In this paper, a bipolar sigmoid function series is proposed to generate 1-D (one dimensional) and 2-D (two dimensional) multi-scroll chaotic attractors. The nonlinear dynamical behaviors of 1-D and 2-D multi-scroll chaotic systems are analyzed, including the equilibrium points, invariance, dissipation, Lyapunov exponents, fractional dimension and Poincaré map. In addition, backstepping controllers are used to stabilize the 1-D and 2-D multi-scroll chaotic systems. The results show that the 1-D and 2-D multi-scroll chaotic attractors can be generated by introducing the bipolar sigmoid function series. The mechanism for generating multi-scroll chaotic attractors is as follows: The equilibrium points with index 2 (type two) are responsible for generating scrolls, and the equilibrium points with index 1 (type one) are responsible for connecting these scrolls. The chaotic behaviors in the 1-D and 2-D multi-scroll chaotic systems are stabilized to some equilibrium points using the backstepping controllers, without determining the desired targeting orbits beforehand.

中文翻译：

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通过引入双极S型函数系列产生的多卷混沌吸引子的混沌控制

在本文中，提出了一个双极乙状结肠功能序列，以生成一维（一维）和二维（二维）多维涡旋混沌吸引子。分析了一维和二维多涡卷混沌系统的非线性动力学行为，包括平衡点，不变性，耗散，李雅普诺夫指数，分数维和庞加莱图。另外，反推控制器用于稳定一维和二维多滚动混沌系统。结果表明，通过引入双极S形函数系列，可以生成一维和二维多涡卷混沌吸引子。产生多涡卷混沌吸引子的机制如下：索引为2（类型2）的平衡点负责产生涡卷，索引为1（类型1）的平衡点负责连接这些涡旋。使用反步控制器可以将一维和二维多滚动混沌系统中的混沌行为稳定到一些平衡点，而无需事先确定所需的目标轨道。