Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators
The execution time that takes to perform numerical simulation of a chaotic oscillator mainly depends on the time-step <i>h</i>. This paper shows that the optimization of chaotic oscillators can be enhanced by estimating the highest <i>h</i> in either one-step or multi-step me...
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MDPI AG
2021-08-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/16/1938 |
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| author | Martín Alejandro Valencia-Ponce Esteban Tlelo-Cuautle Luis Gerardo de la Fraga |
| author_facet | Martín Alejandro Valencia-Ponce Esteban Tlelo-Cuautle Luis Gerardo de la Fraga |
| author_sort | Martín Alejandro Valencia-Ponce |
| collection | DOAJ |
| description | The execution time that takes to perform numerical simulation of a chaotic oscillator mainly depends on the time-step <i>h</i>. This paper shows that the optimization of chaotic oscillators can be enhanced by estimating the highest <i>h</i> 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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mrow><mi>K</mi><mi>Y</mi></mrow></msub></semantics></math></inline-formula>) 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 <i>h</i> is obtained from their stability analysis. The optimization results show the effectiveness of using a high <i>h</i> value for the six numerical methods in reducing execution time while maximizing the positive Lyapunov exponent (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>E</mi><mo>+</mo></mrow></semantics></math></inline-formula>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mrow><mi>K</mi><mi>Y</mi></mrow></msub></semantics></math></inline-formula> of the chaotic oscillators by applying PSO, MOL, and DE algorithms. |
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| language | English |
| last_indexed | 2024-03-10T08:37:24Z |
| publishDate | 2021-08-01 |
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| series | Mathematics |
| spelling | doaj.art-b85991fb669342cfbf695746dfe596592023-11-22T08:34:08ZengMDPI AGMathematics2227-73902021-08-01916193810.3390/math9161938Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic OscillatorsMartín Alejandro Valencia-Ponce 0Esteban Tlelo-Cuautle1Luis Gerardo de la Fraga2Department of Electronics, INAOE, Tonantzintla, Puebla 72840, MexicoDepartment of Electronics, INAOE, Tonantzintla, Puebla 72840, MexicoComputer Science Department, CINVESTAV, Av. IPN 2508, Mexico City 07360, MexicoThe execution time that takes to perform numerical simulation of a chaotic oscillator mainly depends on the time-step <i>h</i>. This paper shows that the optimization of chaotic oscillators can be enhanced by estimating the highest <i>h</i> 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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mrow><mi>K</mi><mi>Y</mi></mrow></msub></semantics></math></inline-formula>) 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 <i>h</i> is obtained from their stability analysis. The optimization results show the effectiveness of using a high <i>h</i> value for the six numerical methods in reducing execution time while maximizing the positive Lyapunov exponent (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>E</mi><mo>+</mo></mrow></semantics></math></inline-formula>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mrow><mi>K</mi><mi>Y</mi></mrow></msub></semantics></math></inline-formula> of the chaotic oscillators by applying PSO, MOL, and DE algorithms.https://www.mdpi.com/2227-7390/9/16/1938chaotic oscillatortime-stepone-step methodmulti-step methodparticle swarm optimization (PSO)many optimizing liaison (MOL) |
| spellingShingle | Martín Alejandro Valencia-Ponce Esteban Tlelo-Cuautle Luis Gerardo de la Fraga Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators Mathematics chaotic oscillator time-step one-step method multi-step method particle swarm optimization (PSO) many optimizing liaison (MOL) |
| title | Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators |
| title_full | Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators |
| title_fullStr | Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators |
| title_full_unstemmed | Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators |
| title_short | Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators |
| title_sort | estimating the highest time step in numerical methods to enhance the optimization of chaotic oscillators |
| topic | chaotic oscillator time-step one-step method multi-step method particle swarm optimization (PSO) many optimizing liaison (MOL) |
| url | https://www.mdpi.com/2227-7390/9/16/1938 |
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