Synchronization of generalized Hénon map using polynomial controller

This Letter presents the chaos synchronization of two discrete-time generalized Hénon map, namely the drive and response systems. A polynomial controller is proposed to drive the system states of the response system to follow those of the drive system. The system stability of the error system formed by the drive and response systems and the synthesis of the polynomial controller are investigated using the sum-of-squares (SOS) technique. Based on the Lyapunov stability theory, stability conditions in terms of SOS are derived to guarantee the system stability and facilitate the controller synthesis. By satisfying the SOS-based stability conditions, chaotic synchronization is achieved. The solution of the SOS-based stability conditions can be found numerically using the third-party Matlab toolbox SOSTOOLS. A simulation example is given to illustrate the merits of the proposed polynomial control approach.     

First-order system least squares and the energetic variational approach for two-phase flow

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen–Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.     

Arnold's method for asymptotic stability of steady inviscid incompressible flow through a fixed domain with permeable boundary

The flow of an ideal fluid in a domain with a permeable boundary may be asymptotically stable. Here the permeability means that the fluid can flow into and out of the domain through some parts of the boundary. This permeability is a principal reason for the asymptotic stability. Indeed, the well-known conservation laws make the asymptotic stability of an inviscid flow impossible, if the usual no flux condition on a rigid wall (or on a free boundary) is employed. We study the stability problem using the direct Lyapunov method in the Arnold's form. We prove the linear and nonlinear Lyapunov stability of a two-dimensional flow through a domain with a permeable boundary under Arnold's conditions. Under certain additional conditions, we amplify the linear result and prove the exponential decay of small disturbances. Here we employ the plan of the proof of the Barbashin-Krasovskiy theorem, established originally only for systems with a finite number of degrees of freedom. (c) 2002 American Institute of Physics.     

Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays

This Letter is concerned with the global asymptotic stability analysis problem for a class of uncertain stochastic Hopfield neural networks with discrete and distributed time-delays. By utilizing a Lyapunov–Krasovskii functional, using the well-known S-procedure and conducting stochastic analysis, we show that the addressed neural networks are robustly, globally, asymptotically stable if a convex optimization problem is feasible. Then, the stability criteria are derived in terms of linear matrix inequalities (LMIs), which can be effectively solved by some standard numerical packages. The main results are also extended to the multiple time-delay case. Two numerical examples are given to demonstrate the usefulness of the proposed global stability condition.     

Adaptive control and synchronization of an uncertain new hyperchaotic Lorenz system

This paper is involved with the adaptive control and synchronization problems for an uncertain new hyperchaotic Lorenz system. Based on the Lyapunov stability theory, the adaptive control law is derived such that the trajectory of hyperchaotic Lorenz system with unknown parameters can be globally stabilized to an unstable equilibrium point of the uncontrolled system. Furthermore, an adaptive control approach is presented to the synchronizations between two identical hyperchaotic systems, particularly between two different uncertain hyperchaotic systems. Numerical simulations show the effectiveness of the presented method.     

Shape optimization towards stability in constrained hydrodynamic systems

Hydrodynamic stability plays a crucial role for many applications. Existing approaches focus on the dependence of the stability properties on control parameters such as the Reynolds or the Rayleigh number. In this paper we propose a numerical method which aims at solving shape optimization problems in the context of hydrodynamic stability. The considered approach allows to guarantee hydrodynamic stability by modifying parts of the underlying geometry within a certain flow regime. This leads to a formulation of a shape optimization problem with constraints on the eigenvalues related to the linearized Navier–Stokes equations. In that context the eigenvalue problem is generally non-symmetric and may involve complex eigenvalues. To validate the proposed numerical approach we consider the flow around a body in a channel. The shape of the body is parameterized and can be changed by means of a discrete number of design variables. It is our aim to find a design which minimizes the drag force and ensures at the same time hydrodynamic stability while keeping the volume of the body constant. The numerical results show that a transition from an unstable design to a stable one is attainable by considering an adequate change of the body shape. The resulting bodies are long and flat which corresponds to common intuition.     

Passive control and synchronization of hyperchaotic Chen system

This paper investigates the control and synchronization of hyperchaotic Chen system based on the passive theory. By using two outputs, novel passive controllers are respectively designed to realize the globally asymptotical stability of the hyperchaotic Chen system and the error dynamical system, which avoids mistakes in Ref.[11], where function W（z） cannot guarantee that fo（z） is globally asymptotically stable via only one output and W（z） is the Lyapunov function of f0（z）. Furthermore, numerical simulations are given to show the effectiveness of our method.     

Stability of piecewise affine systems with application to chaos stabilization

This paper addresses the stability issue for a class of piecewise affine (PWA) systems, where the state spaces are assumed to be dividable into a certain number of hypercuboid subspaces. By constructing appropriate piecewise continuous Lyapunov functions, several numerically tractable stability criteria are developed for four subclasses of such PWA systems, which allow to recast the switching control problem for the PWA systems as a convex optimization problem. Moreover, the proposed method is applied to switching controller design for (globally and locally) stabilizing the unstable equilibrium points of PWA chaotic systems. Numerical simulations on the chaotic Chua's circuit are presented to verify the theoretical results.     

电力电子化电力系统稳定的问题及挑战:以暂态稳定比较为例

随着电力电子技术的进步和环境保护对清洁能源的要求,以同步发电机为主的传统电力系统正向着多样化电力电子装备为主的电力系统转变,由此电力系统正面临着百年来未有之大变局.近年来,国内外不断报道出以电力电子装备为主的新能源基地和传统高压直流等机理不明的电力事故,严重威胁了电力系统安全稳定运行.针对上述问题,本文首先介绍传统电力系统暂态稳定分析的主要方法,接着分析了典型故障场景下简单电力电子化电力系统的动力学行为,并建立了同时包含电力电子设备与传统同步机的多机耦合系统模型,最后总结了电力电子化电力系统的非线性、多时标、复杂性的本质特点,归纳其暂态稳定的基本问题与挑战以及对未来研究方向的展望,希望引起复杂系统和统计物理背景的研究人员的广泛兴趣.     

对流扩散方程的一些单调性差分格式

本文分析了逆风格式、格式和修正Dennis格式随网格Reynolds数变化的性质,构造了相应的半隐差分格式,它们具有无条件稳定性和无条件单调性。数值例子说明这些格式可以计算高Reynolds的定态流动问题。     

Stability analysis of a simple rheonomic nonholonomic constrained system

It is a difficult problem to study the stability of the rheonomic and nonholonomic mechanical systems. Especially it is difficult to construct the Lyapunov function directly from the differential equation. But the gradient system is exactly suitable to study the stability of a dynamical system with the aid of the Lyapunov function. The stability of the solution for a simple rheonomic nonholonomic constrained system is studied in this paper. Firstly, the differential equations of motion of the system are established. Secondly, a problem in which the generalized forces are exerted on the system such that the solution is stable is proposed. Finally, the stable solutions of the rheonomic nonholonomic system can be constructed by using the gradient systems.     

Stability of power grids: An overview

Transient stability and steady-state (small signal) stability in power girds are reviewed. Transient stability concepts are illustrated with simple examples; in particular, we consider three methods for computing region of attraction: time-simulations, extended Lyapunov function, and sum of squares optimization method. We discuss steady state stability in power systems, and present an example of a feedback control via a communication network for the 10?Unit 39 Bus New England Test system.     

Lyapunov instability, local curvature, and the fluid-solid phase transition in two-dimensional particle systems

We study the relation between the Lyapunov spectrum and the multidimensional geometry of the potential energy surface in terms of the distribution of stable and unstable modes for different models. For this purpose we determined Lyapunov exponents for the so-called correlated cell model and its smooth generalization as a function of the density for various energies. In the smooth case averaged structural quantities, such as the fraction of unstable modes, the Gaussian curvature, and the Riemannian curvature were computed and compared to the mechanical instability of the system in the sense of Lyapunov. A similar analysis was also carried out for 36-disk systems representing various fluid and solid states. In all studied cases the most-positive Lyapunov exponent exhibits a maximum at the phase transition in agreement with results for the fluid-solid transition in many-particle systems.     

基于模糊观测器的Chua混沌系统投影同步

研究了Chua混沌系统的广义投影同步问题.基于Takagi-Sugeno(T-S)模糊模型,设计了一种模糊观测器.利用Lyapunov稳定性理论证明了所提方案的可行性和全局稳定性.数值仿真试验进一步验证了该方案可以实现Chua混沌系统的投影同步.方法设计简单,且可以推广到其他的一些混沌系统. 关键词： T-S模糊模型 Chua混沌系统 观测器 投影同步     

Adaptive synchronization of Rossler and Chen chaotic systems

A novel adaptive synchronization method is proposed for two identical Rossler and Chen systems with uncertain parameters. Based on Lyapunov stability theory, we derive an adaptive controller without the knowledge of the system parameters, which can make the states of two identical Rossler and Chen systems globally asymptotically synchronized. Especially, when some unknown uncertain parameters are positive, we can make the controller more simple and, besides, the controller is independent of those positive uncertain parameters. All results are proved using a well-known Lyapunov stability theorem. Numerical simulations are given to validate the proposed synchronization approach.     

Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids

Plane Couette flow of viscoelastic fluids is shown to exhibit a purely elastic subcritical instability at a very small-Reynolds number in spite of being linearly stable. The mechanism of this instability is proposed and the nonlinear stability analysis of plane Couette flow of the Upper-Convected Maxwell fluid is presented. Above a critical Weissenberg number, a small finite-size perturbation is sufficient to create a secondary flow, and the threshold value for the amplitude of the perturbation decreases as the Weissenberg number increases. The results suggest a scenario for weakly turbulent viscoelastic flow which is similar to the one for Newtonian fluids as a function of Reynolds number.     

带有未知非对称控制增益的不确定分数阶混沌系统自适应模糊同步控制

针对带有非对称控制增益的不确定分数阶混沌系统的同步问题设计了模糊自适应控制器. 模糊逻辑系统用来逼近未知的非线性函数, 非对称的控制增益矩阵被分解为一个未知的正定矩阵、一个对角线上元素为+1或-1的已知对角矩阵和 一个未知的上三角矩阵的乘积. 基于分数阶Lyapunov稳定性理论构造了模糊控制器以及分数阶的参数自适应律, 在保证所有变量有界的情况下实现驱动系统和响应系统的同步. 在分数阶系统稳定性分析中给出了一种平方Lyapunov函数的使用方法, 根据此方法很多针对整数阶系统的控制方法可以推广到分数阶系统中. 最后数值仿真结果验证了所提控制方法的可行性.     

Spatiotemporal chaos and noise

Low-dimensional chaotic dynamical systems can exhibit many characteristic properties of stochastic systems, such as broad Fourier spectra. They are distinguishable from stochastic processes through finite values for their dimension, Lyapunov exponents, and Kolmogorov-Sinai entropy. We discuss how these characteristic observables are modified in spatiotemporal chaotic systems like. coupled map lattices. We analyze with the help of Lyapunov concepts how the stochastic limit is approached and how these properties can be observed directly through local dimension measurements from reconstructed time series. Finally, we discuss the interaction of spatiotemporal attractors with external noise and possible connections to problems of pattern selection and stability.     

Hilbert's 17th problem and the quantumness of states

A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them.     

Dynamical analysis of a new multistable chaotic system with hidden attractor: Antimonotonicity,coexisting multiple attractors,and offset boosting

Analyzing chaotic systems with coexisting and hidden attractors has been receiving much attention recently. In this article, we analyze a four dimensional chaotic system which has a plane as the equilibrium points. Also this system is of the group of systems that have coexisting attractors. First, the system is introduced and then stability analysis, bifurcation diagram and Largest Lyapunov exponent of this system are presented as methods to analyze the multistability of the system. These methods reveal that in some ranges of the parameter, this chaotic system has three different types of coexisting attractors, chaotic, stable node and limit cycle. Some interesting dynamics properties such as reversals of period doubling bifurcation and offset boosting are also presented.