#### Cite this article as:

Vdovina G. M. A brief review of the research results of new methods for generating, transmitting and receiving oscillations and waves based on fractal geometry methods. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 1, pp. 8-28. DOI: https://doi.org/10.18500/0869-6632-2020-28-1-8-28

# A brief review of the research results of new methods for generating, transmitting and receiving oscillations and waves based on fractal geometry methods

Purpose of this article is to generalize results of the application of fractal geometry methods in various radiophysic systems and at study of processes occurring in them. Methods. The presentation is built in the form of a brief review of a number of works devoted to new methods for generating, receiving and transmitting signals of various frequency ranges, including microwave frequencies, using fractal geometry approaches. At the same time, it was advisable to give examples of constructing such classical fractals as the Peano curve, Sierpinski triangle and Sierpinski carpet, Koch curve, etc., and indicate the Hausdorff dimension for them. The idea of constructing these fractal curves and objects with some modifications underlies the creation of real physical systems. Results.The review showed that fractal objects are actively used in the design of fractal antennas, fractal resonators and filters built on their basis. Taking into account the fractal surface of the cathode also gives certain advantages and explains some experimental results. Some other fields of application of fractals are indicated, where the complexity of spatial or temporal structures at different scales plays an important role. It should be noted that the artificially created elements in question are self-similar only to a certain extent, representing the first few iterations of constructing fractal curves. In this regard, they are called as quasi-fractal or prefractal objects. Conclusion. Formation of fractal thinking and the fractal view of the world as a whole made it possible to use the principles of self-similarity in the analysis of work and the design of devices. The obvious advantages were a possible reduction in size, the expansion of the frequency range of the device, etc

1. Gulia G. Memoire sur l’Iteration des Fonctions Rationnelles. Journal de Mathematiques Pures et Appliquees. 1918, vol. 1, pp. 47–245.

2. Fatou P. Sur les Equations Fonctionnelles. Bulletin Societe. Math. France. 1919, vol. 47, pp. 161–271.

3. Mandelbrot B. Fractals and the Rebirth of Iteration Theory, pp. 151–160. In: Peitgen H.-O., Richter P.H. The Beauty of Fractals. Berlin, Springer. 1986. 199 p.

4. Crownover R.M. Introduction to Fractals and Chaos. Boston–London, Jones and Bartlett Publishers, 1995. 306 p.

5. Mandelbrot B. The Fractal Geometry of Nature. New York: W.H. Freeman and Company, 1983. 468 p.

6. Demenok S.L. Superfractal. Saint Petersburg: Strata, 2015, 196 p. (in Russian).

7. Art-fractal. Collections of articles. Transl. from eng., fr. by E.B. Nikolaeva. Saint Petersburg: Strata, 2015, 156 p. (in Russian).

8. Rozenberg G.S., Chuprunov E.V., Gelashvili D.B., Iudin D.I. Nature’s geometry has a fractal face. Vestnik of Lobachevsky University of Nizhni Novgorod, 2011, № 1, pp. 411–417 (in Russian).

9. Pineiro G. The Sphere that Wanted to be Infinite: The Paradoxes of Measurement. Barcelona, RBA Coleccionables, 2017, 141 p.

10. Makarenko N.G. Fractals, attractors, neural networks and all that. Scientific session MEPHI – 2002. The 4th Russian Scientific Technical Conference «Neuroinformatics – 2002». Lectures on Neuroinformatics. Part 2. Moscow, MEPHI, 2002, pp. 121–169 (in Russian).

11. Valdivia J.A. The Physics of High Altitude Lightning, Ph.D. Dissertation, University of Maryland, 1998.

12. Duran A.J. The Poetry of Numbers. Revealing Beauty in Maths. Navarra, RBA Coleccionables, 2010, 151 p.

13. Trubetskov D.I. Introduction to Synergetics: Chaos and Structures. Moscow, URSS LIBROKOM, 2014, 240 p. (in Russian).

14. Trubetskov D.I., Trubetskova E.G. Fractal geometry. Izvestiya VUZ. Applied Nonlinear Dynamics, 2016, vol. 24, iss. 6, pp. 4–38 (in Russian).

15. Morozov A.D. Introduction to the Fractal Theory. Nizhny Novgorod: Nizhny Novgorod University, 1999, 140 p. (in Russian).

16. Danilov Iu.A. Fractality. In: The Wonderful World of Science. To the Memory of Iu.A. Danilov. Eds V.I. Saniuk, D.I. Trubetskov. Moscow, Progress-Traditsiia, 2008, 384 p. P. 180–191 (in Russian).

17. Peano G. Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen, 1890, vol. 36, no. 1, pp. 157–160.

18. Torondzhadze M., Bendukidze A. Peano Curves. Kvant,1974, no. 8, pp. 13–16 (in Russian).

19. Hilbert D. Uber die stetige Abbildung einer Linie auf ein Fl ¨ achenst ¨ uck. ¨ Mathematische Annalen, 1891, vol. 38, no. 3, pp. 459–460.

20. Moore E.H. On certain crinkly curves. Transactions American Mathematical Society, 1900, no. 1, pp. 72–90.

21. Koch H. Sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire. Arkiv for Matematik, 1904, no. 1, pp. 681–704.

22. Sierpinski W. Sur une nouvelle courbe continue qui remplit toute une aire plane. Bull. Acad. Sci. de Cracovie (Sci. Math. et Nat., Serie A), 1912, pp. 462–478.

23. Schroeder M. Fractals, Chaos, Power Laws. New York: W.H. Freeman and Company, 1991, 429 p.

24. Slusar V. Fractal antennas. A fundamentally new type of «broken» antennas. Electronics: Science, Technology, Business, 2007, no. 5, pp. 78–83 (in Russian).

25. Kim Y., Jaggard D.L. The fractal random array. Proc. of the IEEE, 1986, vol. 74, no. 9, pp. 1278–1280.

26. Puente-Baliarda C. Fractal Design of Multiband Antenna Arrays. Elec. Eng. Dept. Univ. Illinois, Urbana-Champaign, ECE 477 term project, Dec. 1993.

27. Puente C., Pous R. Diseсo Fractal de Agrupaciones de Antenas. IX Simposium Nacional URSI, Las Palmas, 1994, vol. I, pp. 227–231.

28. Yang X., Chiochetti J., Papadopoulos D., Susman L. Fractal antenna elements and arrays. Applied Microwave & Wireless, 1999, pp. 34–46.

29. Cohen N. Fractal Antennas: Part 1. Communications Quarterly, Summer 1995, pp. 7–22.

30. Vinoy K.J. Fractal Shaped Antenna Elements for Wideand Multi-band Wireless Applications. Thesis of PhD Dissertation. The Pennsylvania State University, 2002, 169 р.

31. Vinoy K.J., Abraham J.K., Varadan, V.K. Reply to comments on «On the Relationship between Fractal Dimension and the Performance of Multi-Resonant Dipole Antennas using Koch Curves». IEEE Transactions on Antennas and Propagation, 2004, vol. 52, iss. 6, pp. 1627–1628.

32. Best S.R. The Koch fractal monopole antenna: The significance of fractal geometry in determining antenna performance. Proceedings of the 2001 Аntenna Аpplications Symposium. Allerton Park Monticello, Illinois, 2001.

33. Gonzalez-Arbesu J.M., Rius J.M., Romeu J. Comments on «On the Relationship between Fractal Dimension and the Performance of Multi-Resonant Dipole Antennas using Koch Curves». IEEE Transactions on Antennas and Propagation, 2004, vol. 52, iss. 6, pp. 1626–1627.

34. Tsachtsiris G., Soras C., Karaboikis M., Makios V. A printed folded Koch monopole antenna for wireless devices. Microwave and Optical Technology Letters, 2004, vol. 40, no. 5, pp. 374–378.

35. Puente-Baliarda C., Rozan E., Jean L., Anguera Pros J. WO Patent № 01/54225 A1. International Patent Classification7 H01Q 1/36. Space-Filling Miniature Antennas. July 26, 2001.

36. Karaboikis M., Soras C., Tsachtsiris G., Makios V. Four-element printed monopole antenna systems for diversity and MIMO terminal devices. Proceedings of the 17th International Conference on Applied Electromagnetics and Communications, Dubrovnik, Oct. 1–3, 2003, pp. 193–196.

37. Karaboikis M., Soras C., Tsachtsiris G., Makios V. Three-branch antenna diversity systems on wireless devices using various printed monopoles. 2003 IEEE International Symposium on Electromagnetic Compatibility, Istanbul, May 11–16, 2003.

38. Tsachtsiris G., Karaboikis M., Soras C., Makios V. A novel fractal rectangular curve printed monopole antenna for portable terminals. URSI International Symposium on Electromagnetic Theory, Pisa, Italy, May 23–27, 2004.

39. Karaboikis M., Soras C., Tsachtsiris G., Papamichael V., Makios V. Multi element antenna systems for diversity and MIMO terminal devices. Laboratory of Electromagnetics. Department of Electrical and Computer Engineering. University of Patras. Patras, Greece, 2004.

40. Slusar V. Fractal antennas. A fundamentally new type of «broken antennas». Part 2. Electronics: Science, Technology, Business, 2007, no. 6, pp. 82–89. (in Russian).

41. Gonzalez-Arbesu J.M., Blanch S., Romeu J. The Hilbert curve as a small self-resonant monopole from a practical point of view. Microwave and Optical Technology Letters, 2003, vol. 39, no. 1, pp. 45–49.

42. Zhu J., Hoorfar A., Engheta N. Bandwidth, cross-polarization, and feed-point characteristics of matched Hilbert antennas. IEEE Antennas and Wireless Propagation Letters, 2003, vol. 2, pp. 2–5.

43. Anguera J., Puente C., Martinez E., Rozan E. The fractal Hilbert monopole: A two-dimensional wire. Microwave and Optical Technology Letters, 2003, vol. 36, no. 2, pp. 102–104.

44. Soler C.J., Anguera Pros J., Puente-Baliarda C., Borja Borau C. WO Patent № 04/042868. International Patent Classification7 H01Q 1/38. Integrated Circuit Package Including Miniature Antenna. 21.05.2004.

45. Lindenmayer systemer. URL: https://allenpike.com/modeling-plants-with-l-systems/ (accessed 19 December 2019).

46. Adaptive Mesh Refinement. Part I. Lecture slides. University of Illinois at Urbana-Champaign. Astronomy 496CAC. Computational Astrophysics and Cosmology, Spring 2003.

47. Slusar V. Fractal antennas. Vishnevskii V.M., Liakhov A.I., Portnoi S.L., Shakhnovich I.V. Broadband Wireless Networks. Moscow, Tekhnosfera, 2005, 592 p. P. 529–542 (in Russian).

48. Sagan H. Space-Filling Curves. Springer-Verlag: New York, 1994. 193 p.

49. Fractal antenna experiment. URL: http://www.m0wwa.co.uk/page/M0WWA_fractal_antenna.html (accessed 19 December 2019).

50. Fractals and GUI . URL: https://habr.com/ru/company/intel/blog/92064/ (accessed 19 December 2019).

51. Potapov A.A. Fractals in Radiophysics and Radar: Sampling Topology. Moscow, Universitetskaia Kniga, 2005, 848 p. (in Russian).

52. Gulyaev Yu. V., Nikitov S.A., Potapov A.A., Davydov A.G. Design of fractal radio systems: Numerical analysis of electromagnetic properties of the Sierpinski fractal antenna. Journal of Communications Technology and Electronics, 2005, vol. 50, no. 9, pp. 988–993.

53. Nesterov D.A., Tsarev V.A. Novel quasi-fractal double-gap multi-beam klystron cavity class. Radioengineering, 2016, no. 7, pp. 87–91 (in Russian).

54. Nesterov D.A., Tsarev V.A. Muchkaev V.Yu. Modeling the interaction of multi-beam electron stream with the microwave field in the output quasi-fractal double-gap resonator of klystron. Radioengineering, 2017, no. 7, pp. 31–36. (in Russian).

55. Miroshnichenko A.Yu., Tsarev V.A., Korchagin A.I. Double-gap fractal type cavities. Antennas, 2011, no. 11 (174), pp. 63–67 (in Russian).

56. Tsarev V.A., Korchagin A.I., Miroshnichenko A.Yu. Investigation of the two-mode interacting fields fractal two-gaps cavity with electrons in the multiple-beam klystrode. Journal of Radio Electronics, 2012, no. 12, pp. 9–12 (in Russian).

57. Gelvich E.A., Borisov L.M., Pugnin V.l. et al. A new generation of power klystrons on the base of multiple-beam design. Int. Microwave Symp. Dig. USA, 1991, vol. 3, pp. 1319–1320.

58. Nesterov D.A., Tsarev V.A. Optimization of parameters dual mode quasi-fractal double-gap cavities for high-power multi-beam klystrons, operating at a frequency of 2.45 GHz. Proceedings of International Conference on Actual Problems of Electron Devices Engineering, 2016, Saratov, September 22–23, 2016, vol. 1, pp. 349–358.

59. Nesterov D.A., Tsarev V.A. Prospects for the use of double-gap quasifractal resonators in highpower multipath klystrons with extremely high efficiency Proceedings of the III Russian Scientific and Technical Conference «Problems of Microwave Electronics V.A. Solntseva 2017», Moscow, 2017, pp. 7–9 (in Russian).

60. Tikhoplavov V.Iu., Tikhoplavov T.S. Harmony of Chaos, or Fractal Reality. Saint Petersburg, Ves, 2003, 340 p. (in Russian).

61. Patent. URL: findpatent.ru/patent/269/2690693.html (accessed 19 December 2019) (in Russian).

62. Ye C.S., Su Y.K., Weng M.H., Wu H.W. Resonant properties of the Sierpinski-based fractal resonator and its application on low-loss miniaturized dual mode bandpass filter. Microwave Opt. Tech. Letters, 2009, vol. 51, no. 5, pp. 1358–1361.

63. Jarry P., Beneat J. Design and Realizations of Miniaturized Fractal Microwave and RF Filters. Hoboken: J. Wiley & Sons: IEEE Press, 2009. 194 p.

64. Zemlyakov К., Crnojevic-Bengin V. Planar low-pass filters based on Hilbert fractal. Microwave and Optical Technology Letters, 2012, vol. 54, iss. 11, pp. 2577–2581.

65. Crnojevic-Bengin V., Zemlyakov К., Jankovic N., Vendik I. Dual-band bandpass filters based on dual-mode Hilbert fractal resonator. Microwave and Optical Technology Letters, 2013, vol. 55, iss. 7, pp. 1440–1443.

66. Zemliakov K.N. Research and Development of Microwave Filters on Multimode Resonators. Ph.D. Dissertation, Saint Petersburg, 2013 (in Russian).

67. Solntsev V.A Nonlinear phenomena in vacuum microelectronic structures. Izvestiya VUZ. Applied Nonlinear Dynamics, 1998, vol. 6, no. 1, p. 54 (in Russian).

68. Trubetskov D.I., Krasnova G.M. About current state high frequency vacuum electronic and microelectronic devices with field emission. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, no. 1, pp. 35–66 (in Russian).

69. Solntsev V.A., Galdetskii A.V., Kleev A.I. Vacuum microwave microelectronics with an average angle of flight. Lectures on Microwave Electronics and Radiophysics. 10 School-seminar. Vol. 1. Saratov, «Kolledzh», 1996, p. 76 (in Russian).

70. Trubetskov D.I., Rozhnev A.G., Sokolov D.V. Lectures on microelectronics. Saratov: «Kolledzh», 1996 (in Russian).

71. Solntsev V.A. The electric field gain in the cathode with fractal multistep surface. 10th Int. Vacuum Microelectronics Conf. Kyongju, Korea, August 17–21, 1997, pp. 730.

72. Gulyaev Yu.V., Sinitsyn N.I., Torgashov G.V. and et al. Emission of low-voltage multi-tip carbon matrices coated by carbon clusters. 9th Int. Vacuum Microelectronics Conf. St. Petersberg, Russia, July 7–12, 1996, pp. 519.

73. Isaeva O.B., Eliseev M.V., Rozhnev A.G., Ryskin N.M. Simulation of field emission from fractal surface. Izvestiya VUZ. Applied Nonlinear Dynamics, 1999, vol. 7, no. 5, pp. 33–43 (in Russian).

74. D’iudni A.K. The Mandelbrot set and its related Julia sets. In the World of Science, 1988, no. 1, pp. 88–92 (in Russian).

75. Iijima S. Helical microtubules of graphitic carbon. Nature, 1991, vol. 354, pp. 56–58.

76. Gulyaev Yu.V., Grigoriev Yu.A., Korol V.N., Rehen G.A. Research of the field emission of fractal carbon structures. Izvestiya VUZ. Applied Nonlinear Dynamics, 2005, vol. 13, no. 1–2, pp. 88–99 (in Russian).

77. Mishchenko S.V., Tkachev A.G. Carbon Nanomaterials. Production, Properties, Application. Moscow, Mashinostroenie, 2008, 320 p. (in Russian).

78. D’iachkov P.N. Carbon Nanotubes: Structure, Properties, Applications. Moscow, BINOM. Laboratoriia znanii, 2006 (in Russian).

79. Lazorenko O.V., Chernogor L.F. Fractal ultra-wideband signals. Radio Physics and Radio Astronomy, 2005, vol. 10, no. 1, pp. 62–84.

80. Bolotov V.N., Tkach Y.V. Fractal Communication system. Technical Physics. The Russian Journal of Applied Physics, 2008, vol. 53, no. 9, pp. 1192–1196.

81. Shashkina A.S., Krivosheikin A.V., Skvortsov N.N., Vorotkov M.V. Fractal properties of LED avalanche breakdown. Nauchno-tekhnicheskie vedomosti SPbGPU. Fiziko-matematicheskie nauki, 2016. no. 4 (253), pp. 85–93 (in Russian).

82. Fractal diffuser URL: https://www.cold-ray.ru/catalog?currentProduct=25838 (accessed 19 December 2019) (in Russian)

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