Simulaciones numéricas de sistemas dinámicos caóticos oscilatorios que conservan su caos
The use of numerical simulations for studying the dynamic of chaotic systems has some associated shortcomings including chaos suppression for true chaotic systems, or the induction of chaos in otherwise non chaotic systems. In consequence, numerical simulations results may substantially depart from...
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| Tipo de recurso: | tesis de maestría |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2018 |
| País: | México |
| Institución: | Instituto Nacional de Astrofísica, Óptica y Electrónica |
| Repositorio: | Repositorio Institucional del INAOE |
| Idioma: | español |
| OAI Identifier: | oai:inaoe.repositorioinstitucional.mx:1009/1609 |
| Acceso en línea: | http://inaoe.repositorioinstitucional.mx/jspui/handle/1009/1609 |
| Access Level: | acceso abierto |
| Palabra clave: | info:eu-repo/classification/Inspec/Dynamic systems info:eu-repo/classification/Inspec/Chaos info:eu-repo/classification/Inspec/Suppression of chaos info:eu-repo/classification/Inspec/Fake chaos info:eu-repo/classification/Inspec/Numerical algorithm info:eu-repo/classification/Inspec/Gautschi method info:eu-repo/classification/cti/1 info:eu-repo/classification/cti/12 info:eu-repo/classification/cti/1203 info:eu-repo/classification/cti/120326 |
| Sumario: | The use of numerical simulations for studying the dynamic of chaotic systems has some associated shortcomings including chaos suppression for true chaotic systems, or the induction of chaos in otherwise non chaotic systems. In consequence, numerical simulations results may substantially depart from the true solution. Although many numerical simulations from Euler or Runge-Ku.a family employ numerical strategies to contain error, they often do not contemplate the special characteristics of chaotic systems. One of these particularities occurs when chaotic systems are oscillatory and hence, their solutions span over many frequencies. In this work, we propose the application of numerical methods based on trigonometric polynomials over traditional choices because they are able to capture oscillatory behaviours. We hypothesize that such choice shall result in the preservation of true chaotic behaviour during longer simulations. Specifically, we show that simulations of chaotic systems based on Gautschi method preserve the chaos for simulations of at least 50,000 time units, comparatively surpassing the performance of other traditional choices. Numerical simulation families considered here for comparison include Euler, Runge-Kutta, Adams-Bashforth and Adams-Moulton strategies. Three experiments were carried out. The first two experiments provided empirical evidence of chaos suppression in chaotic systems with the backward Euler method, and chaos generation in non-chaotic systems with forward Euler method, justifying the need of consider chaotic systems particularities. The third experiment showed that the Gautschi method satisfactorily simulates the test chaotic systems. During evaluation, numerical results were assessed qualitatively through their phase space and periodogram and quantitatively evaluated in terms of the Lyapunov exponents, the Kolmogorov-Sinai entropy and the number of evaluations performed by the numerical method at each step to the model representing the chaotic system as a proxy of computational cost. |
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