This article proposes a unified approach for hidden attractors control in fractional-order chaotic systems. Hidden attractors have small basins of attractions and are very sensitive to initial conditions and parameters. That is, they can be easily drifted from chaotic behavior into another type of dynamics, which is not suitable for encryption applications that require quite wide initial conditions and parameters ranges for encryption key design. Hence, a systematic coordinate affine transformation framework is utilized to construct transformed systems with self-reproducing attractors. Simulation results of two three-dimensional fractional-order chaotic systems with hidden attractors validate that the proposed framework supports attractors geometric structure design and multi-wing generation. Hidden attractor size, polarity, phase, shape and position control while preserving the chaotic dynamics is indicated by strange attractors, spectral entropy, maximum Lyapunov exponent and bifurcation diagrams. Simulations demonstrate the capability of multi-wing generation from fractional-order hidden attractors with no equilibria using non-autonomous parameters as opposed to the classical equilibria extension techniques suitable only for self-excited attractors. The self-reproduced multiple wings can share the same center point or be distributed along an arbitrary line, curve or surface thanks to the non-autonomous translation parameters. Multi-wing attractors widen the basin of attraction and enlarge the state space volume. For practical applications, the proposed technique makes fractional-order systems with hidden attractors suitable for circuit implementations that require specific signal level and polarity conditions. In addition, for digital encryption applications, the relatively wide range of the extra parameters enhances the key space and hence the robustness against brute force attacks.

中文翻译：

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使用仿射变换的分数阶混沌系统中的自复制隐藏吸引子

本文提出了一种分数阶混沌系统中隐藏吸引子控制的统一方法。隐藏的吸引子具有较小的吸引力盆地，并且对初始条件和参数非常敏感。也就是说，它们很容易从混乱的行为转变为另一种动力学，这不适用于需要非常宽的初始条件和参数范围来进行加密密钥设计的加密应用程序。因此，利用系统的坐标仿射变换框架来构建具有自我复制吸引子的变换系统。具有隐藏吸引子的两个二维分数阶混沌系统的仿真结果验证了所提出的框架支持吸引子的几何结构设计和多翼生成。隐藏的吸引子大小，极性，相位，在保持混沌动力学的同时，形状和位置控制由奇怪的吸引子，谱熵，最大Lyapunov指数和分叉图表示。仿真证明了使用非自治参数从没有平衡点的分数阶隐藏吸引子产生多翼的能力，这与仅适用于自激吸引子的经典平衡点扩展技术相反。由于具有非自主的平移参数，自复制的多个机翼可以共享相同的中心点，也可以沿任意线，曲线或曲面分布。多翼吸引子扩大了吸引面，扩大了国家空间的体积。对于实际应用，所提出的技术使具有隐藏吸引子的分数阶系统适用于需要特定信号电平和极性条件的电路实现。此外，对于数字加密应用程序，附加参数的相对较大范围可增强密钥空间，从而增强对暴力攻击的鲁棒性。