The European Physical Journal Plus ( IF 3.4 ) Pub Date : 2021-05-07 , DOI: 10.1140/epjp/s13360-021-01503-y Tianming Liu , Santo Banerjee , Huizhen Yan , Jun Mou
In this paper, the definition of fractional calculus introduces a newly constructed discrete chaotic map. Interestingly, when the order is extended to the improper fractional order, the map still shows a chaotic state with appropriate parameters. The related dynamical behavior is analyzed by a phase diagram, bifurcation model, maximum Lyapunov exponent diagram, and approximate entropy algorithm method. The research results show that both the chaotic range of the fractional-order and the improper fractional-order are larger than the chaotic range of the integer-order, and they have richer dynamical behaviors. Besides, we found that multi-stability phenomena also exist in discrete chaotic systems, and the multiple stability of fractional order is more complicated than that of integer order, and the types of coexisting attractors are more abundant. Finally, the digital circuit and pseudo-random sequence generator of the discrete system are designed. This research guides the application and teaching of discrete fractional-order systems.
中文翻译:
分数阶2D-SCLMM的动态分析及其DSP实现
在本文中,分数阶微积分的定义引入了一种新构造的离散混沌图。有趣的是,当阶数扩展为不正确的分数阶数时,映射图仍显示具有适当参数的混沌状态。通过相图,分叉模型,最大李雅普诺夫指数图和近似熵算法方法分析了相关的动力学行为。研究结果表明,分数阶和不正确分数阶的混沌范围都大于整数阶的混沌范围,并且它们具有更丰富的动力学行为。此外,我们发现离散混沌系统中也存在多重稳定性现象,分数阶的多重稳定性比整数阶的更加复杂,并且共存吸引子的类型更加丰富。最后,设计了离散系统的数字电路和伪随机序列发生器。本研究指导离散分数阶系统的应用和教学。