Silva-Juárez Alejandro, Tlelo-Cuautle Esteban, de la Fraga Luis Gerardo, Li Rui
INAOE, Luis Enrique Erro No. 1. Tonanztintla, Puebla 72840, Mexico.
CINVESTAV, Av. Instituto Politécnico Nacional No. 2508, San Pedro Zacatenco 07360, Mexico.
J Adv Res. 2020 Jun 20;25:77-85. doi: 10.1016/j.jare.2020.05.014. eCollection 2020 Sep.
Fractional-order chaotic oscillators (FOCOs) have been widely studied during the last decade, and some of them have been implemented on embedded hardware like field-programmable gate arrays, which is a good option for fast prototyping and verification of the desired behavior. However, the hardware resources are dependent on the length of the digital word that is used, and this can degrade the desired response due to the finite number of bits to perform computer arithmetic. In this manner, this paper shows the implementation of FOCOs using analog electronics to generate continuous-time chaotic behavior. Charef's method is applied to approximate the fractional-order derivatives as a ratio of two polynomials in the Laplace domain. For instance, two commensurate FOCOs are the cases of study herein, for which we show their dynamical analysis by evaluating their equilibrium points and eigenvalues that are used to estimate the minimum fractional-order that guarantees their chaotic behavior. We propose the use of first-order all-pass and low-pass filters to design the ratio of the polynomials that approximate the fractional-order. The filters are implemented using amplifiers and synthesized on a field-programmable analog array (FPAA) device. Experimental results are in good agreement with simulation results thus demonstrating the usefulness of FPAAs to generate continuous-time chaotic behavior, and to allow reprogramming of the parameters of the FOCOs.
在过去十年中，分数阶混沌振荡器（FOCOs）得到了广泛研究，其中一些已在诸如现场可编程门阵列等嵌入式硬件上实现，这对于快速原型设计和验证期望行为是一个不错的选择。然而，硬件资源取决于所使用数字字的长度，并且由于执行计算机运算的位数有限，这可能会降低期望的响应。以这种方式，本文展示了使用模拟电子学实现分数阶混沌振荡器以生成连续时间混沌行为。应用查雷夫方法在拉普拉斯域中将分数阶导数近似为两个多项式的比值。例如，本文研究了两个同阶分数阶混沌振荡器的情况，通过评估它们的平衡点和特征值来进行动力学分析，这些平衡点和特征值用于估计保证其混沌行为的最小分数阶。我们建议使用一阶全通滤波器和低通滤波器来设计近似分数阶的多项式的比值。这些滤波器使用放大器实现，并在现场可编程模拟阵列（FPAA）器件上进行合成。实验结果与仿真结果吻合良好，从而证明了现场可编程模拟阵列在生成连续时间混沌行为以及允许对分数阶混沌振荡器的参数进行重新编程方面的有用性。
