| Chaotic system is a special kind of nonlinear system,which has the characteristics of high randomness,initial value sensitivity and unpredictability.Due to its inherent properties,chaotic systems are widely used in many fields such as finance,information security and power engineering.In recent years,scholars have applied the fractional calculus theory to chaotic systems and found that the fractional-order chaotic model can not only accurately describe the complex behavior of the nonlinear system,but also has its own unique properties.Fractional-order chaotic systems have become a research hotspot in the field of chaos analysis and its application.Based on this,this paper theoretically analyzes the existing phase space reconstruction methods and parameter identification methods,and studies their application in image encryption.The main content is as follows:Firstly,taking the single-variable fractional-order chaotic time series data as the research object,the mutual information method and the Cao method are used to solve the two key parameters of time delay and embedding dimension,respectively,and the performance of these two methods under different noise levels is also analyzed.The results show that the existing phase space reconstruction method can be applied to the analysis of fractional-order chaotic time series.The key parameters obtained by the mutual information method and Cao method are ideal and have good noise resistance.The reconstructed attractor can well represent the dynamic properties and complex information of the original attractor.Secondly,the sparse identification of nonlinear dynamics algorithm(SINDy)is extended to study the sparse identification problem of fractional-order nonlinear dynamic systems.First,the advantages and disadvantages of the identification parameters are analyzed when the fractional order is known,and then a polynomial fitting method is designed to solve the optimal order for the unknown fractional order.The simulation results show that the method can identify the system parameters from the data and determine the optimal order.Finally,a new image encryption algorithm is proposed by combining the timefractional diffusion system with the fractional-order Chua’s system.The chaotic sequences generated by the fractional Chua’s system are used to scramble the rows and columns of the RGB primary color components of the digital image respectively.Then the initial and boundary conditions of the diffusion system are rewritten according to the chaotic subsequence to generate a diffusion sequence,and the scrambled image is encrypted by the sequence.The algorithm also performs key analysis,histogram analysis,correlation analysis,information entropy analysis,differential attack analysis,noise attack analysis,clipping attack analysis,encryption speed analysis and comparison with other existing algorithms.The results show that the proposed algorithm has the characteristics of large key space,high sensitivity to initial boundary conditions,and fast encryption speed. |
