几类混沌系统的设计及应用研究
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摘要
混沌电路即能产生混沌现象的电路,混沌现象广泛存在于非线性电路中,混沌应用研究离不开混沌电路。传统的混沌电路是典型的模拟电路,使用电容、电感、电阻、集成运放、模拟乘法器等分立元件。随着FPGA (Field Programmable Gate Array,现场可编程门阵列)和DSP (Digital Signal Processing,数字信号处理)等集成电路数字化处理技术的发展,使用FPGA和DSP等技术产生混沌信号正逐渐成为一种趋势。目前基于FPGA和DSP等技术的混沌电路设计还处于个性化设计阶段。设计上存在很大的局限性。其中,最大的局限性体现在不可扩展和复制上。设计者在完成一个混沌电路的设计后,即使电路混沌状态方程只有小小变化,其电路仍需从头设计。因此,研究如何将IP核模块的Soc设计方法引入到基于FPGA的混沌电路化设计中,通过建立IP核资源库,形成一整套参数可配置IP核的混沌电路化设计体系,对促进混沌应用的发展具有重要的意义。
本文在回顾传统的混沌电路产生混沌现象的基础上,对基于IP核的混沌电路设计的可能性,及其组成、IP核的混沌电路设计的基础支撑、IP核的混沌电路设计流程等方面进行了研究。具体研究内容包括：(1)探讨并论证了基于IP核的混沌电路设计的可行性；(2)研究了基于IP核的混沌电路设计的组成,在研究了N阶无量纲连续混沌状态方程的一般形式的基础上,提出基于IP核的混沌电路设计可转换为：基本函数IP核设计和基本运算IP核设计。(3)研究了基于IP核的混沌电路设计的基础支撑,提出了设计IP核管理工具的思想。(4)在传统的FPGA设计流程上,提出基于IP核的混沌电路设计流程,指出基于IP核的混沌电路设计流程将采用IP模块而不是基本逻辑或电路单元作为基础单元,利用现有的IP资源,根据需求选用适当的IP模块,进行系统集成设计。(5)研究了基于采用DES和RSA作为密钥交换方案的数字保密通信系统：混沌加密与常规加密级联的方案。即采用RSA公钥制作算法作为密钥交换方案,加密混沌加密与常规加密算法的私钥,以保证混沌的同步和安全性,使级联方案在网络通信的情况下更容易实现。(6)以软件形式在ARM开发板上实现了基于混沌加密的语音网络传输功能,在开发过程中解决了ARM板与PC之间浮点数运算精度差异的问题。(7)研究了基于地址码的脆弱数字水印技术,并在此基础上给出了一种定位精确的混沌脆弱数字水印框架,该水印不但能检测篡改而且能够完全定位篡改发生的区域。(8)提出了一种有别于传统蔡氏电路设计的新方法。以多项式ax+bx|x|+cx3产生三涡卷蔡氏混沌吸引子为例,通过模块化的系统设计方法,对蔡氏无量纲状态方程进行了电路设计。
本文的研究,取得了如下成果：
(1)提出了基于可配置IP核的混沌系统设计体系
首先论证了IP核的混沌电路设计的可行性,通过研究N阶无量纲连续混沌状态方程的一般形式,提出基于IP核的混沌电路设计实质上可转换为：基本函数IP核设计和基本运算IP核设计。研究了基于IP核的混沌电路设计的基础支撑,提出设计IP核管理工具的思想,作为基于IP核的混沌电路设计的基础支撑。同时,给出基于可配置IP核的混沌电路设计流程。
(2)研究一种混沌加密与常规加密级联的方案并运用于数字保密通信系统中
研究和实现了混沌加密级联DES并采用RSA作为密钥交换方案的保密通信系统。通过两个混沌系统动态比较瞬时值的方法产生伪混沌数字序列,保证每个用户可以通过用户设置不同密码的方式产生不同的伪混沌数字序列；用混沌加密与DES加密级联的方式,解决混沌加密密钥空间小的问题并对抗线性与差分密码分析,同时采用RSA作为密钥交换方案,以保证网络通信的安全性。
(3)以软件形式在无DSP支持的ARM开发板上实现了基于混沌加密的语音网络传输功能
设计并实现了基于混沌理论的实时语音保密通讯系统.该系统采用数字混沌保密通信方案,实现从开发板到计算机的保密语音通讯。该系统的工作流程系统从开发板采集、压缩并使用混沌算法加密语音信息,再将其传送到PC端,然后进行混沌解密、解压,再播放(及保存)语音信息。在此过程中,解决了ARM板与PC之间关于浮点数运算精度差异的问题,由于ARM开发板与PC之间对浮点数运算精度的差异,为此我们通过截取混沌二进制序列的前一部分数据来对语音数据进行加密,事实证明此方案是可行的。
(4)研究了基于混沌的数字水印技术,编写相对应的软件基于离散小波变换的数字水印信息技术是当今信息安全技术邻域的研究热点之一,本文研究一种基于离散小波变换的混沌加密图像水印算法,首先用混沌序列将二值权图像信息加密,再利用混沌序列将其嵌入到原始图像的离散小波变换域系数中。该水印方案不仅使得包含的版权信息量大,而且具有双重的安全性。混沌系列生成器中,采用Logistic映射和Hybrid映射产生两种混沌序列,并对这两者采取了改进的非线性离散化。此外,在程序设计中,根据面向对象的思想,建立了抽象基类,使得我们的算法能适应各种格式的图像。
(5)通过混沌电路的模块设计方法,提出一个新的三维二次自治混沌系统
提出了一个新的三维二次自治混沌系统。该系统具有四个混沌参数和两个非线性项,其中第二个方程中的非线性项为乘积项z2,第三个方程中的非线性项为交叉乘积项xy。分析了该系统的基本动力学问题,包括平衡点特性的分析以及分岔和最大李氏指数的计算等,进行计算机数值模拟,并用模块化方法设计了混沌电路,进行了电路实验。从理论分析、计算机模拟和电路实验三个方面证实了该系统的混沌特性通过理论分析和计算机模拟以及分岔和最大李氏指数的计算,对系统的基本动力学特性进行了研究。利用模块化设计方法对该混沌系统进行了电路设计与实验,给出了实验结果。电路实验与计算机模拟结果完全吻合,从而证实了该方法的可行性。
The chaotic application research cannot be separated from the chaotic circuit, which can generate chaotic phenomena and chaos phenomena is widespread in nonlinear circuits. Traditional chaotic circuit is typical of analog circuits using discrete components, such as capacitors, inductors, resistors, integrated operational amplifier, and analog multiplier and so on. With the development of integrated circuits digital processing technology such as FPGA (Field Programmable Gate) and DSP (Digital Signal Processing) and so on,using the technology of FPGA and DSP to produce chaotic signals is gradually becoming a trend. At present, the chaotic circuit design based on the technology of FPGA and DSP is still in the design phase of the personalization. The design exist a lot of limitations. Among them, the biggest limitation is reflected in the non-expansion and non-replication. After the completion of a chaotic circuit designing, even if chaos state equation has only a little change, the circuit still needs to be designed from scratch. Therefore.research how to introduce the Soc design method of IP core module to the chaotic circuit design system based on the FPGA has great significance. It forms design system of chaos circuit of a set of parameters that can be configured IP core's, and promotes the development of chaos circuit application through the establishment of the IP core repository.
This article study the possibility of design in the chaotic circuit based on IP core, the foundation to support of compositing the IP core chaotic circuit design, and the design process of the IP core chaotic circuit, and so on, based on reviewing that the traditional chaotic circuit produce chaotic phenomenon. Specific research includes:(1) explore and demonstrate the feasibility of design of the chaotic circuit based on IP core.(2)research the composition of the design in chaotic circuit based on the IP core, and put forward the design of chaotic circuit based on the IP core which can convert to the basic function IP core design and basic computing IP core design,based on researching the form of continuous chaotic state equations of the N-order dimensionless.(3) research the foundation to support of design of chaotic circuit based on the IP core,and put forward the thought of designing IP core management tools.(4) put forward the design process of the chaos circuit based on IP core, according to traditional FPGA design process,point that the process of design of the chaos circuit based on IP core will adopt IP module rather than the basic logic or circuit units as a basic unit, and choose appropriate IP module according to the needs to design integrated system using of existing IP resources.(5)research based on the digital communication security system based on adopting DES and RSA as of key exchange program which is the program of chaotic encryption and conventional encryption cascade. That is, using RSA public-key production algorithm as a key exchange scheme to encrypt the private key of the chaotic encryption and conventional encryption algorithm and to ensure the synchronization and security of chaos, to make a cascade scheme in the network communication more easy to implement.(6) realize the voice network transmission function based on chaotic encryption in the ARM development board by the form of software.and solve the problem of the floating-point precision differences between ARM board and PC in the development process. (7) Research the fragile watermarking technique based on the address code, on this basis, providing the framework of a precise positioning Chaotic Fragile digital Watermarking, the watermark not only is able to detect tampering but also perfectly position the area where the tampering occurred. (8) Propose a new method different from the traditional Chua's circuit design. To the polynomial produce chuas'chaotic attractor of three vortex roll as an example, through the approach of modular system design, design circuit to Chua's dimensionless equation of state.
The study of this paper, get the following results:
(1) Propose the design system of the chaos system based on configurable IP core
Firstly, demonstrate the feasibility of the design of the chaotic circuit of IP core through researching the general form of the N order dimensionless continuous chaotic state equations,and put forward the design of chaotic circuit based on the IP core which essentially can convert to the basic function IP core design and basic computing IP core design. Study the foundation to support of chaos circuit design of based on IP core, and propose the idea of designing IP core management tools as the foundation to support of the design of chaotic circuit based on the IP core. At the same time, give the design processes of chaotic circuit based on the configurable IP core.
(2) Research a chaotic encryption and conventional encryption cascade program and apply it to digital secure communication system. Research and realize a chaotic encryption cascade DES and adopt RSA as secure communication system of key exchange scheme. Through the method of two chaotic system dynamic comparative instantaneous value, produce pseudo-chaotic sequence of numbers, and ensure each user to generate different pseudo-chaotic sequence of numbers through the method of setting up different user password; by using the method of chaotic encryption and DES encryption cascade to solve the problem which key space of chaotic encryption is small and to analyze fight against linear and differential cryptanalysis, at the same time, using RSA as key exchange programs, in order to ensure the security of network communications.
(3)implement the voice network transmission function in ARM development board without DSP support in the form of software based on chaos encryption. Design and implement real-time voice secure communication system based on chaos theory and the system uses a digital chaotic secure communication scheme to achieve a secure voice communications from the development board to the computer. The system's workflow system is collecting from the development board, compressing and encrypting voice information using chaotic algorithm, and then transferring to the PC side. Then proceed with the chaotic decryption, decompression, and playing (and saving) the voice message. In this process, solve the problem about floating-point precision differences between ARM board and PC. Due to the difference of floating-point accuracy between the ARM development board and PC, we encrypt speech data through intercepting the first part of chaotic binary sequences. The fact illustrates the program is feasible.
(4)research watermarking technology based on chaotic digital.and write the corresponding software.The digital watermarking IT based on discrete wavelet transformation is one of the focus of today's information security technologies field. This paper researches a chaotic encryption image watermarking algorithm based on discrete wavelet transform, Firstly encrypt two values right image information using chaotic sequence, then embed two values right image information in the discrete wavelet transform domain coefficients of the original image using chaotic sequence. The watermarking scheme not only contains volume the copyright information, but also has a double security. Chaotic series generator using the Logistic map and Hybrid mapping generates two chaotic sequence, and both adopt the improved nonlinear discrete. In addition, according to object-oriented thinking, the establishment of an abstract base class makes our algorithm adapt to a variety of image formats in program design.
(5)by the module design method of chaotic circuits, the paper propose a new three-dimensional quadratic autonomous chaotic system.The system has four chaotic parameters and two nonlinear terms, the nonlinear terms in the second equation is product term,the nonlinear terms in the third equation is the cross-product term. It analyzes the system of basic dynamics problems including the characteristics of analysis of the equilibrium point and calculation of the bifurcation and the maximum lee's index calculation and so on, and takes computer numerical simulation, and then designs chaotic circuits by using modular approach to do the circuit experiments. According to theoretical analysis, computer simulations and circuit experiments confirm the chaotic features of the system by the theory analysis and computer simulation, and bifurcation and maximum lee's index calculation, and study the basic dynamic characteristics of the system. The chaos system of circuit design and experiment is conducted using the modular design, and experimental results are given. Results of circuit experiments and computer simulation are perfect, thus confirm the feasibility of this method.
本文在回顾传统的混沌电路产生混沌现象的基础上,对基于IP核的混沌电路设计的可能性,及其组成、IP核的混沌电路设计的基础支撑、IP核的混沌电路设计流程等方面进行了研究。具体研究内容包括：(1)探讨并论证了基于IP核的混沌电路设计的可行性；(2)研究了基于IP核的混沌电路设计的组成,在研究了N阶无量纲连续混沌状态方程的一般形式的基础上,提出基于IP核的混沌电路设计可转换为：基本函数IP核设计和基本运算IP核设计。(3)研究了基于IP核的混沌电路设计的基础支撑,提出了设计IP核管理工具的思想。(4)在传统的FPGA设计流程上,提出基于IP核的混沌电路设计流程,指出基于IP核的混沌电路设计流程将采用IP模块而不是基本逻辑或电路单元作为基础单元,利用现有的IP资源,根据需求选用适当的IP模块,进行系统集成设计。(5)研究了基于采用DES和RSA作为密钥交换方案的数字保密通信系统：混沌加密与常规加密级联的方案。即采用RSA公钥制作算法作为密钥交换方案,加密混沌加密与常规加密算法的私钥,以保证混沌的同步和安全性,使级联方案在网络通信的情况下更容易实现。(6)以软件形式在ARM开发板上实现了基于混沌加密的语音网络传输功能,在开发过程中解决了ARM板与PC之间浮点数运算精度差异的问题。(7)研究了基于地址码的脆弱数字水印技术,并在此基础上给出了一种定位精确的混沌脆弱数字水印框架,该水印不但能检测篡改而且能够完全定位篡改发生的区域。(8)提出了一种有别于传统蔡氏电路设计的新方法。以多项式ax+bx|x|+cx3产生三涡卷蔡氏混沌吸引子为例,通过模块化的系统设计方法,对蔡氏无量纲状态方程进行了电路设计。
本文的研究,取得了如下成果：
(1)提出了基于可配置IP核的混沌系统设计体系
首先论证了IP核的混沌电路设计的可行性,通过研究N阶无量纲连续混沌状态方程的一般形式,提出基于IP核的混沌电路设计实质上可转换为：基本函数IP核设计和基本运算IP核设计。研究了基于IP核的混沌电路设计的基础支撑,提出设计IP核管理工具的思想,作为基于IP核的混沌电路设计的基础支撑。同时,给出基于可配置IP核的混沌电路设计流程。
(2)研究一种混沌加密与常规加密级联的方案并运用于数字保密通信系统中
研究和实现了混沌加密级联DES并采用RSA作为密钥交换方案的保密通信系统。通过两个混沌系统动态比较瞬时值的方法产生伪混沌数字序列,保证每个用户可以通过用户设置不同密码的方式产生不同的伪混沌数字序列；用混沌加密与DES加密级联的方式,解决混沌加密密钥空间小的问题并对抗线性与差分密码分析,同时采用RSA作为密钥交换方案,以保证网络通信的安全性。
(3)以软件形式在无DSP支持的ARM开发板上实现了基于混沌加密的语音网络传输功能
设计并实现了基于混沌理论的实时语音保密通讯系统.该系统采用数字混沌保密通信方案,实现从开发板到计算机的保密语音通讯。该系统的工作流程系统从开发板采集、压缩并使用混沌算法加密语音信息,再将其传送到PC端,然后进行混沌解密、解压,再播放(及保存)语音信息。在此过程中,解决了ARM板与PC之间关于浮点数运算精度差异的问题,由于ARM开发板与PC之间对浮点数运算精度的差异,为此我们通过截取混沌二进制序列的前一部分数据来对语音数据进行加密,事实证明此方案是可行的。
(4)研究了基于混沌的数字水印技术,编写相对应的软件基于离散小波变换的数字水印信息技术是当今信息安全技术邻域的研究热点之一,本文研究一种基于离散小波变换的混沌加密图像水印算法,首先用混沌序列将二值权图像信息加密,再利用混沌序列将其嵌入到原始图像的离散小波变换域系数中。该水印方案不仅使得包含的版权信息量大,而且具有双重的安全性。混沌系列生成器中,采用Logistic映射和Hybrid映射产生两种混沌序列,并对这两者采取了改进的非线性离散化。此外,在程序设计中,根据面向对象的思想,建立了抽象基类,使得我们的算法能适应各种格式的图像。
(5)通过混沌电路的模块设计方法,提出一个新的三维二次自治混沌系统
提出了一个新的三维二次自治混沌系统。该系统具有四个混沌参数和两个非线性项,其中第二个方程中的非线性项为乘积项z2,第三个方程中的非线性项为交叉乘积项xy。分析了该系统的基本动力学问题,包括平衡点特性的分析以及分岔和最大李氏指数的计算等,进行计算机数值模拟,并用模块化方法设计了混沌电路,进行了电路实验。从理论分析、计算机模拟和电路实验三个方面证实了该系统的混沌特性通过理论分析和计算机模拟以及分岔和最大李氏指数的计算,对系统的基本动力学特性进行了研究。利用模块化设计方法对该混沌系统进行了电路设计与实验,给出了实验结果。电路实验与计算机模拟结果完全吻合,从而证实了该方法的可行性。
The chaotic application research cannot be separated from the chaotic circuit, which can generate chaotic phenomena and chaos phenomena is widespread in nonlinear circuits. Traditional chaotic circuit is typical of analog circuits using discrete components, such as capacitors, inductors, resistors, integrated operational amplifier, and analog multiplier and so on. With the development of integrated circuits digital processing technology such as FPGA (Field Programmable Gate) and DSP (Digital Signal Processing) and so on,using the technology of FPGA and DSP to produce chaotic signals is gradually becoming a trend. At present, the chaotic circuit design based on the technology of FPGA and DSP is still in the design phase of the personalization. The design exist a lot of limitations. Among them, the biggest limitation is reflected in the non-expansion and non-replication. After the completion of a chaotic circuit designing, even if chaos state equation has only a little change, the circuit still needs to be designed from scratch. Therefore.research how to introduce the Soc design method of IP core module to the chaotic circuit design system based on the FPGA has great significance. It forms design system of chaos circuit of a set of parameters that can be configured IP core's, and promotes the development of chaos circuit application through the establishment of the IP core repository.
This article study the possibility of design in the chaotic circuit based on IP core, the foundation to support of compositing the IP core chaotic circuit design, and the design process of the IP core chaotic circuit, and so on, based on reviewing that the traditional chaotic circuit produce chaotic phenomenon. Specific research includes:(1) explore and demonstrate the feasibility of design of the chaotic circuit based on IP core.(2)research the composition of the design in chaotic circuit based on the IP core, and put forward the design of chaotic circuit based on the IP core which can convert to the basic function IP core design and basic computing IP core design,based on researching the form of continuous chaotic state equations of the N-order dimensionless.(3) research the foundation to support of design of chaotic circuit based on the IP core,and put forward the thought of designing IP core management tools.(4) put forward the design process of the chaos circuit based on IP core, according to traditional FPGA design process,point that the process of design of the chaos circuit based on IP core will adopt IP module rather than the basic logic or circuit units as a basic unit, and choose appropriate IP module according to the needs to design integrated system using of existing IP resources.(5)research based on the digital communication security system based on adopting DES and RSA as of key exchange program which is the program of chaotic encryption and conventional encryption cascade. That is, using RSA public-key production algorithm as a key exchange scheme to encrypt the private key of the chaotic encryption and conventional encryption algorithm and to ensure the synchronization and security of chaos, to make a cascade scheme in the network communication more easy to implement.(6) realize the voice network transmission function based on chaotic encryption in the ARM development board by the form of software.and solve the problem of the floating-point precision differences between ARM board and PC in the development process. (7) Research the fragile watermarking technique based on the address code, on this basis, providing the framework of a precise positioning Chaotic Fragile digital Watermarking, the watermark not only is able to detect tampering but also perfectly position the area where the tampering occurred. (8) Propose a new method different from the traditional Chua's circuit design. To the polynomial produce chuas'chaotic attractor of three vortex roll as an example, through the approach of modular system design, design circuit to Chua's dimensionless equation of state.
The study of this paper, get the following results:
(1) Propose the design system of the chaos system based on configurable IP core
Firstly, demonstrate the feasibility of the design of the chaotic circuit of IP core through researching the general form of the N order dimensionless continuous chaotic state equations,and put forward the design of chaotic circuit based on the IP core which essentially can convert to the basic function IP core design and basic computing IP core design. Study the foundation to support of chaos circuit design of based on IP core, and propose the idea of designing IP core management tools as the foundation to support of the design of chaotic circuit based on the IP core. At the same time, give the design processes of chaotic circuit based on the configurable IP core.
(2) Research a chaotic encryption and conventional encryption cascade program and apply it to digital secure communication system. Research and realize a chaotic encryption cascade DES and adopt RSA as secure communication system of key exchange scheme. Through the method of two chaotic system dynamic comparative instantaneous value, produce pseudo-chaotic sequence of numbers, and ensure each user to generate different pseudo-chaotic sequence of numbers through the method of setting up different user password; by using the method of chaotic encryption and DES encryption cascade to solve the problem which key space of chaotic encryption is small and to analyze fight against linear and differential cryptanalysis, at the same time, using RSA as key exchange programs, in order to ensure the security of network communications.
(3)implement the voice network transmission function in ARM development board without DSP support in the form of software based on chaos encryption. Design and implement real-time voice secure communication system based on chaos theory and the system uses a digital chaotic secure communication scheme to achieve a secure voice communications from the development board to the computer. The system's workflow system is collecting from the development board, compressing and encrypting voice information using chaotic algorithm, and then transferring to the PC side. Then proceed with the chaotic decryption, decompression, and playing (and saving) the voice message. In this process, solve the problem about floating-point precision differences between ARM board and PC. Due to the difference of floating-point accuracy between the ARM development board and PC, we encrypt speech data through intercepting the first part of chaotic binary sequences. The fact illustrates the program is feasible.
(4)research watermarking technology based on chaotic digital.and write the corresponding software.The digital watermarking IT based on discrete wavelet transformation is one of the focus of today's information security technologies field. This paper researches a chaotic encryption image watermarking algorithm based on discrete wavelet transform, Firstly encrypt two values right image information using chaotic sequence, then embed two values right image information in the discrete wavelet transform domain coefficients of the original image using chaotic sequence. The watermarking scheme not only contains volume the copyright information, but also has a double security. Chaotic series generator using the Logistic map and Hybrid mapping generates two chaotic sequence, and both adopt the improved nonlinear discrete. In addition, according to object-oriented thinking, the establishment of an abstract base class makes our algorithm adapt to a variety of image formats in program design.
(5)by the module design method of chaotic circuits, the paper propose a new three-dimensional quadratic autonomous chaotic system.The system has four chaotic parameters and two nonlinear terms, the nonlinear terms in the second equation is product term,the nonlinear terms in the third equation is the cross-product term. It analyzes the system of basic dynamics problems including the characteristics of analysis of the equilibrium point and calculation of the bifurcation and the maximum lee's index calculation and so on, and takes computer numerical simulation, and then designs chaotic circuits by using modular approach to do the circuit experiments. According to theoretical analysis, computer simulations and circuit experiments confirm the chaotic features of the system by the theory analysis and computer simulation, and bifurcation and maximum lee's index calculation, and study the basic dynamic characteristics of the system. The chaos system of circuit design and experiment is conducted using the modular design, and experimental results are given. Results of circuit experiments and computer simulation are perfect, thus confirm the feasibility of this method.
引文
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[6]张伟年.动力系统基础[M].北京：高等教育出版社,施普林格出版社,2001.
[7]吕金虎.陆君安.陈十华.馄沌时间序列分析及其应用[M].武汉：武汉大学出版社,2002.
[8]方锦清.驾驭混沌与发展高新技术[M].北京：原子能出版社,2002.
[9]Jinhu Lu, Simin Yu,Henry Leung. Design of 3-D multi-scroll chaotic attractors via basic circuits. Dynamics of Continuous Discrete and Impulsive Systems-Series[J] B-Applications & Algorithms 1: 324-328 Sp. Iss. SI,2005.
[10]Jinhu Lu, Simin Yu, Henry Leung, and Guanrong Chen. Experimental verification of multidirectional multiscroll chaotic attractors[J]. IEEE Transactions on Circuits and Systems-Ⅰ: Regular Papers,2006,53(1):149-165.
[11]Simin Yu, Jinhu Lu, Henry Leung, Guanrong Chen. Design and implementation of n-scroll chaotic attractors from a general Jerk circuit[J]. IEEE Transactions on Circuits and Systems-Ⅰ:Regular Papers,2005, (52)7:1459-1476.
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[13]Simin Yu, Jinhu Lu, Wallace K. S. Tang, and Guanrong Chen. A general multi-scroll Lorenz system family and its realization via digital signal processors[J], Chaos,2006,16(3):033126(1-10).
[14]Simin Yu, Jinhu Lu, and Guanrong Chen. Multi-folded torus chaotic attractors:Design and implementation[J]. Chaos,2007,17(1):013118(1-12)
[15]Simin Yu. Jinhu Lu, and Guanrong Chen. A family of n-scroll hyperchaotic attractors and their realizations[J], Physics Letters A.2007,364(3):244-251.
[16]Simin Yu, Wallace K. S. Tang. Generation of n×m-scroll attractors in a two-port RCL network with hysteresis circuits[J]. Chaos, Solitons & Fractals,2008, (55)9:2089-2098.
[17]Simin Yu, Wallace K. S. Tang, and Guanrong Chen. Generation of nxm-scroll attractors under a Chua-circuit framework[J]. International Journal of Bifurcation And Chaos,2007,17(11): 3951-3964.
[18]Simin Yu. Jinhu Lu, and Guanrong Chen. A module-based and unified approach to chaotic circuit design and its applications[J], International Journal of Bifurcation And Chaos,2007,17(5): 1785-1800.
[19]Lu J H, Chen G R. Multi-scroll chaos generation:Theories, methods and applications[J]. Int. J. Bifurc. Chaos,2006,16(4):775-858.
[20]Suykens J A K, Vandewalle J. Quasilinear approach to nonlinear systems and the design of n-double scroll (n= 1:2; 3; 4;...)[J]. G, IEE Proc.,1991,138(5):595-603.
[21]Yalcin M E. Suykens J A K, Vandewalle J. Families of scroll grid attractors[J]. Int. J. Bifurc. Chaos, 2002,12(1):23-41.
[22]Dequan Li. A three-scroll chaotic attractor.[J] Physics Letters A.2008,365(3):254-259.
[23]Chua L O et all. The double scroll family[J]. IEEE Trans. Circuits Syst. Ⅰ,1986.33(11):1073-1117
[24]Suykens J A K, Chua L O. n-double scroll hypercubes in 1-D CNNs[J], Int. J. Bifurc. Chaos,1997, 7(8):1873-1885.
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[29]Elwakil A S, Ozoguz S. Multiscroll chaotic oscillators:the nonautonomous approach[J]. IEEE Trans. Circuits Syst.Ⅱ,2006,53(9):862-866.
[30]Aziz-Alaoui M A. Differential equations with multispiral attractors[J]. Int. J. Bifurcation Chaos, 1999,9(6):1009-1039.
[31]Yang X S, Li Q D. Generate n-scroll chaotic attractors in linear systems via scalar output feedback[J]. Chaos, Solitons and Fractals.2003,18:25-29.
[32]Li Q D, Yang X S. A new multiple-scrolls chaotic attractor and its circuit implementation[J]. Electronics Letters,2003,39:1306-1307.
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