Q-S synchronization in 3D Hénon-like map and generalized Hénon map via a scalar controller

In this Letter, a generalized, systematic and automatic backstepping scheme is developed to investigate the Q-S synchronization of two identical 3D discrete-time dynamical systems and two different 3D discrete-time dynamical systems. With the aid of symbolic-numeric computation, we use the proposed scheme to illustrate chaos synchronization between two identical 3D generalized Hénon map and Q-S synchronization between two different 3D generalized Hénon map and Hénon-like map via a scalar controller, respectively. Moreover numerical simulations are used to verify the effectiveness of the proposed scheme. In addition, the scheme can be also applied to investigate the tracking problem in the discrete-time systems and to generate automatically the scalar controller in computer with the aid of symbolic-numeric computation.     

Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: a symbolic-numeric computation approach

First, a type of Q-S (complete or anticipated) synchronization is defined in discrete-time dynamical systems. Second, based on backstepping design with a scalar controller, a systematic, concrete and automatic scheme is presented to investigate Q-S (complete or anticipated) synchronization between the discrete-time drive system and response system with strict-feedback form. Finally, the proposed scheme is used to illustrate Q-S (complete or anticipated) synchronization between the two-dimensional discrete-time Lorenz system and Fold system, as well as the three-dimensional hyperchaotic discrete-time Rossler system and generalized discrete-time Rossler system. Moreover numerical simulations are used to verify the effectiveness of the proposed scheme. Our scheme can be also extended to investigate Q-S (complete or anticipated) synchronization between other discrete-time dynamical systems with strict-feedback forms. With the aid of symbolic-numeric computation, the scheme can be performed to yield automatically the scalar controller and to verify its effectiveness in computer.     

Inverse full state hybrid projective synchronization for chaotic maps with different dimensions

A new synchronization scheme for chaotic(hyperchaotic) maps with different dimensions is presented.Specifically,given a drive system map with dimension n and a response system with dimension m,the proposed approach enables each drive system state to be synchronized with a linear response combination of the response system states.The method,based on the Lyapunov stability theory and the pole placement technique,presents some useful features:(i) it enables synchronization to be achieved for both cases of n m and n m;(ii) it is rigorous,being based on theorems;(iii) it can be readily applied to any chaotic(hyperchaotic) maps defined to date.Finally,the capability of the approach is illustrated by synchronization examples between the two-dimensional H′enon map(as the drive system) and the three-dimensional hyperchaotic Wang map(as the response system),and the three-dimensional H′enon-like map(as the drive system) and the two-dimensional Lorenz discrete-time system(as the response system).     

广义Hénon混沌系统的自适应双模控制与同步

提出一种广义Hénon映射的自适应双模控制与同步方法.广义Hénon映射的混沌吸引子比Hénon映射的混沌吸引子更复杂,控制与同步困难,对于保密通信来说具有更高的安全性.该方法采用自适应双模控制,实现了广义Hénon映射的追踪控制与同步,提高了受控系统抑制参数摄动和随机扰动的能力,改善系统的鲁棒性.仿真结果验证了该方法的有效性.     

Control and synchronization of discrete-time chaotic systems via variable structure control technique

We present an input/output linearization control method incorporated in a discrete-time variable structure control technique to resolve the output tracking problem of a class of discrete-time nonlinear systems. The proposed control scheme is then applied to address the control and synchronization problems associated with the Hénon chaotic systems. Numerical simulations demonstrate the feasibility and robustness of the proposed control strategy.     

Hénon混沌系统的广义预测控制与同步快速算法

提出了一种带有终端滑模等式约束的Hénon混沌系统的广义预测控制快速算法.该方法将广义预测控制与离散滑模控制结合起来运用到Hénon混沌系统中，并采用柔化输入信号的方法，避免了广义预测控制算法中的高维矩阵求逆.占用内存小，计算速度快，并且具有较强的跟踪给定信号和抑制系统参数摄动和随机噪声的能力.仿真结果验证了该方法的有效性和快速性. 关键词： 广义预测控制 离散滑模控制 Hénon混沌系统 混沌同步     

A new scheme to generalized (lag, anticipated, and complete) synchronization in chaotic and hyperchaotic systems

In this paper, a generalized (lag, anticipated, and complete) synchronization of a class of continuous-time systems is defined. A systematic, powerful and concrete scheme is developed to investigate the generalized (lag, anticipated, and complete) synchronization between the drive system and response system based on the active control idea. The hyperchaotic R?ssler system, transformed R?ssler and Chen system as well as two coupled nonidentical R?ssler oscillators are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. The scheme can be also extended to research generalized (lag, anticipated, and complete) synchronization between other dynamical systems.     

Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme

In this paper, a bidirectional partial generalized (lag, complete, and anticipated) synchronization of a class of continuous-time systems is defined. Then based on the active control idea, a new systematic and concrete scheme is developed to achieve bidirectional partial generalized (lag, complete, and anticipated) synchronization between two chaotic systems or between chaotic and hyperchaotic systems. With the help of symbolic-numerical computation, we choose the modified Chua system, Lorenz system, and the hyperchaotic Tamasevicius-Namajunas-Cenys system to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. It is interesting that partial chaos synchronization not only can take place between two chaotic systems, but also can take place between chaotic and hyperchaotic systems. The proposed scheme can also be extended to research bidirectional partial generalized (lag, complete, and anticipated) synchronization between other dynamical systems.     

Generalized-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal

In this paper, a systematic and powerful scheme is proposed to address a generalized-type synchronization of a class of continuous-time systems, which includes generalized lag synchronization, generalized anticipated synchronization, and generalized synchronization. The presented scheme is used to investigate the generalized-type synchronization of the 4D hyperchaotic oscillator and the hyperchaotic oscillator with gyrators. Numerical simulations are used to verify the effectiveness of the proposed scheme. The scheme is more powerful than the scalar signal scheme due to Grassi and Mascolo.     

Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hènon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hènon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30     

Generalized-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal

In this paper, a systematic and powerful scheme is proposed to address a generalized-type synchronization of a class of continuous-time systems, which includes generalized lag synchronization, generalized anticipated synchronization, and generalized synchronization. The presented scheme is used to investigate the generalized-type synchronization of the 4D hyperchaotic oscillator and the hyperchaotic oscillator with gyrators. Numerical simulations are used to verify the effectiveness of the proposed scheme. The scheme is more powerful than the scalar signal scheme due to Grassi and Mascolo.     

基于Foliation条件的离散动力系统二维流形计算

研究离散动力系统双曲不动点的二维流形计算,利用不变流形轨道上Jacobian矩阵能够传递导数这一特殊性质,提出一种新的一维流形计算方法,通过预测-校正两个步骤迅速确定流形上新网格点,避免重复计算,并简化精度控制条件.在此基础上,将基于流形面Foliation条件进行推广,推广后的Foliation条件能够控制二维流形上的一维子流形的增长速度,从而实现二维流形在各个方向上的均匀增长.此外,算法可以同时用于二维稳定和不稳定流形的计算.以超混沌三维Hénon映射和具有蝶形吸引子的Lorenz系统为例验证了算法的有效性.     

A three-dimensional dissipative map with three routes to chaos

The classical 2D Hénon map has been generalized to 3D while maintaining the jacobian equal to a constant (?b). A numerical exploration of this map has been conducted. If we fix b and vary the remaining free parameter β, three routes to chaos are observed, illustrated and analyzed.     

基于广义Foliation条件的非线性映射 二维流形计算

主要研究非线性映射函数双曲不动点的二维流形计算问题. 提出了推广的Foliation条件, 以此来衡量二维流形上的一维流形轨道的增长量, 进而控制各子流形的增长速度, 实现二维流形在各个方向上的均匀增长. 此外, 提出了一种一维子流形轨道的递归插入算法, 该算法巧妙地解决了二维流形面上网格点的插入、前像搜索, 以及网格点后续轨道计算问题, 同时插入的轨道不必从初始圆开始计算, 避免了在初始圆附近产生过多的网格点. 以超混沌三维Hénon映射和具有蝶形吸引子的Lorenz系统为例验证了算法的有效性.     

Generalized synchronization between different chaotic maps via dead-beat control

This paper presents a new scheme to achieve generalized synchronization(GS) between different discrete-time chaotic(hyperchaotic) systems.The approach is based on a theorem,which assures that GS is achieved when a structural condition on the considered class of response systems is satisfied.The method presents some useful features:it enables exact GS to be achieved in finite time(i.e.,dead-beat synchronization);it is rigorous,systematic,and straightforward in checking GS;it can be applied to a wide class of chaotic maps.Some examples of GS,including the Grassi-Miller map and a recently introduced minimal 2-D quadratic map,are illustrated.     

Hénon混沌系统广义预测函数控制快速收敛算法

提出一种具有收敛跟踪性能的Hénon混沌系统广义预测函数控制快速算法.基于已有的广义预测函数控制方法,依据混沌系统运动特性和实际中控制量变化趋势选取特殊形式的基函数,避免了控制律中的矩阵求逆计算,并在性能指标函数中引入确定的前馈增益矩阵,实现Hénon混沌系统对参考信号的收敛跟踪.仿真结果验证了该算法的有效性. 关键词： 广义预测控制 预测函数控制 Hénon混沌系统 前馈增益矩阵     

Adaptive Control and Function Projective Synchronization in 2D Discrete-Time Chaotic Systems

This study addresses the adaptive control and function projective synchronization problems between 2D Rulkov discrete-time system and Network discrete-time system. Based on backstepping design with three controllers, a systematic, concrete and automatic scheme is developed to investigate the function projective synchronization of discretetime chaotic systems. In addition, the adaptive control function is applied to achieve the state synchronization of two discrete-time systems. Numerical results demonstrate the effectiveness of the proposed control scheme.     

Dynamical analysis of bounded and unbounded orbits in a generalized Hénon–Heiles system

The Hénon–Heiles potential was first proposed as a simplified version of the gravitational potential experimented by a star in the presence of a galactic center. Currently, this system is considered a paradigm in dynamical systems because despite its simplicity exhibits a very complex dynamical behavior. In the present paper, we perform a series expansion up to the fifth-order of a potential with axial and reflection symmetries, which after some transformations, leads to a generalized Hénon–Heiles potential. Such new system is analyzed qualitatively in both regimes of bounded and unbounded motion via the Poincaré sections method and plotting the exit basins. On the other hand, the quantitative analysis is performed through the Lyapunov exponents and the basin entropy, respectively. We find that in both regimes the chaoticity of the system decreases as long as the test particle energy gets far from the critical energy. Additionally, we may conclude that despite the inclusion of higher order terms in the series expansion, the new system shows wider zones of regularity (islands) than the ones present in the Hénon–Heiles system.     

Hénon混沌系统的追踪控制与同步

基于离散线性系统的稳定性理论,实现了Hénon混沌系统对参考信号的追踪控制.讨论了系统的同步问题,给出了“异结构混沌同步”的新概念.数值仿真表明了本文方法的有效性 关键词： Hénon混沌系统 追踪控制 混沌同步     

Quantifying the synchronizability of externally driven oscillators

This paper is focused on the problem of complete synchronization in arrays of externally driven identical or slightly different oscillators. These oscillators are coupled by common driving which makes an occurrence of generalized synchronization between a driving signal and response oscillators possible. Therefore, the phenomenon of generalized synchronization is also analyzed here. The research is concentrated on the cases of an irregular (chaotic or stochastic) driving signal acting on continuous-time (Duffing systems) and discrete-time (Henon maps) response oscillators. As a tool for quantifying the robustness of the synchronized state, response (conditional) Lyapunov exponents are applied. The most significant result presented in this paper is a novel method of estimation of the largest response Lyapunov exponent. This approach is based on the complete synchronization of two twin response subsystems via additional master-slave coupling between them. Examples of the method application and its comparison with the classical algorithm for calculation of Lyapunov exponents are widely demonstrated. Finally, the idea of effective response Lyapunov exponents, which allows us to quantify the synchronizability in case of slightly different response oscillators, is introduced.