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Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators

Author

Listed:
  • Martín Alejandro Valencia-Ponce

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

  • Esteban Tlelo-Cuautle

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

  • Luis Gerardo de la Fraga

    (Computer Science Department, CINVESTAV, Av. IPN 2508, Mexico City 07360, Mexico
    These authors contributed equally to this work.)

Abstract

The execution time that takes to perform numerical simulation of a chaotic oscillator mainly depends on the time-step h . This paper shows that the optimization of chaotic oscillators can be enhanced by estimating the highest h in either one-step or multi-step methods. Four chaotic oscillators are used as a case study, and the optimization of their Kaplan-Yorke dimension ( D K Y ) is performed by applying three metaheuristics, namely: particle swarm optimization (PSO), many optimizing liaison (MOL), and differential evolution (DE) algorithms. Three representative one-step and three multi-step methods are used to solve the four chaotic oscillators, for which the estimation of the highest h is obtained from their stability analysis. The optimization results show the effectiveness of using a high h value for the six numerical methods in reducing execution time while maximizing the positive Lyapunov exponent ( L E + ) and D K Y of the chaotic oscillators by applying PSO, MOL, and DE algorithms.

Suggested Citation

  • Martín Alejandro Valencia-Ponce & Esteban Tlelo-Cuautle & Luis Gerardo de la Fraga, 2021. "Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators," Mathematics, MDPI, vol. 9(16), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1938-:d:614097
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    References listed on IDEAS

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    1. Donald R. Jones & Joaquim R. R. A. Martins, 2021. "The DIRECT algorithm: 25 years Later," Journal of Global Optimization, Springer, vol. 79(3), pages 521-566, March.
    2. He, Qie & Wang, Ling & Liu, Bo, 2007. "Parameter estimation for chaotic systems by particle swarm optimization," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 654-661.
    3. Li Cui & Chaoyang Chen & Jie Jin & Fei Yu & Chong Fu, 2021. "Dynamic Analysis and FPGA Implementation of New Chaotic Neural Network and Optimization of Traveling Salesman Problem," Complexity, Hindawi, vol. 2021, pages 1-10, April.
    4. R. Cavoretto & A. Rossi & M. S. Mukhametzhanov & Ya. D. Sergeyev, 2021. "On the search of the shape parameter in radial basis functions using univariate global optimization methods," Journal of Global Optimization, Springer, vol. 79(2), pages 305-327, February.
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    Cited by:

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    2. Astrid Maritza González-Zapata & Esteban Tlelo-Cuautle & Brisbane Ovilla-Martinez & Israel Cruz-Vega & Luis Gerardo De la Fraga, 2022. "Optimizing Echo State Networks for Enhancing Large Prediction Horizons of Chaotic Time Series," Mathematics, MDPI, vol. 10(20), pages 1-19, October.
    3. Jiri Petrzela & Miroslav Rujzl, 2022. "Chaotic Oscillations in Cascoded and Darlington-Type Amplifier Having Generalized Transistors," Mathematics, MDPI, vol. 10(3), pages 1-25, February.
    4. Árpád Bűrmen & Tadej Tuma, 2022. "Preface to the Special Issue on “Optimization Theory and Applications”," Mathematics, MDPI, vol. 10(24), pages 1-3, December.
    5. Jiri Petrzela, 2023. "Chaotic States of Transistor-Based Tuned-Collector Oscillator," Mathematics, MDPI, vol. 11(9), pages 1-13, May.
    6. Jiri Petrzela, 2022. "Chaotic and Hyperchaotic Dynamics of a Clapp Oscillator," Mathematics, MDPI, vol. 10(11), pages 1-20, May.

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