This paper is mainly devoted to the investigation of discrete-time fractional systems in three aspects. Firstly, the fractional Bogdanov map with memory effect in Riemann–Liouville sense is obtained. Then, via constructing suitable controllers, the fractional Bogdanov map is shown to undergo a transition from regular state to chaotic one. Meanwhile, the positive largest Lyapunov exponent is calculated by the Jacobian matrix algorithm to distinguish the chaotic areas. Finally, the Grassberger–Procaccia algorithm is employed to evaluate the correlation dimension of the controlled fractional Bogdanov system under different parameters. The main results show that the correlation dimension converges to a fixed value as the embedding dimension increases for the controlled fractional Bogdanov map in chaotic state, which also coincides with the conclusion driven by the largest Lyapunov exponent. Moreover, three-dimensional fractional Stefanski map is considered to further verify the effectiveness and generality of the obtained results.

中文翻译：

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关于离散分数混沌系统的相关维数

本文主要从三个方面研究离散时间分数系统。首先，得到了黎曼-刘维尔意义上的具有记忆效应的分数波格丹诺夫映射。然后，通过构建合适的控制器，分数 Bogdanov 映射显示出经历了从常规状态到混沌状态的转变。同时，通过雅可比矩阵算法计算正最大李雅普诺夫指数，以区分混沌区域。最后，采用Grassberger-Procaccia算法评估不同参数下受控分数Bogdanov系统的相关维数。主要结果表明，对于混沌状态下的受控分数波格丹诺夫图，随着嵌入维数的增加，相关维数收敛到一个固定值，这也与最大的李雅普诺夫指数驱动的结论不谋而合。此外，还考虑了三维分数 Stefanski 图，以进一步验证所得结果的有效性和普遍性。