Abstract
In the article, the possibility of using a bispectrum under the investigation of regular and chaotic behaviour of one-dimensional point mappings is discussed. The effectiveness of the transfer of this concept to nonlinear dynamics was demonstrated by an example of the Feigenbaum mapping. Also in the work, the application of the Kullback-Leibler entropy in the theory of point mappings is considered. It has been shown that this information-like value is able to describe the behaviour of statistical ensembles of one-dimensional mappings. In the framework of this theory some general properties of its behaviour were found out. Constructivity of the Kullback-Leibler entropy in the theory of point mappings was shown by means of its direct calculation for the ”saw tooth” mapping with linear initial probability density. Moreover, for this mapping the denumerable set of initial probability densities hitting into its stationary probability density after a finite number of steps was pointed out.
Highlights
The effectiveness of the transfer of this concept to nonlinear dynamics was demonstrated by an example of the Feigenbaum mapping
It has been shown that this information-like value is able to describe the behaviour of statistical ensembles
Constructivity of the Kullback-Leibler entropy in the theory of point mappings was shown by means of its direct calculation
Summary
При числе итераций N 1 отображения (4) тройная автокорреляционная функция динамической переменной xn может быть вычислена как. Ставшему всемирно известным после выхода работы [13], в которой на основе анализа поведения числовой последовательности (4) с функцией (7) при различных значениях параметра λ [0, 2] было показано, что сценарий перехода к хаосу через бесконечную последовательность бифуркаций удвоения периода универсален для широкого класса динамических систем [13], приведены на рис. 1 и 2 видно, что при значениях параметра λ < λc(= 1.4011...) модуль биспектра отображения Фейгенбаума имеет регулярный характер, соответствующий выходу изображающей точки отображения (7) на периодический режим после первой 4. Фазовая траектория отображения Фейгенбаума и модуль её биспектра при λ = 1.96 Fig. 4.
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