A brief methodological note on chaos theory and its recent applications based on new computer resources

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Chaos theory refers to the behaviour of certain deterministic nonlinear dynamical systems whose solutions, although globally stable, are locally unstable. These chaotic systems describe aperiodic, irregular, apparently random and erratic trajectories, i.e., deterministic complex dynamics. One of the properties that derive from this local instability and that allow characterizing these deterministic chaotic systems is their high sensitivity to small changes in the initial conditions, which can be measured by using the so-called Lyapunov exponents. The detection of chaotic behaviour in the underlying generating process of a time series has important methodological implications. When chaotic behaviour is detected, then it can be concluded that the irregularity of the series is not necessarily random, but the result of some deterministic dynamic process. Then, even if such process is unknown, it will be possible to improve the predictability of the time series and even to control or stabilize the evolution of the time series. This article provides a summary of the main current concepts and methods for the detection of chaotic behaviour from time series.
La teoría del Caos se refiere al comportamiento que muestran ciertos sistemas dinámicos no lineales deterministas cuyas soluciones, aunque globalmente estables, resultan localmente inestables. Estos sistemas caóticos describen trayectorias aperiódicas, e irregulares, aparentemente aleatorias y erráticas, esto es, una dinámica compleja determinista. Una de las propiedades que se derivan de esa inestabilidad local y que permiten caracterizar a estos sistemas caóticos deterministas es su alta sensibilidad a los pequeños cambios en las condiciones iniciales, que puede medirse mediante el uso de los denominados exponentes de Lyapunov. La detección de comportamientos caóticos en el proceso subyacente generador de una serie temporal tiene importantes implicaciones metodológicas. Cuando se detecta comportamiento caótico, entonces se puede concluir que la irregularidad de la serie no es necesariamente aleatoria, sino el resultado de algún proceso dinámico determinista. Entonces, aunque dicho proceso sea desconocido, será posible mejorar las predicciones de la serie temporal e incluso controlar o estabilizar la evolución de dicha serie temporal. Este artículo proporciona un resumen de los principales conceptos y métodos actuales para la detección de comportamientos caóticos a partir de series temporales.
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