STATISTICAL METHODS FOR EVALUATING THE CONNECTIVITY OF DYNAMIC SYSTEMS BASED ON A SEPARATE TIME PROJECTION
Abstract
Based on the approaches of nonlinear dynamics to the estimation of invariants of a dynamical
system, the possibility of determining the degree of coupling of various dynamical systems is
considered. The dynamic coupling of the studied systems is understood as the number of common
components in the systems that determine the time evolution of the observed projections. The proposed
method has been tested on model dynamic systems and used to analyze the behavior of complex
dynamic systems observed in geophysics – apparent electrical resistance in two orthogonal
directions and relative vertical surface displacements. The data of long regime observations used
in the calculations in the seismically active region are interesting by the available facts of sensitivity
to the stress-strain state of the geophysical medium. Assuming a parameter of the state of the
medium as a common component of the observed dynamic processes of various nature, the number
of common components of the systems is estimated based on the proposed methodology. The paper
proposes a statistical method for finding individual samples of synchronous changes in variations
of the dynamic parameters of the observed number of geophysical fields. Assuming the unsteady
nature of the formation of a dynamic system in the presence of a large number of acting external
factors, it is relevant to determine the time intervals for synchronizing the properties of dynamic
systems when a dominant effect appears. The result of the application of the developed method is
the conclusion about synchronization of variations in the correlation dimension of volumetric
deformation at different time scales in the phase of the occurrence of strong seismic events.
References
Gayer L. Nelineynye effekty v khaoticheskikh i stokhasticheskikh sistemakh [Non-linear effects
in chaotic and stochastic systems]. Moscow.-Izhevsk: Institut komp'yuternykh
issledovaniy, 2003, 544 p.
2. Badii R., Broggi G., Derighetti B. et al. Dimension Increase in Filtered Chaotic Signals, Physical
Review Letters, 1988, Vol. 60 (11), pp. 979.
3. Kuznetsov S.P. Dinamicheskiy khaos [Dynamic Chaos]. Moscow: Fizmatlit, 2001, 295 p.
4. Cherepantsev A.S. Kharakteristiki dinamicheskoy sistemy geofizicheskikh poley na razlichnykh
vremennykh masshtabakh [Characteristics of a Dynamic System of Geophysical Fields on Different
Time Scales], Fizika Zemli [Phisica Zemli], 2018, No. 4, Appendix, pp. s3-s19.
5. Cherepantsev A.S. Dimension of the phase space of a dynamic system from geophysical data,
Izvestiya. Physics of the Solid Earth, 2007, Vol. 43, No. 12, pp. 1047-1055.
6. Keylis-Borok V.I., Kosobokov V.G., Mazhkenov S.A. O podobii v prostranstvennom
raspredelenii seysmichnosti [On similarity in the spatial distribution of seismicity],
Vychislitel'naya seysmologiya [Vichislitelnaya Seismologia], 1989, No. 22, pp. 28-40.
7. Cherepantsev A.S. Vydelenie proektsii dinamicheskoy sistemy po dannym nablyudeniy polya
deformatsiy [Determination of the Phase Projection of a Dynamic System from Strain Field
Observations], Fizika Zemli [Phisica Zemli], 2008, No. 2, pp. 39-58.
8. Cherepantsev A.S. Vydelenie dinamicheskoy sostavlyayushchey v variatsiyakh
geofizicheskikh poley na osnove skhodimosti vyborochnogo srednego [Extraction of a dynamic
component from variations in geophysical fields using the convergence of a sample average],
Fizika Zemli [Phisica Zemli], 2008a, No. 11, pp. 31-46.
9. Takens F. Detecting strange attractors in turbulence, Lect. Notes in Math., 1981, pp. 898.
10. Li Q., Nyland E., Is the Dynamics of the Lithosphere Chaotic?, Nonlinear Dynamics and Predictability
of Geophysical Phenomena, 1994. Geophysical Monograph 83, IUUG Vol. 18,
pp. 37-41.
11. Gibson J.F., Farmer J.D., Casdagli M., Eubank S. An analytic approach to practical state
space reconstruction, Physica D, 1992, Vol. 57, pp. 1-30.
12. Parker T., Chua L. Practical numerical algorithms for chaotic systems. New York: Springer-
Verlag, 1989, 348 p.
13. Grassberger P., Procaccia I., On the Characterization of Strange Attractors, Phys.Rev. Lett.,
1983, Vol. 50, pp. 346-349.
14. Tabor M. Khaos i integriruemost' v nelineynoy dinamike [Chaos and integrability in nonlinear
dynamics]. Moscow: Editorial URSS, 2001, 320 p.
15. Shuster G. Determinirovannyy khaos [Deterministic Chaos]. Moscow: Mir, 1988, 240 p.
16. Malinetskiy G.G., Potapov A.B. Sovremennye problemy nelineynoy dinamiki [Modern problems
of nonlinear dynamics]. Moscow: Editorial URSS, 2002, 360 p.
17. Smirnov V.B., Ponomarev A.V., Qian Jiadong, Cherepantsev A.S. Ritmy i determinirovannyy
khaos v geofizicheskikh vremennykh ryadakh [Rhythms and Determined Chaos in Geophysical
Time Series], Fizika Zemli [Phisica Zemli], 2005, No. 6, pp. 6-28.
18. Brillindzher D. Vremennye ryady. Obrabotka dannykh i teoriya [Time series: data analysis and
theory]. Moscow: Mir, 1980, 536 p.
19. Kendall M., Styuart A. T. 2. Statisticheskie vyvody i svyazi [Vol. 2. Statistical conclusions
and links]. Moscow: Nauka, 1973, 899 p.
20. Levin B.R. Teoreticheskie osnovy statisticheskoy radiotekhniki [Theoretical Foundations of
Statistical Radio Engineering]. Moscow: Radio i svyaz', 1989, 656 p.
