Computing Lyapunov dimension and applying it for geomagnetic indices prediction

Authors

DOI:

https://doi.org/10.20535/SRIT.2308-8893.2018.3.11

Keywords:

dynamical system, Lyapunov dimension, manifold, distribution, Lyapunov exponents, geomagnetic indices

Abstract

A method for computing the Lyapunov dimension from the realization of one variable of a dynamical system is proposed. The equality of the information dimension, the Lyapunov dimension, and capacity dimension is noted. The entropy of the distribution of the norms of the tangent vectors of a dynamical system and the Lyapunov dimension are considered together. Theoretical calculations are accompanied by an example of a numerical calculation of the Lyapunov dimension and the mentioned entropy for time series of geomagnetic Kp, Dst, and AE indices. In the considered indices, the entropy is close to the maximum value, and this leads to the closeness of the Lyapunov dimension to the capacity. A variable structure of the Dst index is noted. Using the example of geomagnetic indices, it is confirmed that the Grassberger-Procaccia correlation dimension is smaller than the Lyapunov dimension.

Author Biographies

Serhii M. Ivanov, Space Research Institute NASU-SSAU, Kyiv

Serhii Ivanov,

a Ph.D. student at the Space Research Institute NASU-SSAU, Kyiv, Ukraine.

Research interests: differential equations, manifolds, fractal dimension, Lyapunov stability, modeling.

Vitaliy O. Yatsenko, Space Research Institute NASU-SSAU, Kyiv

Vitaliy Yatsenko,

professor, Doctor of Technical Sciences, the Head of the Department of Remote Sensing and Advanced Instruments of the Space Research Institute NASU-SSAU, Kyiv, Ukraine.

Research interests: the dynamics of nonlinear systems of differential equations, Lyapunov stability, structural stability, topological and orbit-topological equivalence, the diffeomorphism of dynamical systems.

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Published

2018-10-16

Issue

No. 3 (2018)

Section

Methods of system analysis and control in conditions of risk and uncertainty

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