ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Diudin M. S., Kalaidin E. N. Reconstruction of noisy system correlation dimension. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 2, pp. 201-207. DOI: 10.18500/0869-6632-2020-28-2-201-207

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Language: 
Russian
Article type: 
Article
UDC: 
530.182

Reconstruction of noisy system correlation dimension

Autors: 
Diudin M. S., Financial University under the Government of the Russian Federation
Kalaidin E. N., Krasnodar branch of the Financial University under the Government of the Russian Federation
Abstract: 

Purpose. Measurement of correlation dimension is considered in a dynamic system with additive random noise. To determine the correlation dimension correctly, it is necessary to eliminate the shift of horizontal coordinate in the graph of the correlation integral caused by the increase in distance between the points due to addition of random noise. Methods. To calculate the correlation dimension of a dynamic system, it is proposed to use the Grassberger–Procaccia algorithm and then change the calculation results according to the properties of the dynamics’ random component. When an additive normally distributed random noise is added, the distances between attractor points (calculated using the Euclidean norm) become random values distributed over a non-central χ-distribution that has a mathematical expectation greater than the distance before the noise was added. Results. Eliminating the shift of horizontal coordinate allows to get a flat section on the plot of the local slope of correlation integral, which coincides in vertical coordinate with the same section for a system without added noise. Local slope value for this section is close to fractal dimension of the systems under study. Conclusion. Proposed algorithm allows to accurately measure the correlation dimension of dynamic systems with additive (observable) random noise, to refine the estimate of the standard deviation of random noise obtained by other methods.

Reference: 

1. Grassberger P., Procaccia I. Characterization of strange attractors. Phys. Rev. Lett., 1983, vol. 50, p. 346.

2. Ben-Mizrachi A., Procaccia I., Grassberger P. Characterization of experimental (noisy) strange attractors. Phys. Rev. A Gen. Phys., 1984, vol. 29, no. 2, pp. 975–977.

3. Argyris J., Andreadis I., Pavlos G., and Athanasiou M. The influence of noise on the correlation dimension of chaotic attractors. Chaos, Solitons, and Fractals, 1998, vol. 9, no. 3, pp. 343–361.

4. Schreiber T. Determination of the noise level of chaotic time series. Phys. Rev. E, 1993, vol. 48, pp. 13–16.

Received: 
17.11.2019
Accepted: 
14.01.2020
Published: 
30.04.2020