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

• Main Authors: Martín Alejandro Valencia-Ponce , Esteban Tlelo-Cuautle, Luis Gerardo de la Fraga
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Mathematics
• Subjects: chaotic oscillator; time-step; one-step method; multi-step method; particle swarm optimization (PSO); many optimizing liaison (MOL)
• Online Access: https://www.mdpi.com/2227-7390/9/16/1938

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.