Natiq Hayder, Said Mohamad Rushdan Md, Al-Saidi Nadia M G, Kilicman Adem
Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang 43000, Malaysia.
The Branch of Applied Mathematics, Applied Science Department, University of Technology, Baghdad 10075, Iraq.
Entropy (Basel). 2019 Jan 7;21(1):34. doi: 10.3390/e21010034.
Derived from Lorenz-Haken equations, this paper presents a new 4D chaotic laser system with three equilibria and only two quadratic nonlinearities. Dynamics analysis, including stability of symmetric equilibria and the existence of coexisting multiple Hopf bifurcations on these equilibria, are investigated, and the complex coexisting behaviors of two and three attractors of stable point and chaotic are numerically revealed. Moreover, a conducted research on the complexity of the laser system reveals that the complexity of the system time series can locate and determine the parameters and initial values that show coexisting attractors. To investigate how much a chaotic system with multistability behavior is suitable for cryptographic applications, we generate a pseudo-random number generator (PRNG) based on the complexity results of the laser system. The randomness test results show that the generated PRNG from the multistability regions fail to pass most of the statistical tests.
本文从洛伦兹 - 哈肯方程出发，提出了一种具有三个平衡点且仅含两个二次非线性项的新型四维混沌激光系统。研究了动力学分析，包括对称平衡点的稳定性以及这些平衡点上共存多个霍普夫分岔的存在性，并通过数值方法揭示了稳定点和混沌的两个及三个吸引子的复杂共存行为。此外，对激光系统复杂性的研究表明，系统时间序列的复杂性可以定位并确定呈现共存吸引子的参数和初始值。为了研究具有多稳定性行为的混沌系统在密码学应用中的适用程度，我们基于激光系统的复杂性结果生成了一个伪随机数发生器（PRNG）。随机性测试结果表明，从多稳定性区域生成的PRNG未能通过大多数统计测试。
