阅读说明：本技术 一种四维保守混沌系统 (Four-dimensional conservative chaotic system ) 是由 薛薇 张永超 于 2019-01-16 设计创作，主要内容包括：本发明新型提供了一个四维保守混沌系统,具体涉及混沌信号产生和保密通信技术领域。本发明是通过对广义Hamilton系统的研究,理解其中的能量关系构造相应的结构矩阵来保证产生混沌所必要的线性项和非线性项,并利用多次实验的方式获得系统的参数和初值。四维保守混沌系统能够产生复杂的混沌信号,能够有效地掩盖通信交流中的明文信息,达到良好的信息加密效果；同时,系统具有结构简单参数少的优点。(The invention provides a four-dimensional conservative chaotic system, and particularly relates to the technical field of chaotic signal generation and secret communication. According to the invention, through the research on the generalized Hamilton system, the energy relation in the generalized Hamilton system is understood to construct a corresponding structural matrix to ensure that linear terms and nonlinear terms necessary for generating chaos are generated, and parameters and initial values of the system are obtained by utilizing a plurality of experiments. The four-dimensional conservative chaotic system can generate a complex chaotic signal, can effectively cover plaintext information in communication, and achieves a good information encryption effect; meanwhile, the system has the advantages of simple structure and few parameters.)

1. A four-dimensional conservative chaotic system can output four chaotic signals, and a mathematical model corresponding to the four-dimensional chaotic system is as follows:

in the formula, x, y, z and w are respectively system state variables; wherein a is 1.2 and b is 2.

2. A new four-dimensional conservative chaotic system is characterized in that:

the four-dimensional conservative chaotic system comprises 8 terms, wherein the four-dimensional conservative chaotic system comprises 4 nonlinear terms and 4 linear terms, and the adjustable parameters are only a and b. Basic kinetic properties include symmetry, dissipation, equilibrium point, eigenvalues and stability; symmetry generally exists in a dynamic system containing a chaotic attractor, and the symmetry characteristic of the system is verified by changing the variable of the system; the dissipation degree is a standard for judging whether the system is a conservative system or a dissipation system, when the dissipation degree is less than zero, the dissipation chaotic system is adopted, and when the dissipation degree is equal to zero, the dissipation chaotic system is adopted; the balance point is an important characteristic of the system, and whether the hidden chaotic attractor possibly exists can be further judged through the existence of the balance point; the type of the balance point can be judged through the characteristic value, and whether the balance point is a hyperbolic balance point or a non-hyperbolic balance point or the like can be judged; the stability is a state judged by the above balance point and the characteristic value.

3. The new four-dimensional conservative chaotic system as set forth in claim 2, wherein:

the four-dimensional conservative chaotic system finds out that a system equation occurs by substituting (x, -y, -z, -w) into a mathematical model of the systemChanging to show that the system (1) is not symmetrical about the x axis, and sequentially obtaining the system which is not symmetrical about the y axis, the z axis and the w axis by the same principle, namely the system has no symmetry; when the system parameters are respectively a is 1.2 and b is 2, the method is represented by the formulaIndicating that the system is volume-conservative, the L yapunov indices of the system are L₁＝0.206，L₂＝0.021，L₃＝-0.020，L₄-0.207, with only one balance point being 0 (0000), which is a significant feature of this system; its characteristic value is two pairs of pure virtual root lambda₁＝2i，λ₂＝1.2i，λ₃＝-2i，λ₄The obtained balance point is a non-hyperbolic balance point, the chaotic state is obtained when an L yapunov index is larger than zero in the system according to the chaotic judgment condition, so the system is in the chaotic state, and in addition, the sum of the L yapunov indexes of the system is zero, namely L₁+L₂+L₃+L₄0, so the system is a conservative system.

The technical field is as follows:

the invention relates to the technical field of chaotic signal generation and secret communication, in particular to a structure of a complex signal generation system.

Background art:

at present, a low-dimensional dissipative chaotic system is common, but a high-dimensional conservative chaotic system is rare, and because the conservative chaotic system has natural advantages compared with the dissipative chaotic system, a singular attractor does not exist and cannot be reconstructed. The chaos signal is applied to image encryption and video encryption, which is a hot subject of the current domestic and foreign research, and the chaos sequence is utilized to scramble the pixel point positions and change the pixel point sizes in the original image and the original video, thereby achieving the purpose of encryption. At present, the design and application of the chaotic system are not mature, and the requirements of practical application are difficult to meet, so further research work is needed for the application of the chaotic system and the chaotic signal. The invention takes a four-dimensional conservative chaotic system as an example, and adopts a simple system structure to generate a complex chaotic signal.

The invention content is as follows:

the invention provides a novel four-dimensional conservative chaotic system. The four-dimensional conservative chaotic system can generate a complex chaotic signal, can effectively cover plaintext information in communication, and achieves a good information encryption effect.

The invention adopts the following technical means for realizing the purpose of the invention:

the technical means is as follows: a four-dimensional conservative chaotic system can output four chaotic signals, and a mathematical model corresponding to the four-dimensional chaotic system is as follows:

in the formula, x, y, z and w are respectively system state variables; a and b are real numbers;

when a is 1.2 and b is 2, the four-dimensional conservative chaotic system has a chaotic attractor.

The second technical means: a four-dimensional conservative chaotic system is characterized in that:

the four-dimensional conservative chaotic system comprises 8 terms, wherein the four-dimensional conservative chaotic system comprises 4 nonlinear terms and 4 linear terms, and the adjustable parameters are only a and b. Basic kinetic properties include symmetry, dissipation, equilibrium point, eigenvalues and stability; symmetry generally exists in a dynamic system containing a chaotic attractor, and the symmetry characteristic of the system is verified by changing the variable of the system; the dissipation degree is a standard for judging whether the system is a conservative system or a dissipation system, when the dissipation degree is less than zero, the dissipation chaotic system is used, and when the dissipation degree is equal to zero, the dissipation chaotic system is used as the conservative system; the balance point is an important characteristic of the system, and whether the hidden chaotic attractor possibly exists can be further judged through the existence of the balance point; the type of the balance point can be judged through the characteristic value, and whether the balance point is a hyperbolic balance point or a non-hyperbolic balance point or the like can be judged; the stability is a state judged by the above balance point and the characteristic value.

The third technical means: the conservative four-dimensional chaotic system in the technical means II is further limited, and is characterized in that:

the four-dimensional conservative chaotic system is obtained by substituting (x, -y, -z, -w) into the number of the systemIn the mathematical model, the system equation is found to be changed, which indicates that the system (1) is not symmetrical about the x axis, and similarly, the system is not symmetrical about the y axis, the z axis and the w axis in sequence, namely the system has no symmetry; when the system parameters are respectively a-1 and b-2, the method is represented by the formulaIndicating that the system is volume-conservative, the L yapunov indices of the system are L₁＝ 0.206，L₂＝0.021，L₃＝-0.020，L₄-0.207, with only one balance point being 0 (0000) respectively, and having characteristic values of two pairs of pure imaginary roots λ₁＝2i，λ₂＝1.2i，λ₃＝-2i，λ₄The obtained balance point is a non-hyperbolic balance point, the chaotic state is obtained when an L yapunov index is larger than zero in the system according to the chaotic judgment condition, so the system is in the chaotic state, and in addition, the sum of the L yapunov indexes of the system is zero, namely L₁+L₂+L₃+L₄0, so the system is a conservative system.

The technical means of the present invention to solve the above problems is to study the generalized Hamilton system, which can be expressed as follows

Here x ∈ RⁿIs a state variable of the system (2) u ∈ R^m×1Can be regarded as the generalized force to which the system (2) is subjected for the input of the system (2), g (x) ∈ R^n×mIs an input vector; y: r^m×n×R¹|→R^m×1Is the output of the system (2) and can be regarded as the generalized speed of the system;is an antisymmetric matrix satisfying J (x) ═ J^T(x) Often referred to as structural arrays, represent the conserved portion of system energy; h: rⁿI → R is the Hamilton function, i.e. the energy storage function, with respect to the state variable x. Secondly, by constructing a symmetrical arrayAnd obtaining linear and nonlinear coupling terms of state variables in the chaotic system, and finally determining parameters and initial values of the chaotic system through multiple experiments.

Description of the drawings:

FIG. 1 is a L yapunov index chart of the system (1).

Fig. 2 is a poincare cross-sectional view of the system (1).

Fig. 3 is a two-dimensional phase diagram of the state variable x and the state variable y in the system (1).

Fig. 4 is a two-dimensional phase diagram of the state variable x and the state variable z in the system (1).

Fig. 5 is a two-dimensional phase diagram of the state variable x and the state variable w in the system (1).

Fig. 6 is a two-dimensional phase diagram of the state variable y and the state variable z in the system (1).

Fig. 7 is a two-dimensional phase diagram of the state variable y and the state variable w in the system (1).

Fig. 8 is a two-dimensional phase diagram of the state variable z and the state variable w in the system (1).

The specific implementation mode is as follows:

the present invention will be described in detail below with reference to the accompanying drawings.