Abstract
The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.
Highlights
Improvement of Calculation Formula of the Extended Entropic Chaos Degree In Theorem 2, it is assumed that the values of L, M, and Si,j are equal to infinity
I have focused on improving the calculation formula of the extended entropic chaos degree (EECD) and applied the improved calculation formula of the EECD to two-dimensional typical chaotic maps
I have shown that the EECD is almost equal to the total sum of the Lyapunov exponent (LE) for their chaotic maps in many cases
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The Lyapunov exponent (LE) is a widely used measure for quantifying the chaos of a dynamical system It is generally incomputable for time series. The extended entropic chaos degree (EECD) was shown to be the sum of all the Lyapunov exponents of a multidimensional chaotic map under typical chaotic conditions [13]. It was assumed that the number of mapping points and the number of all components of equipartition of the domain are infinity In actual computations, these numbers are treated as finite numbers. I aim to formulate a calculation such that the EECD is equal to the sum of all the Lyapunov exponents of two-dimensional typical chaotic maps in actual numerical computations. I apply the improved calculation formula of the EECD to two-dimensional typical chaotic maps
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