Analysis of chaotic characteristics of trigonometric function system

Abstract

Chaos, as an important subject of nonlinear science, plays an important role in solving problems in both natural sciences and social sciences such as the fields of secure communications, fluid motion, particle motion and so on. Aiming at this problem, this paper proposes a nonlinear dynamic system composed of product trigonometric functions and studies its chaotic characteristics. Through the mathematical derivation of the system’s period, the analysis of the necessary conditions at the fixed point, the experimental drawing of the Lyapunov exponential graph and the branch graph of the system, it is proved that the system has larger chaotic interval and stronger chaotic characteristics. The parameters of the proposed dynamic system are generated randomly, and then the chaotic sequence can be generated. The chaotic sequence is used to encrypt the digital image, a good encryption effect is obtained, and there is a large key space. At the same time, the motion of the particles in the space magnetic field is simulated, which further proves that the trigonometric system has strong chaotic characteristics.

A lossless compression method for random integer based on three-dimensional product- triangular chaotic iterative sequences

“…In this part, [33] we refer to a 3D product triangular chaotic system with 24 parameters to implement data compression. The expression of the 3D chaotic system is shown in equation (2):…”

“…where a , i b , i and c i are frequency parameters, r i is amplitude parameter, and q , i j , xi j , yi and j zi are other parameters. After analyzing the chaotic characteristics of the above system, some scholars point out that the change of chaotic characteristics of iterative sequence is mainly affected by frequency and amplitude [33]. That is, as the frequency increases, the number of attractors increases and the degree of chaos of the system becomes stronger.…”

“…We only need a set of parameter conditions that satisfy the compression requirements to verify the correctness and feasibility of the proposed compression method. Literature [33] specifically points out that when the value of the frequency parameter is between 0 and 7. Meanwhile, when the absolute value of the amplitude is greater than 1, the system is chaotic.…”

“…Therefore, the results of the above research fully confirm the great potential of data compression using triangular chaotic systems. In this regard, [33] some scholars have also proposed a three-dimensional product triangle chaotic system. The three branch functions of the system are all composed of the product of three cosine functions [33].…”

“…In this regard, [33] some scholars have also proposed a three-dimensional product triangle chaotic system. The three branch functions of the system are all composed of the product of three cosine functions [33]. Theoretical calculation in the research shows that the length of its period is closely related to the product of the period of each trigonometric function.…”