On the transfer of a number of concepts of statistical radiophysics to the theory of one-dimensional point mappings

A. M. Agalarov ; A. A. Potapov ; A. E. Rassadin ; A. V. Stepanov

Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 1, pp. 7-17.

Résumé

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.

Détail

Keywords: period doubling bifurcation, B-spline, partition of unity.
Mots-clés : discrete Fourier transform, Frobenius–Perron equation

Bibliographie

[1] Gonchenko S. V., Turaev D. V., “On three types of dynamics and the notion of attractor”, Proceedings of the Steklov Institute of Mathematics, 297:1 (2017), 116–137 | DOI | DOI | MR | MR

[2] Bogomolov S. A., Strelkova G. I., Scholl E., Anishchenko V. S., “Amplitude and phase chimeras in an ensemble of chaotic oscillators”, Technical Physics Letters, 42:7 (2016), 765–768 | DOI

[3] Dmitriev A. S., Efremova E. V., Maksimov N. A., Panas A. I., Generacija haosa, ed. Dmitriev A. S., Tehnosfera, M., 2012, 424 pp. (in Russian) | MR

[4] Potapov A. A., “Chaos Theory, Fractals and Scaling in the Radar: A Look from 2015”, The Foundations of Chaos Revisited: From Poincare to Recent Advancements, ed. Skiadas C., Springer Int. Publ., Switzerland, Basel, 2016, 195-218 | MR

[5] Kashchenko I. S., “Local dynamics of a second-order differential-difference equation with large delay at the first derivative”, Mathematical Notes, 101:1–2 (2017), 379–381 | DOI | DOI | MR

[6] Malahov A. N., Kumuljantnyj analiz sluchajnyh negaussovyh processov i ih preobrazovanij, Sov. radio, M., 1978, 376 pp. (in Russian) | MR

[7] Kul'bak S., Teorija verojatnosti i statistika, Nauka, M., 1967, 408 pp. (in Russian)

[8] Savchenko V. V., “Razlichenie sluchajnyh signalov v chastotnoj oblasti”, Radiotehnika i jelektronika, 42:4 (1997), 426–429 (in Russian)

[9] Gorjachkin O. V., Metody slepoj obrabotki signalov i ih prilozhenija v sistemah radiotehniki i svjazi, Radio i svjaz, M., 2003, 230 pp. (in Russian)

[10] Abdullaev G. O., Potapov A. A., Rabazanov A. K., Rassadin A. Je., “Novyj kriterij razlichenija periodicheskih i haoticheskih rezhimov v dinamicheskih sistemah (na primere modeli Rikitake)”, Materialy XII Mezhdunarodnoj konferencii «Fundamentalnye i prikladnye problemy matematiki i informatiki», priurochennoj k 85-letiju professora M.G. Alishaeva (Rossija, Mahachkala, 19–22 sentjabrja 2017 g.), Mahachkala, 2017, 8–10 (in Russian)

[11] Agalarov A. M., Gadzhimuradov T. A., Potapov A. A., Rassadin A. Je., “Ob evoljucii entropii Kulbaka–Lejblera v stohasticheskih dinamicheskih sistemah”, Aktualnye problemy fizicheskoj i funkcionalnoj elektroniki, materialy 20-j Vserossijskoj molodezhnoj nauchnoj shkoly-seminara (Rossija, Uljanovsk, 19–22 sentjabrja 2017 g.), UlGTU, Uljanovsk, 2017, 84–85 (in Russian)

[12] Junakovskij A. D., Nachala vychislitelnyh metodov dlja fizikov, IPF RAN, Nizhny Novgorod, 2007, 219 pp. (in Russian)

[13] Feigenbaum M. J., “Universal Behaviour in Nonlinear Systems”, Los Alamos Science, 1:1 (1980), 4–27 | MR

[14] Kuznetcov S. P., Dinamicheskij haos, Fizmatlit, 2001, 760 pp. (in Russian)

[15] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaja geometrija. Metody i prilozhenija, Nauka, M., 1979, 760 pp. (in Russian) | MR

[16] Smolencev N. K., Osnovy teorii vejvletov. Vejvlety v MATLAB, DMK Press, M., 2008, 356 pp. (in Russian)

[17] Zaslavskij G. M., Stohastichnost dinamicheskih sistem, Nauka, M., 1984, 272 pp. (in Russian) | MR

[18] Agalarov A. M., Potapov A. A., Rassadin A. Je., Stepanov A. V., “Scenarij perehoda k nereguljarnoj dinamike v generatorah haosa cherez udvoenie perioda i bispektry otobrazhenija Fejgenbauma”, Tezisy dokladov Mezhdunarodnoj nauchnoj konferencii «Novye tendencii v nelinejnoj dinamike» (Rossija, Jaroslavl, 5–7 oktjabrja 2017), JarGU, Jaroslavl, 2017, 11–12 (in Russian)