CN106160999A

CN106160999A - A kind of grid many butterflies wing chaos attractor production method

Info

Publication number: CN106160999A
Application number: CN201610859016.7A
Authority: CN
Inventors: 张鹏; 黄沄; 赵卫峰; 李铮; 张刚
Original assignee: Chongqing University of Post and Telecommunications
Current assignee: Chongqing University of Post and Telecommunications
Priority date: 2016-09-28
Filing date: 2016-09-28
Publication date: 2016-11-23

Abstract

本发明涉及一种网格多蝴蝶翼混沌吸引子产生方法，本发明的方法是对三维Lorenz混沌系统进行比例缩放，并增加两个一阶微分方程，同时在增加的两个一阶微分方程上分别加入一个含有分段线性函数的非线性耦合控制器，从而产生网格多蝴蝶翼混沌吸引子，并且混沌吸引子的个数可由参数控制；本发明可提供更复杂的混沌信号，为混沌保密通信提供更多的选择。The invention relates to a method for generating chaotic attractors with multi-mesh butterfly wings. The method of the invention is to scale the three-dimensional Lorenz chaotic system, and add two first-order differential equations, and at the same time increase the two first-order differential equations A nonlinear coupling controller containing a piecewise linear function is respectively added to generate chaotic attractors with multiple butterfly wings in the grid, and the number of chaotic attractors can be controlled by parameters; the present invention can provide more complex chaotic signals and keep chaos secret Communication offers more options.

Description

A method for generating chaotic attractors with multi-mesh butterfly wings

技术领域technical field

本发明涉及一种网格混沌吸引子产生方法，具体涉及一种网格多蝴蝶翼混沌吸引子产生方法。The invention relates to a method for generating grid chaos attractors, in particular to a method for generating grid chaos attractors with multiple butterfly wings.

背景技术Background technique

如何产生用于混沌保密通信中所需的各种混沌信号是非线性系统科学研究的一个新领域。对于多涡卷混沌吸引子以及两翼蝴蝶混沌吸引子的产生方法的研究已取得了一系列的成果，目前人们又对动力学行为更复杂的多翼以及网格多翼蝴蝶混沌吸引子进行了研究，如在公开号为CN103236920A的中国发明专利申请公开说明书公开了一种改进的四维统一多翼混沌系统；以及在公开号为CN103957098A的中国发明专利申请公开说明书公开了一种产生多蝴蝶形吸引子的混沌电路及实现方法；在公开号为CN104320244A的中国发明专利申请公开说明书公开了一种产生网格多翼蝴蝶混沌吸引子的混沌电路及使用方法。然而，在这些产生多蝴蝶翼以及网格多蝴蝶翼混沌吸引子的发明专利中，并没有提出具体的多蝴蝶翼以及网格多蝴蝶翼混沌吸引子的产生方法。How to generate various chaotic signals required in chaotic secure communication is a new field of nonlinear system science research. A series of achievements have been made in the research on the generation method of the multi-scroll chaotic attractor and the two-wing butterfly chaotic attractor. At present, people are studying the multi-wing and grid multi-wing butterfly chaotic attractor with more complex dynamic behavior. , as disclosed in the publication number CN103236920A Chinese invention patent application publication specification discloses a kind of improved four-dimensional unified multi-wing chaotic system; The chaotic circuit and its implementation method of the sub; the Chinese Invention Patent Application Publication No. CN104320244A discloses a chaotic circuit for generating a grid multi-wing butterfly chaotic attractor and its application method. However, in these invention patents for generating chaotic attractors with multiple butterfly wings and grids with multiple butterfly wings, no specific methods for generating chaotic attractors with multiple butterfly wings and grids with multiple butterfly wings are proposed.

发明内容Contents of the invention

本发明所要解决的技术问题是，针对现有技术还没有具体的产生网格多蝴蝶翼混沌吸引子的方法，提出了一种网格多蝴蝶翼混沌吸引子产生方法。The technical problem to be solved by the present invention is to propose a method for generating chaotic attractors with multiple grid butterfly wings as there is no specific method for generating chaotic attractors with multiple grid butterfly wings in the prior art.

本发明解决上述技术问题的技术方案是，设计了一种网格多蝴蝶翼混沌吸引子产生方法，该方法包括以下步骤：The technical scheme that the present invention solves the above-mentioned technical problem is, has designed a kind of grid multi-butterfly wing chaotic attractor generation method, and this method comprises the following steps:

1)、采用三维Lorenz混沌系统，其无量纲状态方程如下：1), using a three-dimensional Lorenz chaotic system, its dimensionless state equation is as follows:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = a a (({x x}_{22} - - {x x}_{11})),, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = c c {x x}_{11} - - {x x}_{22} - - {x x}_{11} {x x}_{33},, \\ {\overset{\cdot \cdot}{x x}}_{33} = = {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \end{matrix}$

式中a、b、c为系统参数，x₁，x₂和x₃为状态变量；In the formula, a, b, c are system parameters, x ₁ , x ₂ and x ₃ are state variables;

2)、对步骤1)的三维Lorenz混沌系统的状态变量进行比例缩放，令状态变量x₁，x₂和x₃的比例缩放因子分别为γ₁，γ₂和γ₃,则得到一个缩放后的系统：2), scale the state variables of the three-dimensional Lorenz chaotic system in step 1), and set the scaling factors of the state variables x ₁ , x ₂ and x ₃ to be γ ₁ , γ ₂ and γ ₃ respectively, then a scaled system:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{22} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \\ {\overset{\cdot \cdot}{x x}}_{33} = = \frac{{γ γ}_{11} {γ γ}_{22}}{{γ γ}_{33}} {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \end{matrix}$

3)、在步骤2)的系统上，增加一个关于状态变量x₄和一个关于状态变量x₅的一阶微分方程,得到一个五维系统:3), on the system of step 2), add a first-order differential equation about the state variable x ₄ and a state variable x ₅ to obtain a five-dimensional system:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{22} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \\ {\overset{\cdot &Center Dot;}{x x}}_{33} = = \frac{{γ γ}_{11} {γ γ}_{22}}{{γ γ}_{33}} {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \\ {\overset{\cdot \cdot}{x x}}_{44} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{44},, \\ {\overset{\cdot &Center Dot;}{x x}}_{55} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{55} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \end{matrix}$

其中，构造关于状态变量x₄的一阶微分方程的方法是把步骤2)的系统的第一个方程中的状态变量x₁变为x₄；构造关于状态变量x₅的一阶微分方程的方法是把步骤2)的系统的第二个方程中的状态变量x₂变为x₅；Wherein, the method for constructing the first-order differential equation about the state variable x ₄ is to change the state variable x ₁ in the first equation of the system of step 2) into x ₄ ; constructing the first-order differential equation about the state variable x ₅ The method is to change the state variable x in the second equation of the system of step ₂ ) into x ₅ ;

4)、在步骤3)的系统的第四个方程中，加入一个关于状态变量x₄线性耦合控制器u₁；在步骤3)的系统的第五个方程中，加入一个关于状态变量x₅线性耦合控制器u₂，进而建立一个五维系统：4), in the fourth equation of the system in step 3), add a linear coupling controller u ₁ about the state variable x ₄ ; in the fifth equation of the system in step 3), add one about the state variable x ₅ Linearly couple the controller u ₂ , and then build a five-dimensional system:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{11},, \\ {\overset{\cdot \cdot}{x x}}_{22} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{22} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \\ {\overset{\cdot &Center Dot;}{x x}}_{33} = = \frac{{γ γ}_{11} {γ γ}_{22}}{{γ γ}_{33}} {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \\ {\overset{\cdot \cdot}{x x}}_{44} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{44} + + h h {u u}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{55} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{55} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33} + + h h {u u}_{22},, \end{matrix}$

其中，h为增益参数，u₁和u₂是线性耦合控制器；Among them, h is the gain parameter, u ₁ and u ₂ are linear coupling controllers;

5)、设计分段线性函数f(x₄)和f(x₅)，并使用分段线性函数f(x₄)和f(x₅)分别代替步骤4)的系统中的状态变量x₄和x₅，以及线性耦合控制器u₁和u₂中的状态变量x₄和x₅,得到一个新的系统:5), design piecewise linear functions f(x ₄ ) and f(x ₅ ), and use piecewise linear functions f(x ₄ ) and f(x ₅ ) to replace the state variable x ₄ in the system of step 4) respectively and x ₅ , and the state variables x ₄ and x ₅ in the linearly coupled controllers u ₁ and u ₂ , yield a new system:

$\{\begin{matrix} {\overset{\cdot \cdot}{x x}}_{11} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{22} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \\ {\overset{\cdot &Center Dot;}{x x}}_{33} = = \frac{{γ γ}_{11} {γ γ}_{22}}{{γ γ}_{33}} {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \\ {\overset{\cdot &Center Dot;}{x x}}_{44} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a f f (({x x}_{44})) + + h h {u u}_{11}^{' '},, \\ {\overset{\cdot \cdot}{x x}}_{55} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - f f (({x x}_{55})) - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33} + + h h {u u}_{22}^{' '},, \end{matrix}$

其中，u′₁和u′₂是非线性耦合控制器；Among them, u′ ₁ and u′ ₂ are nonlinear coupling controllers;

6)、在步骤5)的系统上，取系统参数a＝10，b＝8/3，c＝28，γ₁＝16，γ₂＝23，γ₃＝40，h＝5，得到混沌信号在x₄和x₅两个方向产生网格多蝴蝶翼混沌吸引子。6), on the system in step 5), take the system parameters a=10, b=8/3, c=28, γ ₁ =16, γ ₂ =23, γ ₃ =40, h=5 to obtain the chaotic signal Generate grid multi-butterfly wing chaotic attractors in two directions of x ₄ and x ₅ .

优选地，所述的线性耦合控制器u₁和u₂分别为：Preferably, the linear coupling controllers u ₁ and u ₂ are respectively:

u₁＝x₁-x₄ u ₁ =x ₁ -x ₄

u₂＝x₂-x₅。u ₂ =x ₂ −x ₅ .

优选地，所述的非线性耦合控制器u′₁和u′₂分别为：Preferably, the nonlinear coupling controllers _u'1 and _u'2 are respectively:

u′₁＝x₁-f(x₄)u′ ₁ ＝x ₁ -f(x ₄ )

u′₂＝x₂-f(x₅)。u′ ₂ =x ₂ −f(x ₅ ).

优选地，所述的分段线性函数f(x₄)和f(x₅)分别为：Preferably, the piecewise linear functions f(x ₄ ) and f(x ₅ ) are respectively:

$f f (({x x}_{44})) = = {x x}_{44} - - {Σ Σ}_{n no = = - - {N N}_{11}}^{{M m}_{11}} sgn sgn (({x x}_{44} + + 22 n no + + 11)) + + (({M m}_{11} - - {N N}_{11} + + 11))$

$f f (({x x}_{55})) = = {x x}_{55} - - {Σ Σ}_{n no = = - - {N N}_{22}}^{{M m}_{22}} sgn sgn (({x x}_{55} + + 22 n no + + 11)) + + (({M m}_{22} - - {N N}_{22} + + 11))$

其中,N₁,M₁,N₂,M₂∈{0,1,2,…}。Among them, N ₁ , M ₁ , N ₂ , M ₂ ∈{0,1,2,…}.

本发明与现有技术相比，给出了一种具体的网格多蝴蝶翼混沌吸引子的产生方法，为混沌保密通信提供更多的混沌信号的选择。Compared with the prior art, the present invention provides a specific method for generating chaotic attractors with multi-grid butterfly wings, and provides more choices of chaotic signals for chaotic secure communication.

附图说明Description of drawings

图1为4×2蝴蝶翼混沌吸引子的数值仿真结果；Fig. 1 is the numerical simulation result of the 4×2 butterfly wing chaotic attractor;

图2为6×3蝴蝶翼混沌吸引子的数值仿真结果；Fig. 2 is the numerical simulation result of the 6×3 butterfly wing chaotic attractor;

图3为8×4蝴蝶翼混沌吸引子的数值仿真结果；Fig. 3 is the numerical simulation result of the 8×4 butterfly wing chaotic attractor;

图4为10×4蝴蝶翼混沌吸引子的数值仿真结果；Fig. 4 is the numerical simulation result of the 10×4 butterfly wing chaotic attractor;

图5为10×5蝴蝶翼混沌吸引子的数值仿真结果；Fig. 5 is the numerical simulation result of the 10×5 butterfly wing chaotic attractor;

图6为12×6蝴蝶翼混沌吸引子的数值仿真结果。Figure 6 shows the numerical simulation results of the 12×6 butterfly wing chaotic attractor.

具体实施方式detailed description

以下针对附图和具体实例对本发明的实施进行具体说明。The implementation of the present invention will be specifically described below with reference to the accompanying drawings and specific examples.

本发明的方法是对三维Lorenz混沌系统进行比例缩放，并增加两个一阶微分方程，同时在增加的两个一阶微分方程上分别加入一个含有分段线性函数的非线性耦合控制器，从而产生网格多蝴蝶翼混沌吸引子。本发明设计的一种网格多蝴蝶翼混沌吸引子产生方法包括以下步骤：The method of the present invention is to scale the three-dimensional Lorenz chaotic system, and increase two first-order differential equations, and simultaneously add a nonlinear coupling controller containing a piecewise linear function to the two increased first-order differential equations, thereby Generate mesh multi-butterfly chaotic attractors. A method for generating chaotic attractors with many grids and butterfly wings designed by the present invention comprises the following steps:

1)、采用三维Lorenz混沌系统，确定或选取其无量纲状态方程如下：1), using the three-dimensional Lorenz chaotic system, determine or select its dimensionless state equation as follows:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = a a (({x x}_{22} - - {x x}_{11})),, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = c c {x x}_{11} - - {x x}_{22} - - {x x}_{11} {x x}_{33},, \\ {\overset{\cdot &Center Dot;}{x x}}_{33} = = {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \end{matrix}$

式中a、b、c为系统参数，x₁，x₂和x₃为状态变量。Where a, b, c are system parameters, x ₁ , x ₂ and x ₃ are state variables.

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{22} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \\ {\overset{\cdot &Center Dot;}{x x}}_{33} = = \frac{{γ γ}_{11} {γ γ}_{22}}{{γ γ}_{33}} {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \end{matrix}$

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{22} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \\ {\overset{\cdot \cdot}{x x}}_{33} = = \frac{{γ γ}_{11} {γ γ}_{22}}{{γ γ}_{33}} {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \\ {\overset{\cdot \cdot}{x x}}_{44} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{44},, \\ {\overset{\cdot &Center Dot;}{x x}}_{55} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{55} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \end{matrix}$

其中，构造关于状态变量x₄的一阶微分方程的方法是把步骤2)的系统的第一个方程中的状态变量x₁变为x₄；构造关于状态变量x₅的一阶微分方程的方法是把步骤2)的系统的第二个方程中的状态变量x₂变为x₅。Wherein, the method for constructing the first-order differential equation about the state variable x ₄ is to change the state variable x ₁ in the first equation of the system of step 2) into x ₄ ; constructing the first-order differential equation about the state variable x ₅ The method is to change the state variable x ₂ in the second equation of the system in step 2) to x ₅ .

$\{\begin{matrix} {\overset{\cdot \cdot}{x x}}_{11} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{22} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \\ {\overset{\cdot &Center Dot;}{x x}}_{33} = = \frac{{γ γ}_{11} {γ γ}_{22}}{{γ γ}_{33}} {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \\ {\overset{\cdot &Center Dot;}{x x}}_{44} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{44} + + h h {u u}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{55} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{55} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33} + + h h {u u}_{22},, \end{matrix}$

其中，h为增益参数，u₁和u₂是线性耦合控制器：where h is the gain parameter and _u1 and _u2 are linearly coupled controllers:

u₁＝x₁-x₄，u ₁ =x ₁ -x ₄ ,

u₂＝x₂-x₅.u ₂ ＝x ₂ -x ₅ .

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a {x x}_{11},, \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - {x x}_{22} - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33},, \\ {\overset{\cdot \cdot}{x x}}_{33} = = \frac{{γ γ}_{11} {γ γ}_{22}}{{γ γ}_{33}} {x x}_{11} {x x}_{22} - - b b {x x}_{33},, \\ {\overset{\cdot \cdot}{x x}}_{44} = = \frac{{aγ aγ}_{22}}{{γ γ}_{11}} {x x}_{22} - - a a f f (({x x}_{44})) + + h h {u u}_{11}^{' '},, \\ {\overset{\cdot &Center Dot;}{x x}}_{55} = = \frac{{cγ cγ}_{11}}{{γ γ}_{22}} {x x}_{11} - - f f (({x x}_{55})) - - \frac{{γ γ}_{11} {γ γ}_{33}}{{γ γ}_{22}} {x x}_{11} {x x}_{33} + + h h {u u}_{22}^{' '},, \end{matrix}$

其中，u′₁和u′₂是非线性耦合控制器：where u′ ₁ and u′ ₂ are nonlinear coupled controllers:

u′₁＝x₁-f(x₄)，u' ₁ =x ₁ -f(x ₄ ),

u′₂＝x₂-f(x₅).u′ ₂ ＝x ₂ -f(x ₅ ).

分段线性函数f(x₄)和f(x₅)分别为：The piecewise linear functions f(x ₄ ) and f(x ₅ ) are:

6)、在步骤5)的系统上，取系统参数a＝10，b＝8/3，c＝28，γ₁＝16，γ₂＝23，γ₃＝40，h＝5，得到混沌信号在x₄和x₅两个方向产生2(N₁+M₁+2)×(N₂+M₂+2)蝴蝶翼混沌吸引子。其中，图1为N₁＝M₁＝N₂＝M₂＝0时，产生的4×2蝴蝶翼混沌吸引子；图2为N₁＝1,M₁＝0,N₂＝1,M₂＝0时，产生的6×3蝴蝶翼混沌吸引子；图3为N₁＝M₁＝N₂＝M₂＝1时，产生的8×4蝴蝶翼混沌吸引子；图4为N₁＝2,M₁＝N₂＝M₂＝1时，产生的10×4蝴蝶翼混沌吸引子；图5为N₁＝2,M₁＝1,N₂＝2,M₂＝1时，产生的10×5蝴蝶翼混沌吸引子；图6为N₁＝M₁＝N₂＝M₂＝2时，产生的12×6蝴蝶翼混沌吸引子。6), on the system in step 5), take the system parameters a=10, b=8/3, c=28, γ ₁ =16, γ ₂ =23, γ ₃ =40, h=5 to obtain the chaotic signal 2(N ₁ +M ₁ +2)×(N ₂ +M ₂ +2) butterfly wing chaotic attractors are generated in two directions of x ₄ and x ₅ . Among them, Figure 1 shows the 4×2 butterfly wing chaotic attractor generated when N ₁ =M ₁ =N ₂ =M ₂ =0; Figure 2 shows N ₁ =1,M ₁ =0,N ₂ =1,M When ₂ = 0, the 6×3 butterfly wing chaotic attractor is generated; Fig. 3 is the 8×4 butterfly wing chaotic attractor generated when N ₁ ＝M ₁ ＝N ₂ ＝M ₂ =1; Fig. 4 is N ₁ ＝2, M ₁ ＝N ₂ ＝M ₂ ＝1, the 10×4 butterfly wing chaotic attractor generated; Figure 5 shows that when N ₁ ＝2, M ₁ ＝1, N ₂ ＝2, M ₂ ＝1, The generated 10×5 butterfly wing chaotic attractor; FIG. 6 shows the generated 12×6 butterfly wing chaotic attractor when N ₁ =M ₁ =N ₂ =M ₂ =2.

从图1—图6可以看出，本发明方法能获得网格多蝴蝶翼混沌吸引子，且产生的蝴蝶翼混沌吸引子的数量可控。It can be seen from Figures 1 to 6 that the method of the present invention can obtain chaotic attractors with multiple grids, and the number of chaotic attractors produced can be controlled.

Claims

1. grid many butterflies wing chaos attractor production method, it is characterised in that the method comprises the following steps:

1), using three-dimensional Lorenz chaos system, its dimensionless state equation is as follows:

\{\begin{matrix} {\overset{\cdot}{x}}_{1} = a (x_{2} - x_{1}), \\ {\overset{\cdot}{x}}_{2} = c x_{1} - x_{2} - x_{1} x_{3}, \\ {\overset{\cdot}{x}}_{3} = x_{1} x_{2} - b x_{3}, \end{matrix}

In above formula, a, b, c are systematic parameter, x₁, x₂And x₃For state variable；

2), to step 1) the state variable of three-dimensional Lorenz chaos system carry out proportional zoom, writ state variable x₁, x₂And x₃ Scale factor be respectively γ₁, γ₂And γ₃, then obtain one scaling after system:

\{\begin{matrix} {\overset{\cdot}{x}}_{1} = \frac{{aγ}_{2}}{γ_{1}} x_{2} - a x_{1}, \\ {\overset{\cdot}{x}}_{2} = \frac{{cγ}_{1}}{γ_{2}} x_{1} - x_{2} - \frac{γ_{1} γ_{3}}{γ_{2}} x_{1} x_{3}, \\ {\overset{\cdot}{x}}_{3} = \frac{γ_{1} γ_{2}}{γ_{3}} x_{1} x_{2} - b x_{3}, \end{matrix}

3), in step 2) system on, structure about state variable x₄Differential equation of first order and structure about state variable x₅'s Differential equation of first order, obtains one five and maintains system:

\{\begin{matrix} {\overset{\cdot}{x}}_{1} = \frac{{aγ}_{2}}{γ_{1}} x_{2} - a x_{1}, \\ {\overset{\cdot}{x}}_{2} = \frac{{cγ}_{1}}{γ_{2}} x_{1} - x_{2} - \frac{γ_{1} γ_{3}}{γ_{2}} x_{1} x_{3}, \\ {\overset{\cdot}{x}}_{3} = \frac{γ_{1} γ_{2}}{γ_{3}} x_{1} x_{2} - b x_{3}, \\ {\overset{\cdot}{x}}_{4} = \frac{{aγ}_{2}}{γ_{1}} x_{2} - a x_{4}, \\ {\overset{\cdot}{x}}_{5} = \frac{{cγ}_{1}}{γ_{2}} x_{1} - x_{5} - \frac{γ_{1} γ_{3}}{γ_{2}} x_{1} x_{3}, \end{matrix}

Wherein, structure is about state variable x₄The method of differential equation of first order be step 2) system first equation in State variable x₁Become x₄；Structure is about state variable x₅The method of differential equation of first order be step 2) system State variable x in two equations₂Become x₅；

4), in step 3) system the 4th equation in, add one about state variable x₄Linear coupling controller u₁；? Step 3) system the 5th equation in, add one about state variable x₅Linear coupling controller u₂, and then set up one Five maintain system:

\{\begin{matrix} {\overset{\cdot}{x}}_{1} = \frac{{aγ}_{2}}{γ_{1}} x_{2} - a x_{1}, \\ {\overset{\cdot}{x}}_{2} = \frac{{cγ}_{1}}{γ_{2}} x_{1} - x_{2} - \frac{γ_{1} γ_{3}}{γ_{2}} x_{1} x_{3}, \\ {\overset{\cdot}{x}}_{3} = \frac{γ_{1} γ_{2}}{γ_{3}} x_{1} x_{2} - b x_{3}, \\ {\overset{\cdot}{x}}_{4} = \frac{{aγ}_{2}}{γ_{1}} x_{2} - a x_{4} + h u_{1}, \\ {\overset{\cdot}{x}}_{5} = \frac{{cγ}_{1}}{γ_{2}} x_{1} - x_{5} - \frac{γ_{1} γ_{3}}{γ_{2}} x_{1} x_{3} + h u_{2}, \end{matrix}

Wherein, h is gain parameter, u₁And u₂It it is linear coupling controller；

5), design piecewise linear function f (x₄) and f (x₅), and use piecewise linear function f (x₄) and f (x₅) replace step respectively 4) state variable x in system₄And x₅, and linear coupling controller u₁And u₂In state variable x₄And x₅, obtain one New system:

\{\begin{matrix} {\overset{\cdot}{x}}_{1} = \frac{{aγ}_{2}}{γ_{1}} x_{2} - a x_{1}, \\ {\overset{\cdot}{x}}_{2} = \frac{{cγ}_{1}}{γ_{2}} x_{1} - x_{2} - \frac{γ_{1} γ_{3}}{γ_{2}} x_{1} x_{3}, \\ {\overset{\cdot}{x}}_{3} = \frac{γ_{1} γ_{2}}{γ_{3}} x_{1} x_{2} - b x_{3}, \\ {\overset{\cdot}{x}}_{4} = \frac{{aγ}_{2}}{γ_{1}} x_{2} - a f (x_{4}) + h u_{1}^{'}, \\ {\overset{\cdot}{x}}_{5} = \frac{{cγ}_{1}}{γ_{2}} x_{1} - f (x_{5}) - \frac{γ_{1} γ_{3}}{γ_{2}} x_{1} x_{3} + h u_{2}^{'}, \end{matrix}

Wherein, u '₁With u '₂It it is Non-linear coupling controller；

6), in step 5) system on, take systematic parameter a=10, b=8/3, c=28, γ₁=16, γ₂=23, γ₃=40, h =5, obtain chaotic signal at x₄And x₅Both direction produces grid many butterflies wing chaos attractor.

Grid many butterflies wing chaos attractor production method the most according to claim 1, it is characterised in that described is linear Coupling Control Unit u₁And u₂It is respectively as follows:

u₁=x₁-x₄

u₂=x₂-x₅。

Grid many butterflies wing chaos attractor production method the most according to claim 1, it is characterised in that described non-thread Property Coupling Control Unit u '₁With u '₂It is respectively as follows:

u′₁=x₁-f(x₄)

u′₂=x₂-f(x₅)。

Grid many butterflies wing chaos attractor production method the most according to claim 1, it is characterised in that described segmentation Linear function f (x₄) and f (x₅) be respectively as follows:

f (x_{4}) = x_{4} - Σ_{n = - N_{1}}^{M_{1}} sgn (x_{4} + 2 n + 1) + (M_{1} - N_{1} + 1)

f (x_{5}) = x_{5} - Σ_{n = - N_{2}}^{M_{2}} sgn (x_{5} + 2 n + 1) + (M_{2} - N_{2} + 1)

Wherein, N₁,M₁,N₂,M₂∈{0,1,2,…}。