Ouannas Adel, Khennaoui Amina-Aicha, Momani Shaher, Pham Viet-Thanh
Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Laarbi Tebessi, Tebessa 12002, Algeria.
Laboratory of Dynamical Systems and Control, University of Larbi Ben M'hidi, Oum El Bouaghi 04000, Algeria.
Chaos. 2020 Aug;30(8):083131. doi: 10.1063/5.0005059.
In this paper, we study the dynamics and control of a Caputo fractional difference form of the Duffing map. We use phase plots, bifurcation diagrams, and Lyapunov exponents to establish the existence of chaos over a wide range of fractional orders and examine the nature of the dynamics. Also, we present the 0-1 test to detect chaos and C complexity, which is an alternative nonlinear statistical measure that can quantify the regularity of a time series. In addition, we measure the approximate entropy to see the performance of our numerical results. Through phase plots and bifurcation diagrams, it is shown that the proposed fractional map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. A one-dimensional feedback stabilization controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Simulation results have been carried out to illustrate the findings of the study.
在本文中，我们研究了达夫映射的卡普托分数阶差分形式的动力学与控制。我们使用相图、分岔图和李雅普诺夫指数来确定在广泛的分数阶范围内混沌的存在，并研究动力学的性质。此外，我们提出了用于检测混沌和C复杂度的0-1检验，C复杂度是一种可量化时间序列规律性的非线性统计量度。另外，我们通过测量近似熵来考察数值结果的性能。通过相图和分岔图表明，所提出的分数阶映射展现出一系列不同的动力学行为，包括混沌和共存吸引子。提出了一种一维反馈镇定控制器。借助线性分数阶离散时间系统的稳定性理论，建立了所提控制器的渐近收敛性。已进行仿真结果以说明该研究的发现。
