Darboux integrability of a generalized Friedmann-Robertson-Walker Hamiltonian system

We study the Darboux first integrals of a generalized Friedmann-Robertson-Walker Hamiltonian system.     

Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems

The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in ${\mathbb{C}}^2$ is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing–van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing–van der Pol system are constructed.     

Local analytic first integrals of planar analytic differential systems

We study the existence of local analytic first integrals of a class of analytic differential systems in the plane, obtained from the Chua?s system studied in L.O. Chua (1992, 1995), N.V. Kuznetsov et al. (2011), G.A. Leonov et al. (2012) , ,  and . The method used can be applied to other analytic differential systems.     

Nonlinear dynamical systems,first integrals,Bose operators,and Lie algebras

A connection between nonlinear autonomous systems of ordinary differential equations, first integrals, Bose operators and Lie algebras is described. An extension to nonlinear partial differential equations is given.     

On the Darboux integrability of a three–dimensional forced–damped differential system

In 2011 Pehlivan proposed a three–dimensional forced–damped autonomous differential system which can display simultaneously unbounded and chaotic solutions. This untypical phenomenon has been studied recently by several authors. In this paper we study the opposite to its chaotic motion, i.e. its integrability, mainly the existence of polynomial, rational and Darboux first integrals through the analysis of its invariant algebraic surfaces and its exponential factors.     

New results for exponential synchronization of linearly coupled ordinary differential systems

This paper investigates the exponential synchronization of linearly coupled ordinary differential systems. The intrinsic nonlinear dynamics may not satisfy the QUAD condition or weak-QUAD condition. First, it gives a new method to analyze the exponential synchronization of the systems. Second, two theorems and their corollaries are proposed for the local or global exponential synchronization of the coupled systems. Finally, an application to the linearly coupled Hopfield neural networks and several simulations are provided for verifying the effectiveness of the theoretical results.     

Limit cycles for a class of second order differential equations

We study the limit cycles of a wide class of second order differential equations, which can be seen as a particular perturbation of the harmonic oscillator. In particular, by choosing adequately the perturbed function we show, using the averaging theory, that it is possible to obtain as many limit cycles as we want.     

New C-integrable and S-integrable systems of nonlinear partial differential equations

A technique to identify new C-integrable and S-integrable systems of nonlinear partial differential equations is reported, with two representative examples displayed and tersely discussed.     

Asymptopic solution for a class of semilinear singularly perturbed fractional differential equation

This paper considers a class of boundary value problems for the semilinear singularly perturbed fractional differential equation.Under the suitable conditions,first,the outer solution of the original problem is obtained;secondly,using the stretched variable and the composing expansion method the boundary layer is constructed;finally,using the theory of differential inequalities the asymptotic behaviour of solution for the problem is studied and the uniformly valid asymptotic estimation is discussed.     

Chaotic synchronization for a class of fractional-order chaotic systems

In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.     

利用标量控制器实现一类混沌系统同步

研究了一类混沌系统的混沌同步，对此类混沌系统，通过设计一个合适标量控制器，可以实现系统的混沌同步.给出了该标量控制器设计的一般方法，并从理论上得到了混沌同步的充分和必要条件，且此充分和必要条件与混沌系统的性质无关. 关键词： 混沌系统 标量控制器 混沌同步     

Impulsive synchronisation of hyperchaotic a class of fractional-order systems

In this paper, an impulsive synchronisation scheme for a class of fractional-order hyperchaotic systems is proposed. The sufficient conditions of a class of integral-order hyperchaotic systems＇ impulsive synchronisation are illustrated. Furthermore, we apply the sufficient conditions to a class of fractional-order hyperchaotic systems and well achieve impulsive synchronisation of these fractional-order hyperchaotic systems, thereby extending the applicable scope of impulsive synchronisation. Numerical simulations further demonstrate the feasibility and effectiveness of the proposed scheme.     

Fractal differential equations and fractal-time dynamical systems

Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate framework. A new calculus calledF ^α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF ^α-integral andF ^α-derivative respectively. TheF ^α-integral is suitable for integrating functions with fractal support of dimension α, while theF ^α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions ofF ^α-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems. We discuss construction and solutions of some fractal differential equations of the form

Impulsive synchronisation of a class of fractional-order hyperchaotic systems

In this paper,an impulsive synchronisation scheme for a class of fractional-order hyperchaotic systems is proposed.The sufficient conditions of a class of integral-order hyperchaotic systems’ impulsive synchronisation are illustrated.Furthermore,we apply the sufficient conditions to a class of fractional-order hyperchaotic systems and well achieve impulsive synchronisation of these fractional-order hyperchaotic systems,thereby extending the applicable scope of impulsive synchronisation.Numerical simulations further demonstrate the feasibility and effectiveness of the proposed scheme.     

一类3维连续混沌系统观测器

讨论了一类连续三维混沌系统的混沌观测器的设计问题.对于此类混沌系统，仅利用系统的 一个状态变量及它对时间的一阶、二阶导数就可以使得观测器的状态变量同被观测系统的状 态变量达到同步. 关键词： 混沌系统 观测器 导数 同步     

Reliable impulsive synchronization for a class of nonlinear chaotic systems

The problem of reliable impulsive synchronization for a class of nonlinear chaotic systems has been investigated in this paper. Firstly a reliable impulsive controller is designed by using the impulsive control theory. Then by the uniform asymptotic stability criteria of systems with impulsive effects, some sufficient conditions for reliable impulsive synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

含有不确定项的混沌系统自适应修正函数投影同步

针对一类含有不确定项的混沌系统修正函数投影同步问题, 提出了自适应修正函数投影同步方法.根据Lyapunov稳定性理论, 给出了两种响应系统的设计方案,并设计了控制器和自适应律. 研究表明,所设计的控制策略对外界干扰有较强的鲁棒性, 且不需要事先已知不确定项的上界,通过引入加速因子,可任意配置同步响应速度, 具有较高的使用价值.理论分析及仿真结果验证了该方法的有效性.     

On the nonexistence of Liouvillian first integrals for generalized Liénard polynomial differential systems

We consider generalized Liénard polynomial differential systems of the form ? = y, ? = -g(x) - f (x) y, with f (x) and g(x) two polynomials satisfying deg(g) ≤ deg(f). In their work, Llibre and Valls have shown that, except in some particular cases, such systems have no Liouvillian first integral. In this letter, we give a direct and shorter proof of this result.