Coupling piecewise power function chaotic system

Abstract

The invention relates to the field of nonlinear dynamics, in particular to a coupled piecewise power function chaotic system. According to the method, compression, translation and segmentation mechanisms are firstly adopted, a segmented power function chaotic system is designed, and the system can generate a random number sequence with good chaotic characteristics. Then, based on the piecewise power function chaotic system, a coupled piecewise power function chaotic system is constructed in a parameter coupling mode, the system has two coupling modes of global coupling and local coupling, and the parameter space is shown in the specification. The invention provides an excellent multi-dimensional random number sequence generation method which can provide powerful support for application in the fields of scientific calculation, engineering control, information technology and the like.

Description

Technical Field

[0003] This invention relates to the field of nonlinear dynamics, specifically to a coupled piecewise power function chaotic system. Background Technology

[0005] Chaotic systems, due to their extreme sensitivity to initial conditions and long-term unpredictability, have shown great application potential in many engineering and information science fields such as secure communication, image encryption, pseudo-random number generation, radar ranging, and fault diagnosis.

[0006] Common chaotic systems are primarily built upon classical mathematical models, such as Logistic systems, Lorenz systems, Chen systems, and various improved systems derived from them. However, with the rapid development of computing power and the continuous advancement of cryptanalysis techniques, these traditional chaotic systems have gradually revealed some inherent flaws:

[0007] First, the complexity of chaotic behavior is limited. The phase space structure of many traditional chaotic systems is relatively simple, and the Lyapunov exponent spectrum is not rich enough, resulting in insufficient complexity of the chaotic sequences they generate, making them relatively vulnerable to powerful phase space reconstruction and prediction attacks.

[0008] Second, the parameter space has a narrow chaotic region. Changes in system parameters can often only maintain a chaotic state within a small range, which limits its robustness and availability in scenarios requiring flexible parameter tuning or where parameters are used as encryption keys.

[0009] Third, it has weak resistance to degradation. During the digital implementation process, due to the finite precision effect, traditional chaotic systems are prone to dynamic degradation, resulting in short-period orbits or the disappearance of chaotic characteristics, which seriously threatens the security of the encryption system.

[0010] Coupling mechanisms, as an important concept in complex systems theory, connect multiple simple dynamical subsystems in a specific way. This is an effective means of generating hyperchaos, expanding parameter spaces, and producing more complex spatiotemporal dynamic behaviors. Introducing coupling concepts into chaotic system design is an important direction for improving system performance.

[0011] Designing a novel "coupled piecewise power function chaotic system" to generate chaotic behavior with higher complexity, better randomness, and stronger resistance to degradation over a wider parameter range has become a pressing technical problem in this field. This invention aims to fill this technological gap. Summary of the Invention

[0013] (a) The technical problem that the invention aims to solve

[0014] The purpose of this invention is to overcome the shortcomings of existing chaotic systems, such as limited complexity and narrow parameter space, and to provide a chaotic system that can generate more complex chaotic behavior and has a wider range of chaotic parameters.

[0015] (II) Technical Solution

[0016] Preferred translation, compression, and piecewise strategies for power functions By performing a transformation, a piecewise power function chaotic system is proposed.

[0017] By optimizing parameter coupling, the piecewise power function chaotic system is extended into a high-dimensional chaotic system. Local and global coupling are employed to achieve local and global perturbations of the parameters.

[0018] (III) Beneficial Effects

[0019] By employing piecewise power function nonlinear terms, this invention can generate more complex nonlinear dynamic structures in phase space compared to traditional nonlinear systems, and the Lyapunov exponent is significantly improved.

[0020] By introducing local and global coupling mechanisms, the system complexity is significantly increased and the parameter space is significantly expanded. The proposed coupling mechanism can be extended to arbitrary dimensions. Attached Figure Description

[0022] To more intuitively describe the actual effects of the present invention, the accompanying drawings involved in the technical solution description are briefly introduced below. The drawings are only used to illustrate the simulation effects of the technical solution and are not intended to limit this application.

[0023] Figure 1 This invention generates Lyapunov exponent diagrams of chaotic sequences of different dimensions.

[0024] Figure 2 This is a probability distribution diagram of chaotic sequences of different dimensions generated by this invention.

[0025] Figure 3 This is a scatter plot of different dimensional combinations generated by the present invention. Detailed Implementation

[0027] The technical solutions in the embodiments of the present invention will be described in detail and completely below. The described embodiments are only some embodiments of the present invention, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the protection scope of the present invention.

[0028] (I) Implementation Examples

[0029] power functions By performing compression, translation, and piecewise segmentation, a novel piecewise power function chaotic system is designed, and its mathematical description is as follows:

[0030]

[0031] in, r This is a segmented control parameter, and its value range is (0, 0.05).

[0032] By employing parameter coupling, the piecewise power function chaotic system is extended into a high-dimensional chaotic system. The mathematical description of the coupled piecewise power function chaotic system is as follows:

[0033]

[0034] here, u This is the coupling parameter, and its value ranges from (0.99, 1). In this system, w and x For local coupling, w and x Mutual coupling influence y and z For global coupling, subject to w, x, y, z The state values ​​of the four parameters have a combined effect.

[0035] (II) Simulation and Effect Analysis

[0036] Numerical simulations were performed on a Lyapunov exponent coupled piecewise power function chaotic system. The results are attached. Figure 1 It can be seen that when the coupling parameter u When the value is greater than 0.6, the Lyapunov exponents of all dimensions are greater than 0, indicating that all dimensions are in a chaotic state.

[0037] set up w 0 = 0.111111 x 0 = 0.444444 y 0 = 0.555555 z 0 = 0.888888 r =0.02、 u =0.99, iterative coupled piecewise power function chaotic system, 1,000,000 iterations, probability density plots of each dimension are as follows. Figure 2 As shown, the probability density of each dimension did not show significant deviations, exhibiting good uniformity.

[0038] Figure 3The graph shows scatter plots of different dimensional combinations. As can be seen, the state values ​​of different dimensional combinations are uniformly distributed in the phase space, with no significant blank or concentrated regions, indicating that each dimension exhibits good randomness and that no synchronization occurs between different dimensions. Therefore, the proposed coupled piecewise power function chaotic system possesses good chaotic characteristics.

Claims

1. Claim 1: An image encryption method with hierarchical decryption capability, characterized in that, The system contains both local and global coupling.

2. Claim 2: The piecewise power function chaotic system, characterized in that, Depending on the different numerical ranges of its input variables, different forms of power functions are used for definition. The definition of a piecewise power function chaotic system is as follows: Where r is the segmented control parameter.

3. Claim 3: The coupled piecewise power function chaotic system according to claim 1, characterized in that, Global coupling parameters affect the system in all dimensions.

4. Claim 4: The coupled piecewise power function chaotic system according to claim 1, characterized in that, Local coupling parameters affect the system in local dimensions.

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