Doungmo Goufo Emile F
Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa.
Chaos. 2019 Feb;29(2):023117. doi: 10.1063/1.5085440.
Not every chaotic system has the particularity of displaying attractors with a fractal structure. That is why strange attractors remain enthralling not only for their fractal structure, but also for their amazing chaotic and multi-scroll dynamics. In this work, we apply the non-local and non-singular kernel operator to a four-dimensional chaotic system with two equilibrium points and show the existence of various types of attractors, including the butterfly type and strange type. Recently, there have been virulent communications related to the validity or not of the index law in fractional differentiation with non-local operators. These discussions resulted in pointing out many important features of the Mittag-Leffler function used as kernel and suitable to describe more complex real world problems. This paper follows the same momentum by pointing out another important feature of the non-local and non-singular kernel operator applied to chaotic models. We solve the model numerically and discuss the bifurcation and period doubling dynamics that eventually lead to chaos (in the form of butterfly attractor). Lastly, we provide related numerical simulations which prove the existence of a chaotic fractal structure (strange attractors).
并非每个混沌系统都具有呈现分形结构吸引子的特性。这就是为什么奇异吸引子不仅因其分形结构，还因其惊人的混沌和多涡卷动力学而一直引人入胜。在这项工作中，我们将非局部和非奇异核算子应用于一个具有两个平衡点的四维混沌系统，并展示了包括蝴蝶型和奇异型在内的各种类型吸引子的存在。最近，出现了与非局部算子分数阶微分中指数律的有效性相关的激烈讨论。这些讨论指出了用作核且适合描述更复杂现实世界问题的米塔格 - 莱夫勒函数的许多重要特征。本文通过指出应用于混沌模型的非局部和非奇异核算子的另一个重要特征，延续了同样的势头。我们对模型进行了数值求解，并讨论了最终导致混沌（以蝴蝶吸引子的形式）的分岔和倍周期动力学。最后，我们提供了相关的数值模拟，证明了混沌分形结构（奇异吸引子）的存在。
