Chaos synchronization of the fractional-order Chen’s system

In this paper, based on the stability theorem of linear fractional systems, a necessary condition is given to check the chaos synchronization of fractional systems with incommensurate order. Chaos synchronization is studied by utilizing the Pecora–Carroll (PC) method and the coupling method. The necessary condition can also be used as a tool to confirm results of a numerical simulation. Numerical simulation results show the effectiveness of the necessary condition.     

On stability and bifurcation of Chen’s system

This article investigates some subtle characteristics of stability and bifurcation of the chaotic Chen’s system, based on rigorous mathematical analysis and symbolic computations.     

Some new insights into a known Chen‐like system

In the paper entitled ‘A novel chaotic system and its topological horseshoe’ in [Nonlinear Analysis: Modelling and Control 18 (1) (2013) 66–77], proposed the 3D chaotic system, , and discussed some of its dynamics according to theoretical and numerical analysis of its parameters . The present work is devoted to giving some new insights into the system for b≥0. Combining theoretical analysis and numerical simulations, some new results are formulated. On the one hand, after some known errors, mainly the distribution of its equilibrium point which is pointed out, correct results are formulated. On the other hand, some of its more rich dynamical properties hiding and not found previously, such as the stability, fold bifurcation, pitchfork bifurcation, degenerated pitchfork bifurcation, and Hopf bifurcation of its isolated equilibria, the dynamics of non‐isolated equilibria, the singularly degenerate heteroclinic cycle, the heteroclinic orbit, and the dynamics at infinity are clearly revealed. Using these results, one can easily explain those interesting phenomena for invariant Lyapunov exponent spectrum and amplitude control that are presented in the known literature. What is more important, we probably demonstrate a new route to chaos. Copyright © 2015 John Wiley & Sons, Ltd.     

On hyperchaos synchronization of a hyperchaotic Lü system

This work presents hyperchaos synchronization of two identical hyperchaotic Lü systems. In this study three methods are applied to achieve hyperchaos synchronization. The sufficient conditions for achieving synchronization of two identical hyperchaotic Lü systems are derived by using Lyapunov stability theory. Numerical simulations are presented to demonstrate the effectiveness of the proposed hyperchaos synchronization schemes.     

Dynamical analysis of the hyperchaos Lorenz system

This article deals with the ultimate bound on the trajectories of the hyperchaos Lorenz system based on Lyapunov stability theory. The innovation of this article lies in that the method of constructing Lyapunov functions applied to the former chaotic systems is not applicable to this hyperchaos system, and moreover, one Lyapunov function can not estimate the bounds of this hyperchaos Lorenz system. We successfully estimate the bounds of this hyperchaos system by constructing three generalized Lyapunov functions step by step. Some computer simulations are also given to show the effectiveness of the proposed scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 440–445, 2016     

Controlling hyperchaos in the new hyperchaotic system

The control problem of a new hyperchaotic system is investigated. The linear, speed, nonlinear doubly-periodic function feedback controls are used to suppress hyperchaos to unstable equilibrium. Limit cases of doubly-periodic function are considered and the hyperbolic function and trigonometric function feedback control laws are derived. The Routh–Hurwitz criterion is applied to study the conditions of the asymptotic stability of the controlled hyperchaotic system. Based on Mathematica program, numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.     

Stabilization of parameters perturbation chaotic system via adaptive backstepping technique

The work of Yassen [M.T. Yassen, Chaos control of chaotic dynamical systems using backstepping design, Chaos Soliton Fract. 27 (2006) 537–548] which mainly investigated the stabilization problem for a class of chaotic systems without the parameters perturbation. This paper is concerned with stabilization problem for a class of parameters perturbation chaotic systems via both backstepping design method and adaptive technique. The proposed controllers can guarantee that the parameters perturbation systems will be stabilized at a fixed bounded point. Furthermore, the paper also proposes controllers to stabilize the uncertain chaotic system at equilibrium point with only backstepping design method. Finally, numerical simulations are given to illustrate the effectiveness of the proposed controllers.     

A homotopy perturbation algorithm to solve a system of Fredholm–Volterra type integral equations

In this paper, an application of He’s homotopy perturbation (HPM) method is applied to solve the system of Fredholm and Volterra type integral equations, the results revealing that the HPM is very effective and simple.     

Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

We consider a singular perturbation of the generalized viscous Cahn–Hilliard equation based on constitutive equations introduced by Gurtin. This equation rules the order parameter ρ, which represents the density of atoms, and it is given on a n‐rectangle (n?3) with periodic boundary conditions. We prove the existence of a family of exponential attractors that is robust with respect to the perturbation parameter ε>0, as ε goes to 0. In a similar spirit, we analyze the stability of the global attractor. If n=1, 2, then we also construct a family of inertial manifolds that is continuous with respect to ε. These results improve and generalize the ones contained in some previous papers. Finally, we establish the convergence of any trajectory to a single equilibrium via a suitable version of the ?ojasiewicz–Simon inequality, provided that the potential is real analytic. Copyright © 2007 John Wiley & Sons, Ltd.     

Normal deviations from the averaged motion for some reaction–diffusion equations with fast oscillating perturbation

We study the normalized difference between the solution u of a reaction–diffusion equation in a bounded interval [0,L], perturbed by a fast oscillating term arising as the solution of a stochastic reaction–diffusion equation with a strong mixing behavior, and the solution of the corresponding averaged equation. We assume the smoothness of the reaction coefficient and we prove that a central limit type theorem holds. Namely, we show that the normalized difference converges weakly in C([0,T];L²(0,L)) to the solution of the linearized equation, where an extra Gaussian term appears. Such a term is explicitly given.     

Homotopy perturbation method for the mixed Volterra–Fredholm integral equations

This article presents a numerical method for solving nonlinear mixed Volterra–Fredholm integral equations. The method combined with the noise terms phenomena may provide the exact solution by using two iterations only. Two numerical illustrations are given to show the pertinent features of the technique. The results reveal that the proposed method is very effective and simple.     

A New Class of Graphs That Satisfies the Chen‐Chvátal Conjecture

A well‐known combinatorial theorem says that a set of n non‐collinear points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work, we prove a slightly stronger version of Chen and Chvátal conjecture for a family of graphs containing chordal graphs and distance‐hereditary graphs.     

Robust controlling hyperchaos of the Rössler system subject to input nonlinearities by using sliding mode control

A sliding mode control is designed to stabilize the well-known hyperchaos of Rössler system to equilibrium points subject to sector nonlinear input. The proposed control law is robust against both the input nonlinearity and external disturbance. The error bound can be arbitrarily set by assigning the corresponding dynamics to the sliding surfaces when the desired state is not an equilibrium point. Simulation results show that the system state can be regulated to an equilibrium point in the state space. It is also seen that the system still possesses advantage of fast response and good transient performance even though the control input is nonlinear.     

Chaotic bursting lag synchronization of Hindmarsh–Rose system via a single controller

Chaotic bursting lag synchronization of Hindmarsh–Rose system is investigated. Two lag synchronization schemes with only a single controller are proposed to synchronize Hindmarsh–Rose chaotic system via back stepping method. Especially in the second scheme, only one state variable is contained in the controller. Based on Lyapunov stability theory, the sufficient conditions for synchronization are obtained analytically in both cases. Finally, numerical simulations are provided to show the effectiveness of the developed methods.     

From the Klein–Gordon–Zakharov system to the Klein–Gordon equation

In a singular limit, the Klein–Gordon (KG) equation can be derived from the Klein–Gordon–Zakharov (KGZ) system. We point out that for the original system posed on a d‐dimensional torus, the solutions of the KG equation do not approximate the solutions of the KGZ system. The KG system has to be modified to make correct predictions about the dynamics of the KGZ system. We explain that this modification is not necessary for the approximation result for the whole space with d≥3. Copyright © 2016 John Wiley & Sons, Ltd.     

Global sharp interface limit of the Hele–Shaw–Cahn–Hilliard system

In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two‐phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by ? , and the mobility coefficient (the diffusional Peclet number) is ? ^α . We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as ? →0 in the case of 0≤α  < 1. Copyright © 2016 John Wiley & Sons, Ltd.     

Weights in Serre's conjecture for via the Bernstein–Gelfand–Gelfand complex

Let be an n-dimensional mod p Galois representation. If ρ is modular for a weight in a certain class, called p-minute, then we restrict the Fontaine–Laffaille numbers of ρ; in other words, we specify the possibilities for the restriction of ρ to inertia at p. Our result agrees with the Serre-type conjectures for GL_n formulated by Ash, Doud, Pollack, Sinnott, and Herzig; to our knowledge, this is the first unconditional evidence for these conjectures for arbitrary n.     

Vibration of a Reissner–Mindlin–Timoshenko plate–beam system

In this paper, we consider a plate–beam system in which the Reissner–Mindlin plate model is combined with the Timoshenko beam model. Natural frequencies and vibration modes for the system are calculated using the finite element method. The interface conditions at the contact between the plate and beams are discussed in some detail. The impact of regularity on the enforcement of certain interface conditions is an important feature of the paper.     

Sliding mode control of hyperchaos in Rössler systems

In this paper, an extended sliding mode controller is applied to control a hyperchaotic motion in Rössler system. The sliding surface of this paper used is one dimension higher than traditional sliding surface and guarantees it passing through the activated initial states of controlled system. Therefore, using the characteristic of this sliding mode to design a controller not only can meet the desired specification but also without chattering phenomenon and abrupt state change. By comparing with the result in the existed literatures, the results show that the proposed controller can steer Rössler system to the desired state accurately. It also provides a good characteristic for disturbance rejection.