In the applied research of nonlinear system, the low degree of chaos in the dynamical system leads to the limitation of using the chaos method to solve some practical problems. In this paper, we use the product trigonometric function and ternary polynomial to build a dynamical system, which has strong chaotic characteristics. The dynamical system is constructed by two product trigonometric functions and a ternary linear equation, and its chaotic properties are verified by bifurcation diagrams, Lyapunov exponents, fractal dimensions, etc. The system has many parameters and large parameter intervals and is not prone to cycles. The conditions for the non-divergence of this system are given by mathematical derivation, and it is found that the linear part of the system can be replaced by an arbitrary ternary polynomial system and still not diverge, and the bifurcation diagram is drawn to verify it. Finally, the chaotic sequence is distributed more uniformly in the value domain space by adding the modulo operation. Then, the bit matrix of multiple images is directly permuted by the above system, and the experiment confirms that the histogram, information entropy, and pixel correlation of its encrypted images are satisfactory, as well as a very large key space.

中文翻译：

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乘积三角函数和多项式构建的混沌系统应用于彩色图像加密

在非线性系统的应用研究中，动力系统的低混沌度导致了使用混沌方法解决一些实际问题的局限性。在本文中，我们使用乘积三角函数和三元多项式来构建具有强混沌特性的动力系统。动力系统由两个乘积三角函数和一个三元线性方程组构成，并通过分岔图、李雅普诺夫指数、分形维数等验证了其混沌特性。系统参数多、参数区间大，不易产生循环。数学推导给出了该系统不发散的条件，发现该系统的线性部分可以用任意三元多项式系统代替，仍然不发散，并绘制分岔图进行验证。最后，通过加模运算，混沌序列在值域空间中分布更均匀。然后，通过上述系统直接对多幅图像的位矩阵进行置换，实验证实其加密图像的直方图、信息熵和像素相关性均令人满意，密钥空间非常大。