D.V. Terin - Ph.D. (Phys.-Math.), Associate Professor, Yuri Gagarin State Technical University of Saratov E-mail: terinden@mail.ru S.A. Korchagin - Post-graduate Student, Yuri Gagarin State Technical University of Saratov E-mail:korchaginser@gmail.com

A model constructed hierarchically layered composite structure which exhibits properties of similarity at different scales. For the pro-posed model is used to composite fractal analysis, which includes: evaluation of the permissible extent of the interval; fractal calculation capacity; Hausdorff dimension, Minkowski dimensional; Hurst exponent calculation. To quantify the complexity of fractal composite model is calculated fractal capacity. Since the developed model can be used to describe the physical properties of real objects, then we are dealing with an object that has the properties of similarity in a limited range of scales. On fractal superimposed region of existence - max and min sizes at which the fractal properties. Above this range limited size of the object. Below, as limiting the scope of the existence, in this case, a Van- der-Waals radius. It was found that the fractal model, studied in this work, in the case of Si-Al composite allowable scale range, at an initial rate of 1 cm layer, bounded above 12th level fractal. For self-similar sets, what is studied hierarchically built composite Hausdorff dimension can be defined explicitly. To estimate the number of speakers of the fractal dimension values we calculate the Hurst exponent. From the calculations of the fractal capacity, it follows that the minimum deviation from the regression line of the material 1 is the seventh level of fractal, the material 2 - 6th and 7th levels. Using the least-squares method allows you to calculate the expected value of the cell size dependency, covering the research object of their number to an arbitrary level fractal. The study found: 1. For the composite of fractal dimension Minkowski morphologies xx does not coincide with the Hausdorff dimension. 2. The first and second levels of fractal morphologies xx composites demonstrate the presence of the final dispersion of a random process, the lack of obvious trends in a number of speakers, the fractal properties of the object are mild. 3. The fourth level fractal shows the closest value to the Hurst exponent of natural processes. Thus, the probability of occurrence of such a fractal in nature is higher than with composites of these morphologies other levels fractal. 4. Hierarchically structured composite fractal level 3-12 has the property of persistence (the ongoing trend is supported).

 

1. Volchuk V.N. Opredelenie chuvstvitelnosti multifraktalnykh kharakteristik metalla // Vestnik Pridneprovskojj gosudarstvennojj akademii stroitelstva i arkhitektury. 2015. № 12(213). S. 10-14.
2. Kuznecov S.P. Dinamicheskijj khaos. M.: Izd-vo fiz.-mat. lit-ry. 2001. S. 80-82, 170-171.
3. West B.J., Bologna M., Grigolini P. Physics of Fractal Operators. N.Y.: Springer-Verlag. 2003. 354 p.
4. Mehaute A., Nigmatullin R.R., Nivanen L. Fleches du Temps et Geometrie Fractale. Paris: Hermes. 1998 (in French).
5. Korchagin S.A., Terin D.V. Issledovanie ehlektrodinamicheskikh svojjstv sloistogo kompozita fraktalnojj struktury // Trudy XIII Mezhdunar. nauchno-tekhnich. konf. «Aktualnye problemy ehlektronnogo priborostroenija (APEHP-2016). V 12 tomakh. 2016. S. 272-276.
6. Balankin A.S., Bory-Reyes J., Shapiro M. Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric // Physica A: Statistical Mechanics and its Applications. February 2016. V. 444. № 15. P. 345-359.
7. Dunne G.V. Heat kernels and zeta functions on fractals // J. Phys. A. 2012. V. 45. № 37. R. 374016. 22.
8. Reuter M., Saueressig F. Asymptotic Safety, Fractals, and Cosmology // Lect. Notes Phys. 2013. V. 863. R. 185-223.
9. Korchagin S.A., Klinaev JU.V., Terin D.V., Romanchuk S.P. Vychislitelnyjj ehksperiment s modeljami fraktalnykh nanokompozitnykh struktur // Vestnik Saratovskogo gosudarstvennogo tekhnicheskogo universiteta. 2015. T. 3. № 1(80). S. 33-40.
10. Potapov A.A. i dr. Fraktalnye modeli i metody na osnove skejjlinga v fundamentalnykh i prikladnykh problemakh sovremennojj fiziki // Sb. nauch. trudov «Neobratimye processy v prirode i tekhnike» / Pod red. V.S. Gorelika i A.N. Morozova. M.: MGTU im. N.EH. Baumana. 2008. Vyp. II. S. 5-107.
11. Benoît Mandelbrot How Long Is the Coast of Britain - Statistical Self-Similarity and Fractional Dimension // Science. New Series. May 5 1967. V. 156. № 3775. R. 636-638.
12. Bolshojj ehnciklopedicheskijj slovar / Gl. red. A.M. Prokhorov. Izd. 2-e, pererab. i dop. M.-SPb. 2000.
13. Starchenko N.V. Indeks fraktalnosti i lokalnyjj analiz khaoticheskikh vremennykh rjadov: Avtoref. diss. - k.f.-m.n. M.: MIFI. 2005.
14. Kuznecov P.V., Petrakova I.V., SHrajjber JU. Fraktalnaja razmernost kak kharakteristika ustalosti polikristallov metallov // Fizicheskaja mezomekhanika 7. Spec. vypusk CH.1. 2004. S. 389-392.
15. Hausdorff F. Dimesion und Ausseres Mass // Matematishe Annalen. 1919. № 79. R. 157-179.
16. KHasanov M.M., Bulgakova G.T. Nelinejjnye i neravnovesnye ehffekty v reologicheski slozhnykh sredakh. Moskva-Izhevsk: Institut kompjuternykh issledovanijj. 2003. S. 19.
17. SHuplecov JU.V., Ampilova N.B. Algoritm vychislenija razmernosti Minkovskogo dlja polutonovykh izobrazhenijj // Izvestija Rossijjskogo gosudarstvennogo pedagogicheskogo universiteta im. A.I. Gercena. 2014. № 165. S. 99-106.
18. Kronover R.M. Fraktaly i khaos v dinamicheskikh sistemakh M.: POSTMARKET. 2000.
19. Feder E. Fraktaly. M.: Mir. 1991.
20. Andreev S.D., Ivlev L.S. Vremennaja i prostranstvennaja izmenchivost polejj opticheskikh i aehrozolnykh kharakteristik v atmosfere. CH. I. Opticheskie kharakteristiki atmosfery // Optika atmosfery i okeana. 1997. T. 10. № 12. S. 1440-1449.
21. Feller W. The asymptotic distribution ofthe range ofs ums of independent variables // Ann. Math. Statist. 1951. V. 22. P. 427-432.
22. Kalush JU.A., Loginov V.M. Pokazatel KHjorsta i ego skrytye svojjstva // Sibirskijj zhurnal industrialnojj matematiki. T. 5. Vyp. 4. 2002. S. 29-37.
23. Klikushin JU.N. Metod fraktalnojj klassifikacii slozhnykh signalov // ZHurnal radioehlektroniki. T. 4. 2000. S. 52-54.