Calculation of Coefficients of Simplest Normal Forms of Hopf and Generalized Hopf Bifurcations

The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.

Numerical analysis of a simplest fractional-order hyperchaotic system

In this paper,a simplest fractional-order hyperchaotic(SFOH)system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system,which possesses seven terms without any quadratic or higher-order polynomials.The numerical solution of the SFOH system is investigated based on the Adomian decomposition method(ADM).The methods of segmentation and replacement function are proposed to solve this system and analyze the dynamics.Dynamics of this system are demonstrated by means of phase portraits,bifurcation diagrams,Lyapunov exponent spectrum(LEs)and Poincarésection.The results show that the system has a wide chaotic range with order change,and large Lyapunov exponent when the order is very small,which indicates that the system has a good application prospect.Besides,the parameter a is a partial amplitude controller for the SFOH system.Finally,the system is successfully implemented by digital signal processor(DSP).It lays a foundation for the application of the SFOH system.

Simplest Normal Forms of Generalized Neimark-Sacker Bifurcation

The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.

THE WEAK DETONATION SOLUTIONS TO THE SIMPLEST COMBUSTION MODEL

In the present paper the authors prove that all the generalised entropy,solutions of the CJ-model, which for the given Riemann initial data are a oneparameter family u(eta) characterised by weak detonation discontinuity point eta, are just the limits of the admissible solutions u(ek) Of the selfsimilar combustion model. In fact, the authors prove that for any possible eta, there exists a constant B>0 s.t.

Application of the Generalized Simplest Equation Method to the Burgers Equation

We successfully constructed wide classes of exact solutions for the Burgers equation by using the generalized simplest equation method. This method yielded a B&aumlcklund transformation between the Burgers equation and a related constraint equation. By dealing with the constraint equation, we obtained the traveling wave solutions and non-traveling wave solutions of the Burgers equation.

Hilbert空间中伪核逆的扰动与最简表示

从空间拓扑直和分解角度研究Hilbert空间中有界线性算子伪核逆的扰动问题,给出伪核逆具有最简表示的充要条件和稳定扰动下伪核逆的表达式.

Deriving the New Traveling Wave Solutions for the Nonlinear Dispersive Equation,KdV-ZK Equation and Complex Coupled Kd V System Using Extended Simplest Equation Method

In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov–Kuznetsov(Kd V-ZK) equation and complex coupled Kd V system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations(NLEEs) in mathematical physics.

The Simplest General Formula of Approximate Calculationfor Total π- Bond Energies of Randomly Fused Polycyclic Aromatic Hydrocarbons

The total π-bond energy shows a quantum-chemical characteristic of the conjugated molecules. Based on HMO methods, a lot of people try to search for the simple general formulas of an approximated calculation for a wide use. So far, a large number of calculation methods have been suggested, but not the best. On

The Effects of Simplest Little Higgs Model on the Spin Correlations in the Top Quark Pair Production at the LHC and ILC

In the simplest little Higgs model （SLH）, we study the spin correlations in the top quark pair production at the LHC and ILC. We find that the SLH always suppresses the tt spin correlations compared to the SM values. At the LHC, the suppression can be over 10% for mz, 〈 750 GeV. The SLH prediction value is outside the 1σ range of the experimental data from ATLAS, and within 1σ range of the experimental data from CMS. At the ILC, the SLH can sizably suppress the tt spin correlation for mz, approaching the center-of-mass energy √s. For example, the maximal suppression can reach -22.5%, -14.5%, and -9.5% for √s = 500 Ge V, 800 Ge V, and 1000 GeV, respectively. Therefore, the tt spin correlation at the ILC can be a sensitive probe for the SLH.

初等行变换求齐次线性方程组通解的教学探讨

求齐次线性方程组的通解在“线性代数”与“高等代数”的教学中占据着重要地位。教材的解法是利用初等行变换,将系数矩阵化为行阶梯形矩阵,从而确定基本未知量和自由未知量,然后根据行阶梯形矩阵写出对应的齐次线性方程组,并用自由未知量表示基本未知量,从而得到齐次线性方程组的通解。本文通过利用初等行变换将系数矩阵化为行最简形矩阵,直接产生基础解系,进而获得齐次线性方程组的通解。

一类混沌系统的高增益观测器同步方法

针对SQCF(simplest quadratic chaotic flow)模型类的不确定混沌性系统,基于高增益观测器的方法,讨论了混沌同步设计方法.通过在观测器中加入滑模项来抑制未知干扰的影响,以此设计了一个鲁棒高增益观测器.高增益的选取基于一代数Riccati方程的解,为此讨论了Riccati方程解的存在性.设计滑模增益,使系统达到并保持在滑模面上.基于坐标变换和Lyapunov稳定性理论,证明了同步的收敛性.针对一个SQCF混沌系统进行了仿真设计,仿真结果表明了该方法的有效性.

雨水泵站调蓄池三维数值模拟

建立调蓄池底部沟槽排沙模型,利用三维紊流数值模拟的方法,模拟了调蓄池底部沟槽在不同工况下的水流流态,特别是流道曲率变化处的流场,预测调蓄池在各个运行模式下,是否达到冲淤流速,并通过物理模型试验来论证.结论表明,数模计算与物理模型试验吻合得很好,降低了试验成本.

基于威胁传播的多节点网络安全态势量化评估方法

传统的安全态势评估方法主要面向小规模网信息系统,忽略了网络节点间风险的关联性.针对能源互联网中复杂的网络结构,提出基于威胁传播的多节点网络安全态势量化评估方法,1)该方法提出能源互联网网络节点的概念和相关定义,并利用图理论对能源互联网的网络结构进行建模;2)提出基于威胁传播概率的安全态势量化方法,计算网络节点安全态势,并提出多节点网络的权重计算方法 LR-NodeRank,进而评估整个网络的融合安全态势;3)提出一种基于最简威胁图的安全态势改进方法,计算需要开展安全加固的网络边界.实验结果表明:该方法能够准确评估多节点网络的安全态势,能够有效计算边界连接关系.

忆阻器的电路实现与应用

通过常用的电器元件,实现了一种有源忆阻器的等效电路,通过计算得到了该忆阻器等效电路中元件参数,基于MULTISIM的仿真证明了该等效电路的有效性。将忆阻器应用于电路,成功地构成了一个最简混沌系统,该电路仅由一个有源忆阻器,一个电感和一个电容构成。通过MATLAB仿真得到了该电路的系统相图、Lyapunov指数和分数维,证实了该电路的混沌特性。

高维Hopf分岔系统的最简规范形

针对高维Hopf动态分岔问题,研究了不经计算其传统规范形,直接计算高维任意阶数的Hopf分岔系统的最简规范形.利用中心流形定理,将原n维动力系统降为二维的中心流形,根据规范形理论,对中心流形上流的方程进一步化简,在不经过计算传统规范形的情况下,直接计算出其最简规范形中只包含的三阶和五阶项.编写了Mathematica程序,利用该程序,可直接由原n维动力系统计算出其最简规范形.通过3个算例验证了该方法的正确性和计算程序的高效性.

研究强非线性振动问题的最简规范形方法

提出了一种应用最简规范形理论研究强非线性振动问题稳态渐近解的方法。最简规范形理论可以有效地化简传统规范形中的高阶项成分,复规范形方法的特点是简化矩阵分析的繁杂性,引入待定瞬时固有频率法则是将规范形理论的适用范围拓展至用于获取强非线性振动问题的稳态渐近解。结合上述特点,本文的改进方法克服了原有待定瞬时固有频率法在处理高阶传统规范形问题上遇到的计算复杂性问题。数值结果也表明本文方法更为简便、有效且具有更高的计算精度。

临床医生的个案研究设计

个案研究设计适合于提出研究假说 ,这种假说以后可以通过其他方法得到临床确认 ,这也适用于先导性临床试验 (pilotstudy) ,即在设计正式的临床试验之前 ,对要研究的药物进行一个初步的评估。其设计类型包括简单的A B时序设计、经典的A B A反转设计、替代治疗设计、多基线设计。该设计应用的前提是治疗在较短时间能表现出最大效应 ,并且效应是短暂的 ,一旦治疗停止 ,则病情可逆 ,如果不是这种情况 ,则必须考虑成组设计。个案研究的方法一般依赖于“A”阶段的稳定基线期。该研究方法不适宜对急性或易变的疾病状况进行研究 ,而适用于对具有稳定病症表现的慢性或复发性疾病的研究。

高维非线性动力系统最简规范形的计算

运用可逆线性变换和近恒同变换,研究了不经计算传统规范形,直接计算高维非线性动力系统的最简规范形。引进可逆线性变换,将非线性动力系统的线性矩阵拓扑等价于符合实际研究需求的分块对角线矩阵:相伴矩阵分布在对角线上,其余元素均为0。利用低阶项来化简高阶项,得到了高维非线性动力系统的最简规范形。在该最简规范形中,对应于每一个相伴矩阵的非线性系数矩阵,只有最后一行含有非0元素,其余各行元素均为0。借助M athem atica语言,编制了计算任意高维非线性动力系统的最简规范形的通用程序。运行该程序,分别计算了4维、6维和7维非线性动力系统的直到4阶的最简规范形。

汽车最简转弯态的研究

通过对汽车低速转弯状态的研究,提出且定义了汽车最简转弯态的概念,并建立了前驱和后驱力学模型,分别通过解析法和图解法进行分析,得到了各车轮作用力随转向轮转角增加的变化规律及它们的图解表达式。

电力系统二阶共振点稳定模式的最简正规形分析

电力系统在强非线性情况下会发生共振。运用最简正规形思想,推导了电力系统在二阶共振点处的最简正规形,解决了传统正规形在共振点处系数奇异的问题;通过计算近似解析解,修正了非线性参与因子,并且对主导稳定模式的变化过程进行分析,揭示了二阶共振点主导稳定模式形成的物理机理。仿真结果指出,靠近鞍结分岔的二阶共振是电压稳定主导模式形成的临界域,这种变化过程由共振项引起。所提方法对于深刻认识电压稳定与功角稳定的关系具有一定价值。