关于一类四阶非线性系统李雅普诺夫函数构造的研究

本文导出了四维线性系统的李雅普诺夫函数公式，研究了一类四阶非线性系统平凡解的稳定性。     

一类四阶非线性系统的稳定性

本文利用“类比法”构造了一类四阶非线性系统的李雅普诺夫函数 ,导出了该系统的平凡解的稳定性条件     

一类四阶非线性系统的稳定性

本利用“类经法”构造了一类四阶非线性系统的李雅普诺夫函数,导出了该系统的平凡解的稳定性条件。     

四阶非线性系统的全局渐近稳定性

运用构造李雅普诺夫函数的方法 ,研究了一类四阶非线性系统的全局渐近稳定性 ,给出了该系统零解全局渐近稳定的充分条件     

四阶非线性系统的全局部渐近稳定性

运用构造李雅普诺夫函数的方法,研究了一类四阶非线性系统的全局部渐近稳定性,给出了该系统零解全局渐近稳定的充分条件。     

一类四阶非线性系统的全局渐近稳定性

运用类比法,构造了一类四阶非线性系统的李雅普诺夫函数,得到其平凡解全局渐近稳定的充分性准则,推广并改进了有关这类系统的其它结果.     

论李雅普诺夫函数的构造

А.М.Ляпунов稳定性理论中的一个核心问题,就是李雅普诺夫函数的构造问题。尽管近卅年来人们作了不少的努力,但直到现在为止,对于一般非线性系统而言,还是没有构造其李雅普诺夫函数的通用而有效的方法。虽然如此,但针对具体问题进行具体分析,也就是针对实际中出现的各种非线性系统,通过定性分析,然后根据实际情况,构造出恰当的李雅普诺夫函数,就这一点而论,还是取得了极其丰富的成果。因此在探索非线性     

时变离散大系统的结构与关联稳定性

本文应用模型降阶的集结法与李雅普诺夫函数分解法,研究了线性、非线性时变离散大系统的关联稳行性。同时给出了分解系数及非线性项的估计公式。     

一类三阶非线性系统的全局稳定性分析

本文运用“类比法”构造了一类三阶非线性系统的李雅普诺夫函数,从而推出了该系统全局稳定的条件．     

非线性非定常系统在经常作用干扰下的稳定性定理——李雅普诺夫间接法

研究在经常作用干扰下的稳定性是研究李雅普诺夫意义下稳定性的深入和发展,并且更有实际意义,因为干扰往往并不是一次的.关于经常作用干扰下的稳定性,马尔金提出了一个一般定理;但实用它必须找出满足一定条件的李雅普诺夫函数.这对于非线性非定常系统是很困难的.本文根据李雅普诺夫间接法的原理,将之推广应用到判别非线性非定常系统在经常作用干扰下的稳定性,证明了一个重要结论.     

Nonsingular decoupled terminal sliding-mode control for a class of fourth-order nonlinear systems

This paper presents a nonsingular decoupled terminal sliding mode control (NDTSMC) method for a class of fourth-order nonlinear systems. First, the nonlinear fourth-order system is decoupled into two second-order subsystems which are referred to as the primary and secondary subsystems. The sliding surface of each subsystem was designed by utilizing time-varying coefficients which are computed by linear functions derived from the input–output mapping of the one-dimensional fuzzy rule base. Then, the control target of the secondary subsystem was embedded to the primary subsystem by the help of an intermediate signal. Thereafter, a nonsingular terminal sliding mode control (NTSMC) method was utilized to make both subsystems converge to their equilibrium points in finite time. The simulation results on the inverted pendulum system are given to show the effectiveness of the proposed method. It is seen that the proposed method exhibits a considerable improvement in terms of a faster dynamic response and lower IAE and ITAE values as compared with the existing decoupled control methods.     

Completeness theorem for singular differential pencils

A theorem completeness theorem of special vector functions induced by the products of the so-called Weyl solutions of a fourth-order differential equation and by their derivatives on the semiaxis is presented. We prove that such nonlinear combinations of Weyl solutions and their derivatives constitute a linear subspace of decreasing (at infinity) solutions of a linear singular differential system of Kamke type. We construct and study the Green function of the corresponding singular boundary-value problems on the semiaxis for operator pencils defining differential systems of Kamke type. The required completeness theorem is proved by using the analytic and asymptotic properties of the Green function, operator spectral theory methods, and analytic function theory.     

A variant of Steffensen’s method of fourth-order convergence and its applications

In this paper, a variant of Steffensen’s method of fourth-order convergence for solving nonlinear equations is suggested. Its error equation and asymptotic convergence constant are proven theoretically and demonstrated numerically. The derivative-free method only uses three evaluations of the function per iteration to achieve fourth-order convergence. Its applications on systems of nonlinear equations and boundary-value problems of nonlinear ODEs are showed as well in the numerical examples.     

A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule

Darvishi and Barati [M.T. Darvishi, A. Barati, Super cubic iterative methods to solve systems of nonlinear equations, Appl. Math. Comput., 2006, 10.1016/j.amc.2006.11.022] derived a Super cubic method from the Adomian decomposition method to solve systems of nonlinear equations. The authors showed that the method is third-order convergent using classical Taylor expansion but the numerical experiments conducted by them showed that the method exhibits super cubic convergence. In the present work, using Ostrowski’s technique based on point of attraction, we show that their method is in fact fourth-order convergent. We also prove the local convergence of another fourth-order method from 3-node quadrature rule using point of attraction.     

Bifurcations of traveling wave solutions for the nonlinear schrodinger equation with fourth-order dispersion and cubic-quintic nonlinearity

For the nonlinear schrodinger equation with fourth-order dispersion and cubic-quintic nonlinearity, by using the method of dynamical systems, the dynamics and bifurcations of the corresponding traveling wave system are studied. Under different parametric conditions, twenty exact parametric representations of the traveling wave solutions are obtained.     

Fourth-Order Symmetric DIRK Methods for Periodic Stiff Problems

New symmetric DIRK methods specially adapted to the numerical integration of first-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion conditions for symmetric DIRK methods as well as symmetric stability functions with real poles and maximal dispersion order. Two new fourth-order symmetric methods with four and five stages are obtained. One of the methods is fourth-order dispersive whereas the other method is symplectic and sixth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with the symplectic DIRK method derived by Sanz-Serna and Abia (SIAM J. Numer. Anal. 28 (1991) 1081–1096).     

Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters

In this paper, we investigate the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results.     

Positive solutions of nonlinear beam equations with time and space singularities

We consider the existence and multiplicity of positive solutions to a nonlinear fourth-order two-point boundary value problem. The nonlinear term may be singular with respect to both the time and space variables. In mechanics, the problem describes the deformation of an elastic beam fixed at the left and supported at the right by sliding clamps. By introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions, several local existence theorems are obtained.     

Fourth-order compact finite difference method for fourth-order nonlinear elliptic boundary value problems

A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.     

A family of Steffensen type methods with seventh-order convergence

In this paper, based on some known fourth-order Steffensen type methods, we present a family of three-step seventh-order Steffensen type iterative methods for solving nonlinear equations and nonlinear systems. For nonlinear systems, a development of the inverse first-order divided difference operator for multivariable function is applied to prove the order of convergence of the new methods. Numerical experiments with comparison to some existing methods are provided to support the underlying theory.