CN110098917A - The building method of new type of chaotic system based on fractal algorithm - Google Patents

Abstract

The invention discloses a construction method of a new chaotic system based on a fractal algorithm, which includes selecting a three-dimensional chaotic system and taking the chaotic sequence generated by it as the input sequence of the fractal algorithm; applying the obtained chaotic sequence to the ternary fractal algorithm to obtain the final A new type of fractal chaotic system. The construction method of this new type of chaotic system based on fractal algorithm provided by the present invention can construct many different chaotic systems by selecting different chaotic subsystems and increasing the number of fractals, which can not only improve the key space of the system, but also improve the applicability Strong, high flexibility.

Description

The Construction Method of New Chaotic System Based on Fractal Algorithm

technical field

The invention belongs to the field of electronic communication, and in particular relates to a construction method of a novel chaotic system based on a fractal algorithm.

Background technique

With the development of economy and technology, chaos science, as a kind of nonlinear dynamics, has attracted more and more attention from researchers from all walks of life. The good randomness, ergodicity, and unpredictability of chaotic signals have high application value in the field of information security. At the same time, in order to better improve the dynamic characteristics of chaotic systems, people are committed to research on strengthening chaotic systems. From single-scroll, single-wing chaotic system to multi-scroll, multi-wing chaotic system, from a low-dimensional chaotic system with a positive Lyapunov exponent to an ultra-chaotic system with multiple positive Lyapunov exponents, all are to obtain better performance Strengthen the chaotic system.

Different from previous studies on constructing new chaotic systems from the internal mechanism of the system, the fractal algorithm starts from the external mechanism and expands the construction approach of chaotic system modeling. The new chaotic system based on fractal algorithm has more complex dynamic behavior, and has a greater application prospect in chaotic secure communication. Therefore, chaotic system modeling based on fractal algorithm is a hot spot in chaos theory and application research.

However, the current construction process of chaotic system based on fractal algorithm has poor applicability and poor flexibility, which restricts the popularization of this algorithm.

Contents of the invention

The purpose of the present invention is to provide a method for constructing a novel chaotic system based on a fractal algorithm with good applicability and high flexibility.

The construction method of this novel chaotic system based on fractal algorithm provided by the invention comprises the following steps:

S1. Select the three-dimensional chaotic system model, and use the chaotic sequence generated by the system as the input sequence of the fractal algorithm;

S2. Apply the chaotic sequence obtained in step S1 to the ternary fractal algorithm, so as to obtain the final new fractal chaotic system.

Applying the chaotic sequence obtained in step S1 to the ternary fractal algorithm described in step S2 is specifically applying the chaotic sequence to the ternary fractal algorithm by using the following formula:

where x _n , y _n , z _n are chaotic sequences; R _n , Q _n and S _n are the output fractal sequences; Δ ₁ and Δ ₂ are the set variable factors of the fractal algorithm.

Values of fractal algorithm variable factors Δ ₁ and Δ ₂ are: Δ ₁ =x _n , Δ ₂ =y _n .

The construction method of this new type of chaotic system based on fractal algorithm provided by the present invention can construct many different chaotic systems by selecting different chaotic subsystems and increasing the number of fractals, which can not only improve the key space of the system, but also improve the applicability Strong, high flexibility.

Description of drawings

Fig. 1 is a schematic flow chart of the method of the present invention.

Fig. 2 is the attractor phase diagram before and after the fractal when the method of the present invention adopts the Lorenz system as the three-dimensional chaotic system model.

Fig. 3 is the bifurcation diagram before and after the fractal when the method of the present invention adopts the Lorenz system as the three-dimensional chaotic system model.

Fig. 4 is a schematic diagram of spectrum distribution characteristics before and after fractal when the method of the present invention adopts the Lorenz system as the three-dimensional chaotic system model.

Fig. 5 is a schematic diagram of the complexity before and after fractal when the method of the present invention adopts the Lorenz system as the three-dimensional chaotic system model.

Detailed ways

As shown in Figure 1, it is the method flow diagram of the inventive method: the construction method of this novel chaotic system based on the fractal algorithm provided by the present invention comprises the following steps:

S1. Select the three-dimensional chaotic system model, and use the chaotic sequence generated by the system as the input sequence of the fractal algorithm;

S2. Apply the chaotic sequence obtained in step S1 to the ternary fractal algorithm to obtain the final new fractal chaotic system; specifically, apply the chaotic sequence to the ternary fractal algorithm by using the following formula:

where x _n , y _n , z _n are three-dimensional chaotic sequences; R _n , Q _n and S _n are the output fractal sequences; Δ ₁ and Δ ₂ are the set variable factors of the fractal algorithm.

In a specific implementation, a preferred solution for the values of the fractal algorithm variable factors Δ ₁ and Δ ₂ is: Δ ₁ =x _n , Δ ₂ =y _n .

Taking the Lorenz system as an example, the fractal Lorenz system is constructed based on the ternary fractal algorithm. Through the attractor phase diagram, bifurcation diagram, permutation entropy and spectrum analysis, the dynamic characteristics of the system are analyzed to verify the validity of the method. The present invention will be further described in detail in conjunction with the drawings and technical solutions, and the implementation of the present invention will be described in detail through preferred examples.

Example: Fractal Lorenz system, the specific structure is as follows:

Step 1: Select the Lorenz system as the fractal system model, and its system equation is:

Step 2: Solve the continuous chaotic system to obtain its three subsequences, and use the subsequences as the input sequence of the three-dimensional fractal algorithm, taking Δ ₁ =x _n , Δ ₂ =y _n ;

Step 3: Analyze the dynamic characteristics of the system by using the attractor phase diagram, bifurcation diagram, spectral distribution and permutation entropy.

The dynamical properties of chaotic systems are often evaluated by attractor phase diagrams, bifurcation diagrams, spectral distributions and permutation entropy.

Figure 2(a)-(d) are the attractor phase diagrams of the Lorenz system, after the first fractal, after the second fractal, and after the third fractal, respectively. Obviously, compared with the Lorenz system, the fractal algorithm can effectively replicate the phase diagram of the original system. After each fractal, the number of cavities in the new system doubles compared to the previous one. The ternary fractal algorithm continues the characteristics of the binary fractal algorithm. Through this method, a construction method of a multi-cavity system is undoubtedly added. At the same time, due to the flexibility of fractal algorithms, discrete systems are equally applicable to ordinary time series. Figure 3(a)-(d) are the bifurcation diagrams of the Lorenz system, after the first fractal, after the second fractal, and after the third fractal, respectively. It can be seen that many periodic windows in the Lorenz system become chaotic state after fractal operation. Figure 4(a)-(d) are the comparison of the spectrum distribution characteristics of the Lorenz system, after the first fractal, after the second fractal, and after the third fractal, respectively.

Figure 2(a)-(d) are the attractor phase diagrams of the Lorenz system, after the first fractal, after the second fractal, and after the third fractal, respectively. Obviously, compared with the Lorenz system, the fractal algorithm can effectively replicate the cavity of the original system. After each fractal, the number of cavities in the new system doubles compared to the previous one. The ternary fractal algorithm continues the characteristics of the binary fractal, but it is more beneficial to improve the characteristics of the system. Through this method, a construction approach of a multi-chamber system is undoubtedly added. At the same time, due to the flexibility of fractal algorithms, discrete systems are equally applicable to ordinary time series.

The dynamical behavior of chaotic systems can be evaluated using bifurcation diagrams. Figure 3(a)-(d) are the bifurcation diagrams of the Lorenz system, after the first fractal, after the second fractal, and after the third fractal, respectively. It can be seen that many periodic windows in the Lorenz system become chaotic in the fractal Lorenz system after fractal operation. The chaotic system modeling based on the ternary fractal algorithm can effectively improve the chaotic performance of the system.

Figure 4(a)-(d) shows the spectrum distribution characteristics of Lorenz system before and after fractal. For the general frequency spectrum, the more uniform the distribution, the better the distribution characteristics. Comparing the simulation results, it can be clearly seen that the distribution of the Lorenz system itself is relatively concentrated, while the distribution of the fractal Lorenz system based on the triple fractal algorithm is more uniform. Therefore, the chaotic system modeling based on the ternary fractal algorithm can effectively improve the distribution characteristics of the chaotic system.

In the application of cryptography, the better the randomness of the time series generated by the chaotic system, the higher the security of the cryptography system. The randomness of the sequence can be measured by the complexity algorithm, among which the permutation entropy (PE) algorithm is widely used because of its simple implementation. In this experiment, the permutation entropy algorithm is used to calculate the complexity, and the simulation results are shown in Figure 5. It can be seen from the figure that the highest permutation entropy complexity of the Lorenz system is only 0.25, while the permutation entropy complexity of the fractal Lorenz system after one fractal reaches 0.75, and the permutation entropy complexity of the fractal Lorenz system after three fractals increases to 0.98. Therefore, this construction method can effectively improve the complexity of the chaotic system.

Claims (3)

1. A kind of construction method of the novel chaotic system based on fractal algorithm, comprises the steps: S1. Select a three-dimensional chaotic system, and use the chaotic sequence generated by the system as the input sequence of the fractal algorithm; S2. Apply the chaotic sequence obtained in step S1 to the ternary fractal algorithm, so as to obtain the final new fractal chaotic system. 2. the construction method of the novel chaotic system based on fractal algorithm according to claim 1, it is characterized in that described in step S2 the chaotic sequence that step S1 obtains is applied to ternary fractal algorithm, specifically for adopting following formula to subsequence Apply to ternary fractal algorithm: where x _n , y _n , z _n are three-dimensional chaotic sequences; R _n , Q _n and S _n are the output fractal sequences; Δ ₁ and Δ ₂ are the set variable factors of the fractal algorithm. 3. according to the construction method of the novel chaotic system based on fractal algorithm described in claim 1 or 2, it is characterized in that the value of fractal algorithm variable factor Δ ₁ and Δ ₂ is: Δ ₁ =x _n , Δ ₂ =y _n .