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Issues on Applying One- and Multi-Step Numerical Methods to Chaotic Oscillators for FPGA Implementation

Author

Listed:
  • Omar Guillén-Fernández

    (Department of Electronics, INAOE, Tonantintla, Puebla 72840, Mexico
    These authors contributed equally to this work.)

  • María Fernanda Moreno-López

    (Department of Electronics, INAOE, Tonantintla, Puebla 72840, Mexico
    These authors contributed equally to this work.)

  • Esteban Tlelo-Cuautle

    (Department of Electronics, INAOE, Tonantintla, Puebla 72840, Mexico
    These authors contributed equally to this work.)

Abstract

Chaotic oscillators have been designed with embedded systems like field-programmable gate arrays (FPGAs), and applied in different engineering areas. However, the majority of works do not detail the issues when choosing a numerical method and the associated electronic implementation. In this manner, we show the FPGA implementation of chaotic and hyper-chaotic oscillators from the selection of a one-step or multi-step numerical method. We highlight that one challenge is the selection of the time-step h to increase the frequency of operation. The case studies include the application of three one-step and three multi-step numerical methods to simulate three chaotic and two hyper-chaotic oscillators. The numerical methods provide similar chaotic time-series, which are used within a time-series analyzer (TISEAN) to evaluate the Lyapunov exponents and Kaplan–Yorke dimension ( D K Y ) of the (hyper-)chaotic oscillators. The oscillators providing higher exponents and D K Y are chosen because higher values mean that the chaotic time series may be more random to find applications in chaotic secure communications. In addition, we choose representative numerical methods to perform their FPGA implementation, which hardware resources are described and counted. It is highlighted that the Forward Euler method requires the lowest hardware resources, but it has lower stability and exactness compared to other one-step and multi-step methods.

Suggested Citation

  • Omar Guillén-Fernández & María Fernanda Moreno-López & Esteban Tlelo-Cuautle, 2021. "Issues on Applying One- and Multi-Step Numerical Methods to Chaotic Oscillators for FPGA Implementation," Mathematics, MDPI, vol. 9(2), pages 1-14, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:151-:d:478846
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    References listed on IDEAS

    as
    1. J. M. Muñoz-Pacheco & D. K. Guevara-Flores & O. G. Félix-Beltrán & E. Tlelo-Cuautle & J. E. Barradas-Guevara & C. K. Volos, 2018. "Experimental Verification of Optimized Multiscroll Chaotic Oscillators Based on Irregular Saturated Functions," Complexity, Hindawi, vol. 2018, pages 1-17, March.
    2. Esteban Tlelo-Cuautle & Antonio de Jesus Quintas-Valles & Luis Gerardo de la Fraga & Jose de Jesus Rangel-Magdaleno, 2016. "VHDL Descriptions for the FPGA Implementation of PWL-Function-Based Multi-Scroll Chaotic Oscillators," PLOS ONE, Public Library of Science, vol. 11(12), pages 1-32, December.
    Full references (including those not matched with items on IDEAS)

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