Chaotic system model construction method based on state variables and uncertain variables

Abstract

The invention relates to the technical field of nonlinear dynamics and chaotic systems, in particular to a chaotic system model construction method based on state variables and uncertain variables. Starting from the angle of combination of a state variable and an uncertain variable, a construction method with a unified form is proposed in combination with an amplitude undetermined matrix, the feasibility of the construction method is verified by using a lyapunov index definition, and the dynamic behavior change of the system under the condition that the system is influenced by unknown parameters is explored. The permutation entropy is used to explore system chaotic behavior changes caused by random natural numerical values and mathematical functions in random amplitudes, a three-dimensional continuous chaotic system model structure is used as a basis to expand into a four-dimensional state, and different complexities are used to display system complex characteristics in the four-dimensional state. A three-dimensional continuous system and a four-dimensional continuous system are applied to a classical chaotic system DNA color image encryption system. The method provided by the invention not only has theoretical innovation in the aspect of constructing a chaotic system, but also shows a relatively high practical value in practical application.

Description

Technical Field

[0001] The invention relates to the technical field of nonlinear dynamics and chaotic systems, and in particular to a method for constructing a chaotic system model based on state variables and uncertain variables. Background Art

[0002] Nonlinear dynamic systems have broad application prospects in physics, engineering, biology, and cryptography. In particular, chaotic systems have become the focus of researchers due to their sensitive dependence on initial conditions and complex dynamic behavior. Although chaotic systems exhibit complex and irregular behaviors in terms of dynamic behavior, they are deterministic and their behavior is completely determined by deterministic equations. Due to the determinism of its equations, combining chaotic systems with confidential communications or biomedicine has advantages that cannot be ignored. However, how to construct chaotic systems with complex dynamic characteristics is still a challenging topic. There are many ways to construct chaotic systems. The more common ones are through the promotion and expansion of classical chaotic systems, constructing chaotic systems by introducing nonlinear feedback control items, constructing piecewise linear systems, using delayed feedback methods to introduce time delays in the feedback link of the system, parameter-driven methods, nonlinear control parameters, random perturbations, composite systems, switching of dynamic systems, symbolic dynamic systems, and memristors. These methods of constructing chaotic systems are the current mainstream construction methods, which have greatly promoted the construction of system models for exploring chaotic systems.

[0003] The delayed feedback scheme in the chaotic system construction method is a type of construction scheme that generates chaotic behavior by introducing time delay in the system feedback. This scheme is usually applied to biological, physical and engineering systems. Its basic idea is to use the delayed feedback term to make the current state of the system not only depend on the current input, but also be affected by the past state. However, the delayed feedback system pays more attention to emphasizing the dependence of the system state on the past state, is suitable for systems with time-delayed feedback, and is usually used in deterministic systems. Another random perturbation method usually introduces random perturbations or noise into the system to make the system exhibit chaotic characteristics. This type of method is not a strictly deterministic system, but can exhibit complex dynamic behaviors in a statistical sense. The random perturbation method emphasizes the complex behavior of the system under random perturbations by introducing random noise, and is suitable for random systems.

[0004] Inspired by the delayed feedback scheme and the random perturbation scheme, we are committed to exploring a new method for constructing chaotic systems. From the perspective of combining state variables and uncertain variables, we combine unknown variables with uncertain variables and propose a new construction method. This scheme focuses on the fact that random parameters are not just a time delay, which makes the selection range of random parameters wider. At the same time, the random parameters are combined with state variables to construct chaotic systems, which greatly demonstrates the sensitivity of chaotic systems. The addition of random amplitudes also diversifies the model structure of chaotic systems, enriching the construction methods of chaotic systems to a certain extent.

[0005] For the verification of the feasibility of a specific scheme, in addition to conducting a large number of simulation experiments based on existing theoretical verification methods, it is also necessary to conduct specific simulation applications. The combination of chaotic systems and image encryption systems for innovative chaotic image encryption schemes is a hot topic in current confidential communications, and many practical research advances have been made. Therefore, a classic chaotic image encryption scheme is selected, and two specific instantiated chaotic systems under the requirements of the construction scheme of the present invention are included. The chaotic characteristics of these two instantiated system models can be verified from the perspective of image encryption, and the feasibility of the system construction scheme in the present invention can be more intuitively explained from another perspective. Summary of the invention

[0006] In order to expand the modeling methods of chaotic systems, the present invention focuses on the construction of nonlinear dynamic models, with continuous chaotic systems as the main research object. Starting from the perspective of combining state variables and uncertain variables, the idea of ​​combining unknown variables with uncertain variables to construct a continuous chaotic system is proposed. Based on the three-dimensional continuous chaotic system model structure, it is expanded to a four-dimensional state. Different complexities are used to demonstrate the complex characteristics of the system in the four-dimensional state, illustrating the universal applicability of the construction method. Finally, the present invention applies the constructed three-dimensional and four-dimensional continuous systems to the classic chaotic system DNA color image encryption system.

[0007] To achieve the above object, the present invention provides the following technical solutions: According to a first aspect of the present invention, the present invention claims a method for constructing a chaotic system model based on state variables and uncertain variables, comprising: S1, obtain state variables, combine with the amplitude matrix to be determined, adopt a unified form of construction method, and use the Lyapunov index definition to verify the feasibility of the construction method; S2, obtaining a three-dimensional chaotic system with a single nonlinear structure, using the Lyapunov exponent to screen numerical parameters, bifurcation diagram, attractor diagram, 0-1 test and dual-parameter Lyapunov exponent diagram method to obtain the dynamic behavior changes of the three-dimensional chaotic system under the influence of unknown parameters; S3, using permutation entropy to analyze the changes in the dynamic behavior caused by random natural values ​​and mathematical functions in the random amplitude, based on the three-dimensional continuous chaotic system model structure, expanding it to a four-dimensional state, using different complexities to demonstrate the complex characteristics of the system in the four-dimensional state, and determining the universal applicability of the construction method; S4, the constructed three-dimensional and four-dimensional continuous systems are applied to the classic chaotic system DNA color image encryption system and perform robustness and security comparative analysis.

[0008] Furthermore, the S1 further includes: Starting from the construction of nonlinear dynamics model, for continuous chaotic system ,in , P express s dimensional vector field, Indicates accompanying time t Changing state variables; The construction method is proposed from the perspective of state variables and unknown parameters. From the perspective of state variables, a continuous system is obtained: ; The small disturbance is , is an ordinary differential equation, It is a state variable and participates in the construction of the system model; Multiply the matrices with unknown amplitudes to obtain the matrix equation: ; in, is the state variable, To determine the random amplitude within the interval; Output is about the system construction method of combining state variables with unknown parameters and random parameters as amplitudes.

[0009] Furthermore, the S1 further includes: Definition: A continuous system has two motion trajectories with similar but different starting points. The initial point is and , the trajectory line is set to and ; Set Existence , taking the derivative of the continuous function, we get: ; In the formula, the Jacobian matrix of the system is , represents a high-order small term, and the approximation ignores the The impact of higher powers; Further processing the above formula, we get the linearized equations for the continuous system: ; Performing integral transformation, we get: .

[0010] Definition of the Lyapunov exponent for continuous systems, along with time t The discretization process of the change of S The average of the time period for: ; For the system model, let , As the eigenvalue, we can get the system's Lyapunov index The expression is: ; It is assumed that there are qualified random amplitude intervals and unknown parameters involved in the construction of the continuous chaotic system. The different types of random amplitudes are: ; In the formula is the magnitude matrix, Take the value of amplitude.

[0011] With the combination of parameters and state variables to construct the system, the construction expression of the chaotic system is obtained: ; The M in the formula is an ellipsis; In the formula is a state variable that changes with time.

[0012] In this way, the relevant calculation of the Lyapunov exponent is carried out when the system undergoes multiple iterations. When , we can get the above formula in Jacobian matrix at iteration : ; From the above formula, it can be seen that the construction of the system model involving state variables and random amplitudes has no effect on the Lyapunov exponent, and the construction of the new system does not change the chaotic characteristics of the original system.

[0013] Furthermore, the S2 further includes: Adopting the logical idea of ​​screening initial conditions, the relevant initial conditions of the system are screened, and the initial conditions of the parameters are obtained after a large number of iterations; Randomly select initial conditions within the range of values ​​that meet the conditions, use bifurcation diagrams and Lyapunov index diagrams for comparative analysis, and observe the dynamic behavior of the system model; The initial value is selected as (0.1, 0.1, 0.1), and the parameter selection range is selected , , ;Analyze the dynamic behavior of the system under the influence of parameters through bifurcation diagrams and Lyapunov index diagrams; Based on the system model, application; for the four-dimensional system, detailed dynamic behavior exploration is required. Starting from the parameters participating in the construction of the system model structure, the random amplitude is set as a fixed reference condition to obtain the specific simulation results of the unknown parameters for the dynamic behavior of the system model; Set the amplitude to , the numerical range of the unknown parameters is set to , and obtain the bifurcation diagram and attractor diagram under the corresponding parameter conditions.

[0014] For dynamic observable data , choose an arbitrary random constant , and define the following equation: ; in, . and are two cumulative sums, representing different projection components of the signal. They are usually used to describe the cumulative distribution of the signal in certain specific directions. n represents a discrete time index, represents the signal amplitude and weight, Indicates phase.

[0015] According to the above formula, the root mean square displacement is defined as for: ; By observation The change trend of the root mean square displacement with time is to judge the motion state; When in Brownian motion, the root mean square displacement increases linearly with time; When the state of motion is bounded, the RMS displacement is also bounded; Test asynchronous growth rate : ; pass Whether the value of is close to 0 or 1, we can intuitively judge whether the motion is in a periodic or quasi-periodic state or a chaotic state.

[0016] Furthermore, the S3 further includes: SE and C0 complexity are used as important methods to describe and analyze chaotic systems; It plays an important role in analyzing the transformation of the dynamic behavior of the system. SE complexity quantitatively analyzes the degree of chaos of the system by analyzing the time series of the system. High Shannon entropy indicates that the state of the system changes dramatically over time, with high uncertainty and complexity, indicating that the system is in a chaotic state. Low Shannon entropy indicates that the state of the system changes more regularly over time, with low information content, indicating that the system is in a periodic or stable state; C0 complexity evaluates the complexity of a sequence by calculating the frequency of occurrence of different permutation patterns in the time series. This allows the system's time series to contain different permutation patterns; SE complexity and C0 complexity are used to analyze the changes in dynamic behavior caused by changes in random amplitudes in the system. The four-dimensional system is combined with the initial conditions to obtain the effect diagram of the system under different conditions.

[0017] Furthermore, the encryption in S4 further includes: Plain text image settings, select the size MN The color image is the plaintext image, which is divided into three two-dimensional matrices of R, G, and B. In order to ensure the rationality of the image size, a 0-filling operation is performed.

[0018] ; In the formula, W is the block size.

[0019] Initial setup and parameter setting, select one-dimensional Generating chaotic sequences as chaotic maps , the sequence length is equal to the image size MN ; The selection of the initial value is based on the initial value in the three-dimensional system.

[0020] Transformation of chaotic sequence. Transform one-dimensional chaotic sequence Convert to a two-dimensional matrix to ensure that the sequence size is the same as the image size, and normalize the sequence to between 0 and 255 for subsequent DNA operations.

[0021] Use the Runge-Kutta method to solve the four-dimensional system and generate four sequences of length MN , , , , taking the two-dimensional matrix R as an example, the data value range is 0 to 255, and the data is represented by 8-bit binary numbers. Using the initial value of the four-dimensional , , , , and the initial conditions are calculated from different bits of the image; DNA coding and operation, converting image pixel values ​​into DNA sequences. Each pixel value corresponds to a DNA code. The generated chaotic sequence is used to perform XOR operation on the DNA sequence. The DNA coding rules are given by Determined by; DNA operations are determined by Decide; The encoded matrix is ​​permuted in rows and columns, and the permutation order is a chaotic sequence generated by a three-dimensional system. Decide; The permuted matrix is ​​subjected to DNA operation again to obtain the final ciphertext image.

[0022] Furthermore, the encryption in S4 is followed by decryption, which specifically includes: Decompose the ciphertext image into three channels: R, G, and B; Generate a chaotic sequence. Generate the same chaotic sequence about the three-dimensional chaotic system as that during encryption, and set the parameters to the same initial value as during encryption; Inverse permutation operation: perform inverse permutation operation on the matrices of the three channels of R, G, and B. The permutation order is opposite to that during encryption, from the last column to the first column and from the last row to the first row. Solve the four-dimensional system. Use the Runge-Kutta method to continue solving the four-dimensional system and generate the same four sequences as when encrypting. , , , .

[0023] Perform DNA inverse operation on the inverse permuted matrix, and the decoding rule is given by Determined by; the inverse operation rule is determined by It is decided to remove the zero elements added during encryption and restore the original pixel value.

[0024] The decoded R, G, and B two-dimensional matrices are combined into a three-dimensional matrix to obtain the decrypted color image.

[0025] The present invention relates to the technical field of nonlinear dynamics and chaotic systems, and in particular to a method for constructing a chaotic system model based on state variables and uncertain variables; starting from the perspective of combining state variables and uncertain variables, a construction method with a unified form is proposed in combination with an amplitude to-be-determined matrix, the feasibility of the construction method is verified by using the Lyapunov exponent definition, the dynamic behavior changes of the system under the influence of unknown parameters are explored, the chaotic behavior changes of the system caused by random natural values ​​and mathematical functions in the random amplitude are explored by using permutation entropy, the three-dimensional continuous chaotic system model structure is expanded to a four-dimensional state, different complexities are used to display the complex characteristics of the system in the four-dimensional state, and the three-dimensional and four-dimensional continuous systems are applied to the classic chaotic system DNA color image encryption system. The method of the present invention is not only theoretically innovative in constructing chaotic systems, but also shows high practical value in practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 A work flow chart of a method for constructing a chaotic system model based on state variables and uncertain variables as claimed in an embodiment of the present invention; Figure 2 A schematic diagram of the complexity of a four-dimensional system under different amplitude conditions for a chaotic system model construction method based on state variables and uncertain variables as claimed in an embodiment of the present invention. DETAILED DESCRIPTION

[0027] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0028] The terms "first", "second" and "third" in the present invention are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of indicated technical features. Therefore, the features defined as "first", "second" and "third" may explicitly or implicitly include at least one of the features. In the description of the present invention, the meaning of "multiple" is at least two, such as two, three, etc., unless otherwise clearly and specifically defined. All directional indications (such as up, down, left, right, front, back...) in the embodiments of the present invention are only used to explain the relative position relationship, movement, etc. between the components under a certain specific posture (as shown in the accompanying drawings). If the specific posture changes, the directional indication also changes accordingly. In addition, the terms "including" and "having" and any of their variations are intended to cover non-exclusive inclusions. For example, a process, method, system, product or device that includes a series of steps or units is not limited to the listed steps or units, but optionally also includes steps or units that are not listed, or optionally also includes other steps or units inherent to these processes, methods, products or devices.

[0029] Reference to an "embodiment" herein means that a particular feature, structure, or characteristic described in conjunction with the embodiment may be included in at least one embodiment of the present invention. The appearance of the phrase in various places in the specification does not necessarily refer to the same embodiment, nor is it an independent or alternative embodiment that is mutually exclusive with other embodiments. It is explicitly and implicitly understood by those skilled in the art that the embodiments described herein may be combined with other embodiments.

[0030] According to the first embodiment of the present invention, the present invention claims a method for constructing a chaotic system model based on state variables and uncertain variables, referring to Figure 1 ,include: S1, obtain state variables, combine with the amplitude matrix to be determined, adopt a unified form of construction method, and use the Lyapunov index definition to verify the feasibility of the construction method; S2, obtaining a three-dimensional chaotic system with a single nonlinear structure, using the Lyapunov exponent to screen numerical parameters, bifurcation diagram, attractor diagram, 0-1 test and dual-parameter Lyapunov exponent diagram method to obtain the dynamic behavior changes of the three-dimensional chaotic system under the influence of unknown parameters; S3, using permutation entropy to analyze the changes in the dynamic behavior caused by random natural values ​​and mathematical functions in the random amplitude, based on the three-dimensional continuous chaotic system model structure, expanding it to a four-dimensional state, using different complexities to demonstrate the complex characteristics of the system in the four-dimensional state, and determining the universal applicability of the construction method; S4, the constructed three-dimensional and four-dimensional continuous systems are applied to the classic chaotic system DNA color image encryption system and perform robustness and security comparative analysis.

[0031] Furthermore, the S1 further includes: Starting from the construction of nonlinear dynamics model, for continuous chaotic system ,in , P express s dimensional vector field; The construction method is proposed from the perspective of state variables and unknown parameters. From the perspective of state variables, a continuous system is obtained: ; in, Participate in the construction of system models simultaneously with state variables; Multiply the matrices with unknown amplitudes to obtain the matrix equation: ; in, To determine the random amplitude within the interval; Output is about the system construction method of combining state variables with unknown parameters and random parameters as amplitudes.

[0032] Furthermore, the S1 further includes: Definition: A continuous system has two motion trajectories with similar but different starting points. The initial point is and , the trajectory line is set to and ; Set Existence , taking the derivative of the continuous function, we get: ; In the formula, the Jacobian matrix of the system is ; Further processing of equation (2) yields the linearized equations for the continuous system: ; Performing integral transformation, we get: .

[0033] Definition of the Lyapunov exponent for continuous systems, along with time t The discretization process of the change of S The average for each time period is: ; For the system model, let , As the eigenvalue, the expression of the system's Lyapunov exponent is obtained as: ; It is assumed that there are qualified random amplitude intervals and unknown parameters involved in the construction of the continuous chaotic system. The different types of random amplitudes are: ; With the combination of parameters and state variables to construct the system, the construction expression of the chaotic system is obtained: ; The M in the formula is an ellipsis; In this way, the relevant calculation of the Lyapunov exponent is carried out when the system undergoes multiple iterations. When , we can get the above formula in Jacobian matrix during iteration: ; From the above formula, it can be seen that the construction of the system model involving state variables and random amplitudes has no effect on the Lyapunov exponent, and the construction of the new system does not change the chaotic characteristics of the original system.

[0034] Among them, in this implementation, the Lyapunov index plays an important role in determining the dynamic behavior of the system. It is an indicator to measure the sensitivity of the system to the initial conditions. Specifically, the Lyapunov index describes the divergence or convergence speed of adjacent orbits in the phase space of the system. If the Lyapunov index is positive, it means that the system is chaotic, that is, the adjacent orbits will separate rapidly; if the Lyapunov index is negative, it means that the system is stable and the adjacent orbits will gradually approach; if the Lyapunov index is zero, it means that the system is critical, that is, the adjacent orbits remain within a certain distance. Therefore, it is feasible to use the lyapunov index value at different initial conditions as the idea of ​​determining the dynamic behavior of the system model. According to the number of positive, zero and negative values ​​contained in the lyapunov index, the dynamic behavior of the system is reasonably judged. However, for the judgment of the chaotic behavior of the system model, the lyapunov index cannot be used as a sufficient and necessary condition. It is necessary to consider the dynamic characteristics of the system, the complexity of the trajectory, the spectrum analysis and other aspects of information. However, this does not prevent the lyapunov index from being used to screen the given interval and make a preliminary judgment on whether the system is in a chaotic state to a certain extent.

[0035] Relying on computer simulation tests and taking the Lyapunov exponent as the basis for judgment, a process framework diagram for screening the rational parameters of the model was designed. Algorithm 1 can select the appropriate parameter value interval to a certain extent through a large number of iterations to ensure the significance of the dynamic behavior of the system model. This can also provide a certain reference value for studying a given dynamic model.

[0036] Determine the parameter range: First, set the minimum and maximum values ​​of the parameters to define the initial parameter range.

[0037] Randomly select initial parameters: Within the set parameter range, randomly select a set of parameters as starting values ​​and calculate their corresponding initial Lyapunov exponents.

[0038] Gradually narrow the parameter range: In each iteration, the parameter range is gradually narrowed to more accurately find reasonable parameters.

[0039] Verify the rationality of the parameters: After each parameter range is updated, randomly select parameters within the new range and calculate their corresponding Lyapunov exponents. If this exponent is less than zero, the parameter group is considered unreasonable and the attempt is terminated; otherwise, continue to verify other parameters.

[0040] Check results: When parameters that meet the requirements are found (i.e., parameters whose Lyapunov exponent is greater than or equal to zero), the parameter group is marked as reasonable and the screening process ends.

[0041] Output reasonable parameter range: Finally, the selected parameter interval is output as a reasonable parameter range. This range ensures the stability and feasibility of the system model under the selected parameter conditions. Furthermore, the S2 further includes: Adopting the logical idea of ​​screening initial conditions, the relevant initial conditions of the system are screened, and the initial conditions of the parameters are obtained after a large number of iterations; Randomly select initial conditions within the range of values ​​that meet the conditions, use bifurcation diagrams and Lyapunov index diagrams for comparative analysis, and observe the dynamic behavior of the system model; The initial value is selected as (0.1, 0.1, 0.1), and the parameter selection range is selected , , ;Analyze the dynamic behavior of the system under the influence of parameters through bifurcation diagrams and Lyapunov index diagrams; Table 1 shows the corresponding initial conditions provided by the present invention; Table 1 Initial conditions table

[0042] Based on the system model, application; for the four-dimensional system, detailed dynamic behavior exploration is required. Starting from the parameters participating in the construction of the system model structure, the random amplitude is set as a fixed reference condition to obtain the specific simulation results of the unknown parameters for the dynamic behavior of the system model; Set the amplitude to , the numerical range of the unknown parameters is set to , and obtain the bifurcation diagram and attractor diagram under the corresponding parameter conditions.

[0043] For dynamic observable data , choose an arbitrary random constant , and define the following equation: ; in, .

[0044] According to the above formula, the root mean square displacement is defined as: ; By observation The change trend of the root mean square displacement with time is to judge the motion state; When in Brownian motion, the root mean square displacement increases linearly with time; When the state of motion is bounded, the RMS displacement is also bounded; Test the asynchronous growth rate: ; pass Whether the value of is close to 0 or 1, we can intuitively judge whether the motion is in a periodic or quasi-periodic state or a chaotic state.

[0045] In this embodiment, permutation entropy is used as an algorithm for calculating the complexity of nonlinear time series, and the specific implementation steps are as follows: Generate a time series about a system. Use the system's differential equations to generate the time evolution of state variables.

[0046] Select the delay time and embedding dimension. Select the appropriate delay time according to the time series generated by the system. and embedding dimension m Delay time represents the time interval between each element in the delay vector, embedded in the dimension m Indicates the number of state variables used to construct the delay vector.

[0047] Create delay vectors. For the time series generated by the system, construct a delay vector sequence based on the selected delay time and embedding dimension. Assume that the selected time series is , then the i-th element of the delay vector is .

[0048] Calculate the permutation. For each delay vector , generating a permutation based on the size relationship of the elements.

[0049] Calculate the frequency distribution of permutations. Count the frequencies of all possible permutations and get a frequency distribution , where π represents the permutation.

[0050] Calculate the permutation entropy. Use the formula , where π traverses all possible permutations.

[0051] The entire time series is divided into multiple overlapping subsequences, and the permutation entropy of each subsequence is calculated. Finally, the average value is taken.

[0052] Since permutation entropy shows good sensitivity in dealing with nonlinear phenomena, the permutation entropy will be used to explore the system dynamics behavior caused by the change of amplitude.

[0053] Furthermore, the S3 further includes: SE and C0 complexity are used as important methods to describe and analyze chaotic systems; It plays an important role in analyzing the transformation of the dynamic behavior of the system. SE complexity quantitatively analyzes the degree of chaos of the system by analyzing the time series of the system. High Shannon entropy indicates that the state of the system changes dramatically over time, with high uncertainty and complexity, indicating that the system is in a chaotic state. Low Shannon entropy indicates that the state of the system changes more regularly over time, with low information content, indicating that the system is in a periodic or stable state; C0 complexity evaluates the complexity of a sequence by calculating the frequency of occurrence of different permutation patterns in the time series. This allows the system's time series to contain different permutation patterns; Use SE complexity and C0 complexity to analyze the changes in dynamic behavior caused by changes in random amplitude in the system. Combine the four-dimensional system with the initial conditions in the initial condition table to obtain the effect diagram of the system under different conditions. Figure 2 It is a schematic diagram of the complexity of the four-dimensional system provided by the present invention under different amplitude conditions. Figure 2It is shown that under different amplitude conditions, the numerical variation trend of SE complexity and C0 complexity of the four-dimensional system is obvious. The color comparison diagram can illustrate that the four-dimensional system constructed with the three-dimensional system is also applicable to the system construction method proposed in the present invention that combines state variables with unknown parameters and uses random parameters as amplitudes. Under these conditions, the four-dimensional system model also has complex dynamic behavior.

[0054] In this embodiment, information entropy is a method for calculating the uncertainty of image information and is often used to reflect the uncertainty of image information. Therefore, using information entropy to calculate the information entropy of the plaintext image and the ciphertext image can intuitively verify the uncertainty of the ciphertext image. The calculation formula of information entropy is: ; In the above formula Indicates the possible grayscale values ​​in the image. Since the grayscale values ​​of the color images used in the article test are all 256, the theoretical value of information entropy is 8. The information entropy test is performed on the above plaintext image and ciphertext image, and the results are shown in Table 2.

[0055] Table 2 Image information entropy table

[0056] Usually, there is a close correlation between the pixels in all directions of the original image, while the encrypted image loses this correlation and presents a weaker correlation. Therefore, we can evaluate the effectiveness of the encryption system by comparing the pixel correlation between the original image and the encrypted image in the horizontal, vertical and diagonal directions. The affinity of adjacent pixel pairs can be calculated using a specific formula.

[0057] ; ; ; ; The above formula is used to test the correlation of adjacent pixels of the plaintext image and the ciphertext image in different directions. The test results are shown in Table 3.

[0058] Table 3 Pixel correlation test results

[0059] Furthermore, the encryption in S4 further includes: Plain text image settings, select the size MN The color image is the plaintext image, which is divided into three two-dimensional matrices of R, G, and B. In order to ensure the rationality of the image size, a 0-filling operation is performed.

[0060] ; In the formula, W is the block size.

[0061] Initial setup and parameter setting, select one-dimensional Generating chaotic sequences as chaotic maps , the sequence length is equal to the image size MN ; The selection of the initial value is based on the initial value in the three-dimensional system.

[0062] Transformation of chaotic sequence. Transform one-dimensional chaotic sequence Convert to a two-dimensional matrix to ensure that the sequence size is the same as the image size, and normalize the sequence to between 0 and 255 for subsequent DNA operations.

[0063] Use the Runge-Kutta method to solve the four-dimensional system and generate four sequences of length MN , , , , taking the two-dimensional matrix R as an example, the data value range is 0 to 255, and the data is represented by 8-bit binary numbers. Using the initial value of the four-dimensional , , , , and the initial conditions are calculated from different bits of the image; DNA coding and operation, converting image pixel values ​​into DNA sequences. Each pixel value corresponds to a DNA code. The generated chaotic sequence is used to perform XOR operation on the DNA sequence. The DNA coding rules are given by Determined by; DNA operations are determined by Decide; The encoded matrix is ​​permuted in rows and columns, and the permutation order is a chaotic sequence generated by a three-dimensional system. Decide; The permuted matrix is ​​subjected to DNA operation again to obtain the final ciphertext image.

[0064] Furthermore, the encryption in S4 is followed by decryption, which specifically includes: Decompose the ciphertext image into three channels: R, G, and B; Generate a chaotic sequence. Generate the same chaotic sequence about the three-dimensional chaotic system as that during encryption, and set the parameters to the same initial value as during encryption; Inverse permutation operation: perform inverse permutation operation on the matrices of the three channels of R, G, and B. The permutation order is opposite to that during encryption, from the last column to the first column and from the last row to the first row. Solve the four-dimensional system. Use the Runge-Kutta method to continue solving the four-dimensional system and generate the same four sequences as when encrypting. , , , .

[0065] Perform DNA inverse operation on the inverse permuted matrix, and the decoding rule is given by Determined by; the inverse operation rule is determined by It is decided to remove the zero elements added during encryption and restore the original pixel value.

[0066] The decoded R, G, and B two-dimensional matrices are combined into a three-dimensional matrix to obtain the decrypted color image.

[0067] It can be seen from the above embodiments that the expected benefits and commercial value of the technical solution of the present invention after transformation are as follows: a. Improve information security: By utilizing three-dimensional and four-dimensional continuous chaotic systems in DNA color image encryption, the present invention can significantly improve the complexity and robustness of data encryption, effectively prevent information leakage and tampering, and has broad application prospects in the field of information security.

[0068] b. Optimize the efficiency of encryption algorithms: This solution uses the unpredictability and complexity of chaotic systems to make the encryption process more resistant to cracking while maintaining high encryption and decryption efficiency. It is suitable for scenarios with high requirements for encryption speed, such as real-time image and video transmission.

[0069] c. Wide range of application scenarios: This technology is not only suitable for image encryption, but can also be applied to other data encryption scenarios that require high security, such as financial data transmission, confidential document protection, Internet of Things communication, etc., expanding the scope of its commercial application.

[0070] The technical solution of the present invention fills the technical gap in the industry: a. Combining unknown and uncertain variables to construct chaotic systems: This paper proposes a method to combine unknown variables with uncertain variables to construct continuous chaotic systems, which is different from the traditional single state variable construction method. This innovative method makes the dynamic behavior of the system more complex and diverse, filling the relevant research gap in the field of chaotic system construction.

[0071] b. Parameter screening and system analysis based on Lyapunov index: The numerical parameters are screened by Lyapunov index, and the system is further analyzed in detail by combining bifurcation diagram, attractor diagram, 0-1 test and dual-parameter Lyapunov index diagram, providing a comprehensive dynamic behavior research method. This method is innovative in dynamic system analysis and can effectively analyze the impact of parameters on the chaotic behavior of the system.

[0072] c. Scalability and applicability from three-dimensional to four-dimensional systems: This invention demonstrates a method for expanding the three-dimensional continuous chaotic system model to a four-dimensional system, and demonstrates its complex characteristics in higher dimensions. The scalability and universal applicability of this method fills the technical gap in the construction and analysis of high-dimensional chaotic systems in nonlinear dynamics.

[0073] d. Innovative image encryption method: By applying the constructed three-dimensional and four-dimensional chaotic systems to the DNA color image encryption system, the present invention proposes a new solution in the field of information encryption. Compared with traditional encryption methods, this solution has significant advantages in robustness and anti-attack capabilities, filling the gap in the field of high-security chaotic encryption.

[0074] d. Using permutation entropy to evaluate the random behavior changes of the system: Using permutation entropy, a detailed study was conducted on the chaotic behavior changes of the system under random amplitudes, which provided a new method and perspective for evaluating the impact of randomness on the behavior of chaotic systems and supplemented the deficiencies of existing chaotic system analysis methods.

[0075] The technical solution of the present invention overcomes the following technical prejudices: a. Limitations of single state variable construction: Traditionally, the construction of chaotic systems mainly relies on fixed state variables, which has certain limitations on the flexibility and complexity of the system. This invention overcomes this bias by combining unknown variables with uncertain variables to construct chaotic systems, making the constructed system more adaptable and diverse in dealing with complex environments.

[0076] b. Inherent conservatism in the construction of high-dimensional systems: In the field of nonlinear dynamics, many studies usually focus only on three-dimensional systems, believing that the construction and analysis of high-dimensional systems are too complex and difficult to implement. This paper overcomes the conservative attitude towards the construction of high-dimensional systems by expanding the three-dimensional continuous chaotic system into a four-dimensional system and demonstrating its performance at different complexities, thus expanding the research space of high-dimensional chaotic systems.

[0077] c. Single application of Lyapunov exponent: Traditionally, Lyapunov exponent is mainly used to judge the chaotic nature of the system, but its potential in parameter screening and system behavior analysis is often not fully explored. This invention overcomes this single perspective by screening parameters through Lyapunov exponent and combining other analysis methods (such as bifurcation diagram, attractor diagram, 0-1 test, etc.), and presents a more comprehensive system dynamics behavior research method.

[0078] d. Preference for linear methods in information encryption: The field of information encryption tends to use encryption methods based on algebraic operations or linear transformations, believing that these methods are easier to implement and more secure. This invention overcomes the preference for traditional linear encryption methods by applying the unpredictability and complexity of chaotic systems to DNA color image encryption, and proves the advantages of nonlinear chaotic methods in the field of encryption.

[0079] e. Ignoring the influence of randomness on chaotic behavior: In traditional research, the influence of randomness on the behavior of chaotic systems is often not given enough attention and is often regarded as an uncontrollable factor. This paper overcomes this bias by using permutation entropy to conduct an in-depth study of the changes in chaotic behavior of the system under random amplitudes, revealing the important role of randomness in system complexity and providing new ideas for more accurate system behavior control.

[0080] In the several embodiments provided by the present invention, it should be understood that the disclosed systems, devices and methods can be implemented in other ways. For example, the device embodiments described above are only schematic, for example, the division of units is only a logical function division, and there may be other division methods in actual implementation, such as multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be an indirect coupling or communication connection through some interfaces, devices or units, which can be electrical, mechanical or other forms.

[0081] In addition, each functional unit in each embodiment of the present invention may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above integrated unit may be implemented in the form of hardware or in the form of software functional units. The above is only an implementation mode of the present invention, and does not limit the patent scope of the present invention. Any equivalent structure or equivalent process transformation made by using the contents of the present invention specification and drawings, or directly or indirectly used in other related technical fields, is also included in the patent protection scope of the present invention.

[0082] The specific implementation methods of the invention are described in detail above, but they are only examples, and the invention is not limited to the specific implementation methods described above. For those skilled in the art, any equivalent modification or substitution of the invention is also within the scope of the invention, therefore, the equalization, modification, improvement, etc. made without departing from the spirit and principle of the invention should be included in the scope of the invention.

Claims

1. A method for constructing a chaotic system model based on state variables and uncertain variables, characterized in that: include: S1, obtain state variables, combine with the amplitude matrix to be determined, adopt a unified form of construction method, and use the Lyapunov index definition to verify the feasibility of the construction method; S2, obtaining a three-dimensional chaotic system with a single nonlinear structure, using the Lyapunov exponent to screen numerical parameters, bifurcation diagram, attractor diagram, 0-1 test and dual-parameter Lyapunov exponent diagram method to obtain the dynamic behavior changes of the three-dimensional chaotic system under the influence of unknown parameters; S3, using permutation entropy to analyze the changes in the dynamic behavior caused by random natural values ​​and mathematical functions in the random amplitude, based on the three-dimensional continuous chaotic system model structure, expanding it to a four-dimensional state, using different complexities to demonstrate the complex characteristics of the system in the four-dimensional state, and determining the universal applicability of the construction method; S4, the constructed three-dimensional and four-dimensional continuous systems are applied to the classic chaotic system DNA color image encryption system and perform robustness and security comparative analysis.

2. The method for constructing a chaotic system model based on state variables and uncertain variables according to claim 1, characterized in that: Said S1 further comprises: Starting from the construction of nonlinear dynamics model, for continuous chaotic system ,in , P express s dimensional vector field; The construction method is proposed from the perspective of state variables and unknown parameters. From the perspective of state variables, a continuous system is obtained: ； in, Participate in the construction of system models simultaneously with state variables; Multiply the matrices with unknown amplitudes to obtain the matrix equation: ； in, To determine the random amplitude within the interval; Output is about the system construction method of combining state variables with unknown parameters and random parameters as amplitudes.

3. The method for constructing a chaotic system model based on state variables and uncertain variables according to claim 1, characterized in that: Said S1 further comprises: Definition: A continuous system has two motion trajectories with similar but different starting points. The initial point is and The trajectory is set to and ; Set Existence , taking the derivative of the continuous function, we get: ； In the formula, the Jacobian matrix of the system is ; Further processing of equation (2) yields the linearized equations for the continuous system: ； Performing integral transformation, we get: , Definition of the Lyapunov exponent for continuous systems, along with time t The discretization process of the change of S The average for each time period is: ； For the system model, let , As the eigenvalue, the expression of the system's Lyapunov exponent is: ； It is assumed that there are qualified random amplitude intervals and unknown parameters involved in the construction of the continuous chaotic system. The different types of random amplitudes are: ； With the combination of parameters and state variables to construct the system, the construction expression of the chaotic system is obtained: ； The M in the formula is an ellipsis; In this way, the relevant calculation of the Lyapunov exponent is carried out when the system undergoes multiple iterations. When , we can get the above formula in Jacobian matrix during iteration: ； From the above formula, it can be seen that the construction of the system model involving state variables and random amplitudes has no effect on the Lyapunov exponent, and the construction of the new system does not change the chaotic characteristics of the original system.

4. The method for constructing a chaotic system model based on state variables and uncertain variables according to claim 1, characterized in that: Said S2 further comprises: Adopting the logical idea of ​​screening initial conditions, the relevant initial conditions of the system are screened, and the initial conditions of the parameters are obtained after a large number of iterations; Randomly select initial conditions within the range of values ​​that meet the conditions, use bifurcation diagrams and Lyapunov index diagrams for comparative analysis, and observe the dynamic behavior of the system model; The initial value is selected as (0.1, 0.1, 0.1), and the parameter selection range is selected , , ; The dynamic behavior of the system under the influence of parameters is analyzed through bifurcation diagrams and Lyapunov index diagrams; Based on the system model, application; for the four-dimensional system, detailed dynamic behavior exploration is required, starting from the parameters participating in the construction of the system model structure, setting the random amplitude as a fixed reference condition, and obtaining the specific simulation results of the unknown parameters for the dynamic behavior of the system model; Set the amplitude to , the numerical range of the unknown parameters is set to , we can get the bifurcation diagram and attractor diagram under the corresponding parameter conditions. For dynamic observable data , choose an arbitrary random constant , and define the following equation: ； in , The root mean square displacement is defined as: ； By observation The change trend of the root mean square displacement with time is to judge the motion state; When in Brownian motion, the root mean square displacement increases linearly with time; When the state of motion is bounded, the RMS displacement is also bounded; Test the asynchronous growth rate: ； pass Whether the value of is close to 0 or 1, we can intuitively judge whether the motion is in a periodic or quasi-periodic state or a chaotic state.

5. The method for constructing a chaotic system model based on state variables and uncertain variables according to claim 1, characterized in that: The S3 further includes: SE and C0 complexity are used as important methods to describe and analyze chaotic systems; It plays an important role in analyzing the transformation of the dynamic behavior of the system. SE complexity quantitatively analyzes the degree of chaos of the system by analyzing the time series of the system. A high Shannon entropy indicates that the state of the system changes dramatically over time, with high uncertainty and complexity, indicating that the system is in a chaotic state. A low Shannon entropy indicates that the state of the system changes more regularly over time, with low information content, indicating that the system is in a periodic or stable state. C0 complexity evaluates the complexity of a sequence by calculating the frequency of occurrence of different permutation patterns in the time series, so that the system's time series contains different permutation patterns; SE complexity and C0 complexity are used to analyze the changes in dynamic behavior caused by changes in random amplitudes in the system. The four-dimensional system is combined with the initial conditions to obtain the effect diagram of the system under different conditions.

6. The method for constructing a chaotic system model based on state variables and uncertain variables according to claim 1, characterized in that: The encryption in S4 also includes: Plain text image settings, select the size MN The color image is a plaintext image, which is divided into three two-dimensional matrices of R, G, and B. To ensure the rationality of the image size, a 0-filling operation is performed. ； In the formula, W is the block size, Initial setup and parameter setting, select one-dimensional Generating chaotic sequences as chaotic maps , the sequence length is equal to the image size MN ; The selection of the initial value is based on the initial value in the three-dimensional system. The transformation of chaotic sequence is to transform one-dimensional chaotic sequence into Convert to a two-dimensional matrix to ensure that the sequence size is the same as the image size, and normalize the sequence to between 0 and 255 for subsequent DNA operations. Use the Runge-Kutta method to solve the four-dimensional system and generate four sequences of length MN , , , , taking the two-dimensional matrix R as an example, the data value range is 0 to 255, the data is represented by 8-bit binary numbers, and the initial value of the four-dimensional , , , , and the initial conditions are calculated from different bits of the image; DNA coding and operation, converting image pixel values ​​into DNA sequences. Each pixel value corresponds to a DNA code. The generated chaotic sequence is used to perform XOR operation on the DNA sequence. The DNA coding rules are given by Determined by; DNA operations are determined by Determine; permute the rows and columns of the encoded matrix, and the permutation order is the chaotic sequence generated by the three-dimensional system Decide; The permuted matrix is ​​subjected to DNA operation again to obtain the final ciphertext image.

7. The method for constructing a chaotic system model based on state variables and uncertain variables according to claim 1, characterized in that: The encryption in S4 is followed by decryption, which specifically includes: Decompose the ciphertext image into three channels: R, G, and B; Generate a chaotic sequence, generate the same chaotic sequence about the three-dimensional chaotic system as that during encryption, and set the parameters to be consistent with the initial values ​​during encryption; Inverse permutation operation: perform inverse permutation operation on the matrices of the three channels of R, G, and B. The permutation order is opposite to that during encryption, from the last column to the first column and from the last row to the first row. Solve the four-dimensional system and use the Runge-Kutta method to continue solving the four-dimensional system, generating the same four sequences as when encrypting , , , , Perform DNA inverse operation on the inverse permuted matrix, and the decoding rule is given by Determined by; the inverse operation rule is determined by Determine, remove the zero elements added during encryption, and restore the original pixel value. The decoded R, G, and B two-dimensional matrices are combined into a three-dimensional matrix to obtain the decrypted color image.

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