, abrupt and periodical variations are considered in the numerical simulations of LTV chainlike[2] and nonchainlike systems as well as weakly nonlinear systems such as Duffing oscillators and Van der Pol oscillators with white noise added in the system responses to demonstrate the effectiveness, accuracy and robustness of the proposed method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust intelligent sliding model control using recurrent cerebellar model articulation controller for uncertain nonlinear chaotic systems

In this paper, a robust intelligent sliding model control (RISMC) scheme using an adaptive recurrent cerebellar model articulation controller (RCMAC) is developed for a class of uncertain nonlinear chaotic systems. This RISMC system offers a design approach to drive the state trajectory to track a desired trajectory, and it is comprised of an adaptive RCMAC and a robust controller. The adaptive RCMAC is used to mimic an ideal sliding mode control (SMC) due to unknown system dynamics, and a robust controller is designed to recover the residual approximation error for guaranteeing the stable characteristic. Moreover, the Taylor linearization technique is employed to derive the linearized model of the RCMAC. The all adaptation laws of the RISMC system are derived based on the Lyapunov stability analysis and projection algorithm, so that the stability of the system can be guaranteed. Finally, the proposed RISMC system is applied to control a Van der Pol oscillator, a Genesio chaotic system and a Chua’s chaotic circuit. The effectiveness of the proposed control scheme is verified by some simulation results with unknown system dynamics and existence of external disturbance. In addition, the advantages of the proposed RISMC are indicated in comparison with a SMC system.