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

For citation:

Gonchenko S. V., Gonchenko A. S., Kazakov A. O., Самылина Е. А. Mixed dynamics: elements of theory and examples. Izvestiya VUZ. Applied Nonlinear Dynamics, 2024, vol. 32, iss. 6, pp. 722-765. DOI: 10.18500/0869-6632-003138, EDN: PFFDTK

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon and_2024-6_gonchenko_et-al_722-765.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
Article type: 
Article
UDC: 
517.925 + 517.93
DOI: 
10.18500/0869-6632-003138
EDN: 
PFFDTK

Mixed dynamics: elements of theory and examples

Autors: 
Gonchenko Sergey Vladimirovich, Lobachevsky State University of Nizhny Novgorod
Gonchenko Aleksandr Sergeevich, Lobachevsky State University of Nizhny Novgorod
Kazakov Alexey Olegovich, National Research University "Higher School of Economics"
Самылина Евгения Александровна, National Research University "Higher School of Economics"
Abstract: 

The main goal of the paper is to present recent results obtained in the mathematical theory of dynamical chaos and related to the discovery of its new, third, form, the so-called mixed dynamics. This type of chaos is very different from its two classical forms, conservative and dissipative chaos, and its main difference is that attractors and repellers can intersect without coinciding.

The main results of the paper are related to construction of theoretical schemes aimed to mathematical justification of this phenomenon using the most general methods of topological dynamics. The paper also provides a number of examples of systems from applications in which mixed dynamics is observed. It is shown that such dynamics can be of different types, from close to conservative to strongly dissipative, and also that it can arise as a result of various bifurcation mechanisms.

Key words: 
dynamical chaos
mixed dynamics
CRH-attractor
full attractor
absolute Newhouse domain
Acknowledgments: 
The work was supported by the Russian Science Foundation grant No. 24-11-00339 (section 1), the Russian Science Foundation grant No. 23-71-30008 (section 2). Research in section 3.1 was carried out within the framework of the grant FSWR-2020-0036 of the Ministry of Science and Higher Education of the Russian Federation. The studies in sections 3.2 and 3.3 were carried out with the financial support of the fundamental research program of the National Research University Higher School of Economics.
Reference: 
  1. Anosov DV, Bronshtein IU. Smooth dynamical systems. Dynamical systems – 1, Itogi Nauki i
    Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Moscow: VINITI; 1985. P. 204–229.
  2. Dynamical systems – 9, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Moscow:
    VINITI; 1991. 248 p.
  3. Gonchenko S. V. Reversible mixed dynamics: A concept and examples // Discontinuity, Nonlinearity, and Complexity. 2016. Vol. 5, no. 4. P. 365–374. DOI: 10.5890/DNC.2016.12.003.
  4. Gonchenko SV, Turaev DV. On three types of dynamics and concept of attractor. Proc. Steklov
    Math. Inst. 2017;297:116–137. DOI: 10.1134/S0081543817040071.
  5. Gonchenko SV, Turaev DV, Shilnikov LP. On Newhouse regions for two-dimensional diffeomorphisms close to diffeomorphism with non-rough heteroclinic contour. Proc. Steklov Math. Inst.
    1997;216:76–125.
  6. Turaev D. V. Richness of chaos in the absolute Newhouse domain // Proceedings of the International Congress of Mathematicians, 2010. P. 1804–1815. DOI: 10.1142/9789814324359_0122.
  7. Turaev D. V. Maps close to identity and universal maps in the Newhouse domain // Communications in Mathematical Physics. 2015. Vol. 335, no. 3. P. 1235–1277. DOI: 10.1007/s00220-015-2338-4.
  8. Gonchenko S. V., Shilnikov L. P., Turaev D. V. Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps // Nonlinearity. 2007. Vol. 20, no. 2. P. 241–275. DOI: 10.1088/0951-7715/20/2/002.
  9. Turaev D. V. On dimension of non-local bifurcational problems // Bifurcation and Chaos. 1996. Vol. 6, no. 5. P. 919–948. DOI: 10.1142/S0218127496000515.
  10. Ruelle D. Small random perturbations of dynamical systems and the definition of attractors // Comm. Math. Phys. 1981. Vol. 82. P. 137–151. DOI: 10.1007/BF01206949.
  11. Newhouse S. Diffeomorphisms with infinitely many sinks // Topology. 1974. Vol. 13. P. 19–18.
  12. Afraimovich V. S., Shilnikov L. P. Strange attractors and quaiattractors // In.: G. I. Barenblatt, G. Iooss, D. D. Joseph (eds) Nonlinear Dynamics and Turbulence. Boston: Pitmen, 1983. P. 1–34.
  13. Tedeschini-Lalli L., Yorke J. A. How often do simple dynamical processes have infinitely many coexisting sinks? // Commun. Math. Phys. 1986. Vol. 106. P. 635–657. DOI: 10.1007/BF01463400.
  14. Lai Y. C., Grebogi C., Yorke J. A., Kan I. How often are chaotic saddles nonhyperbolic // Nonlinearity. 1993. Vol. 6, no. 5. P. 779–797.
  15. Pikovsky A., Topaj D. Reversibility vs. synchronization in oscillator latties // Physica D. 2002. Vol. 170. P. 118–130.
  16. Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Turaev D. V. On the phenomenon of mixed dynamics in Pikovsky-Topaj system of coupled rotator // Physica D: Nonlinear Phenomena. 2017. Vol. 350. P. 45–57. DOI: 10.1016/j.physd.2017.02.002.
  17. Gonchenko A. S., Gonchenko S. V., Kazakov A. O. Richness of chaotic dynamics in nonholonomic models of a Celtic stone // Regular and Chaotic Dynamics. 2013. Vol. 18, no. 5. P. 521–538. DOI: 10.1134/S1560354713050055.
  18. Delshams A., Gonchenko S. V., Gonchenko V. S., Lazaro J. T., Sten’kin O. Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps // Nonlinearity. 2013. Vol. 26. P. 1–33.
  19. Gonchenko SV, Gonchenko MS, Sinitsky IO. On mixed dynamics of two-dimensional reversible
    diffeomorphisms with symmetric non-transversal heteroclinic cycles. Izv. RAN. Ser. Mat. 2020;
    84(1):23–51. DOI: 10.1070/IM8786.
  20. Kazakov A. O. Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane // Regular and Chaotic Dynamics. 2013. Vol. 18, no. 5. P. 508–520. DOI: 10.1134/S1560354713050043.
  21. Kuznetsov S. P. Regular and chaotic motions of the Chaplygin sleigh with periodically switched location of nonholonomic constraint // EPL (Europhysics Letters). 2017. Vol. 118, no. 1. P. 10007. DOI: 10.1209/0295-5075/118/10007.
  22. Gonchenko AS, Gonchenko SV, Kazakov AO, Samylina EA. Chaotic dynamics and multistability
    in nonholonomic model of Celtic stone. Izvestia of universities. Radiophysics. 2018;61(10):
    867-882.
  23. Kazakov A. O. On the appearance of mixed dynamics as a result of collision of strange attractors and repellers in reversible systems // Radiophysics and Quantum Electronics. 2019. Vol. 61, no. 8–9. P. 650–658. DOI: 10.1007/s11141-019-09925-6.
  24. Kazakov A. O. Merger of a Henon-like attractor with a Henon-like repeller in a model of vortex dynamics // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2020. Vol. 30, no. 1. P. 011105. DOI: /10.1063/1.5144144.
  25. Emelianova A. A., Nekorkin V. I., On the intersection of a chaotic attractor and a chaotic repeller in the system of two adaptively coupled phase oscillators // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019. Vol. 29, no. 11. P. 111102. DOI: 10.1063/1.5130994.
  26. Emelianova A. A., Nekorkin V. I., The third type of chaos in a system of two adaptively coupled phase oscillators // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2020. Vol. 30, no. 5. P. 051105. DOI: 10.1063/5.0009525.
  27. Ariel G., Schiff J. Conservative, dissipative and super-diffusive behavior of a particle propelled in a regular flow // Physica D: Nonlinear Phenomena. 2020. Vol. 411. P. 132584. DOI: 10.1016/j.physd.2020.132584.
  28. Chigarev V., Kazakov A., Pikovsky A. Kantorovich-Rubinstein-Wasserstein distance between overlapping attractor and repeller // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2020. Vol. 30, no. 7. P. 073114. DOI: 10.1063/5.0007230.
  29. Gonchenko AS, Gonchenko SV, Kazakov AO. Three types of attractors and mixed dynamics
    of nonholonomic models of rigid body motion. Proc. Steklov Inst. Math. 2020;308:125–140.
    DOI: 10.1134/S0081543820010101.
  30. Gonchenko SV. Three Forms of Dynamical Chaos. Radiophys Quantum El. 2021;63:756–775.
  31. Gonchenko SV, Gonchenko AS, Morozov KE. The third type of dynamics and Poincare homoclinic ´
    trajectories. Radiophysics and Quantum Electronics. 2024;66(9):693–719. DOI: 10.1007/s11141-
    024-10329-4.
  32. Conley C. C. Isolated Invariant Sets and the Morse Index. Regional conference series in mathematics, vol. 38. American Mathematical Soc, 1978. 89 p. DOI: 10.1090/cbms/038.
  33. Hurley M. Attractors: persistency, and density of their basins // Trans. Amer. Math. Soc. 1982. Vol. 269, no. 1. P. 247–271. DOI: 10.1090/S0002-9947-1982-0637037-7.
  34. Galias Z., Tucker W. Is the Henon attractor chaotic? // Chaos. 2015. Vol. 25, iss. 3. 033102. ´ DOI: 10.1063/1.4913945.
  35. Tucker W. A rigorous ode solver and Smale’s 14th problem // Foundations of Computational Mathematics. 2002. Vol. 2. P. 53–117. DOI: 10.1007/s002080010018.
  36. Delshams A., Gonchenko M., Gonchenko S. V., Lazaro J. T. Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies // Discrete Continuous Dyn. Sys. A. 2018. Vol. 38, no. 9. P. 4483–4507. DOI: 10.3934/dcds.2018196.
  37. Turaev DV, Shilnikov LP. An example of a wild strange attractor. Math. Sb. 1998;189(2):291–314.
    DOI: 10.1070/sm1998v189n02ABEH000300.
  38. Gonchenko S. V., Kazakov A. O., Turaev D. Wild pseudohyperbolic attractors in a four-dimensional Lorenz system // Nonlinearity. 2021. Vol. 34, no. 4. P. 2018–2047. DOI: 10.1088/1361-6544/abc794.
  39. Gavrilov NK, Shilnikov LP. On three-dimensional dynamical systems close to systems with a
    structurally unstable homoclinic curve. Part 1. Math.USSR Sb. 1972;17:467–485 DOI: 10.1070/
    SM1972v017n04ABEH001597.
  40. Gavrilov NK, Shilnikov LP. On three-dimensional dynamical systems close to systems with a
    structurally unstable homoclinic curve. Part 2. Math.USSR Sb. 1973;19(1):139–156 DOI: 10.1070/
    SM1973v019n01ABEH001741.
  41. Kuznetsov SP. Dynamical chaos and hyperbolic attractors. From mathematics to physics. MoscowIzhevsk; 2013. 488 p.
  42. Tucker W. The Lorenz attractor exists // Comptes Rendus de l’Academie des Sciences-Series I-Mathematics. 1999. Vol. 328, № 12. P. 1197–1202. DOI: 10.1016/S0764-4442(99)80439-X.
  43. Afraimovich VS, Bykov VV, Shilnikov LP. The origin and structure of the Lorenz attractor. Sov.
    Phys. Dokl. 1977;22:253–255.
  44. Afraimovich VS, Bykov VV, Shilnikov LP. On attracting structurally unstable limit sets of Lorenz
    attractor type. Trans. Mosc. Math. Soc. 1982;44:153–216.
  45. Lorenz E. Deterministic nonperiodic flow // Journal of the Atmospheric Sciences. 1963. Vol. 20, no. 2. P. 130–141. DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.C.
  46. Gonchenko S., Ovsyannikov I., Simo C., Turaev D. Three-dimensional Henon-like maps and wild Lorenz-like attractors // Int. J. of Bifurcation and chaos. 2005. Vol. 15, no. 11. P. 3493–3508. DOI: 10.1142/S0218127405014180.
  47. Gonchenko S. V., Gonchenko A. S., Ovsyannikov I. I., Turaev D. V. Examples of Lorenz-like attractors in Henon-like maps // Math. Model. Nat. Phen. 2013. Vol. 8, no. 5. P. 48–70.
  48. Gonchenko S., Gonchenko A., Kazakov A., Samylina E. On discrete Lorenz-like attractors // Chaos. 2021. Vol. 31, iss. 2. P. 023117. DOI: 10.1063/5.0037621.
  49. Gonchenko AS, Gonchenko SV, Shilnikov LP. Towards scenarios for emergence of chaos in
    three-dimensional maps. Rus. Nonlinear Dynamics. 2012;8(1):3–28.
  50. Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Turaev D. Simple scenarios of onset of chaos in three-dimensional maps // Int. J. Bif. and Chaos. 2014. Vol. 24, no. 8. 25 p. DOI: 10.1142/S0218127414400057.
  51. Gonchenko S., Karatetskaia E., Kazakov A., Kruglov V. Conjoined Lorenz twins — a new pseudohyperbolic attractor in three-dimensional maps and flows // Chaos. 2022. Vol. 32. P. 121107. DOI: 10.1063/5.0123426.
  52. Borisov A. V., Kazakov A. O., Sataev I. R. The reversal and chaotic attractor in the nonholonomic model of Chaplygin’s top // Regular and Chaotic Dynamics. 2014. Vol. 19. P. 718–733. DOI: 10.1134/S1560354714060094.
  53. Borisov AV, Kazakov AO, Sataev IR. The reversal and chaotic attractor in the nonholonomic model
    of Chaplygin’s top. Regular and Chaotic Dynamics. 2014;19:718–733. DOI: 10.1134/S15603
    54714060094.
  54. Gonchenko S. V., Simo C., Vieiro A. Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight // Nonlinearity. 2013. Vol. 26, no. 3. P. 621–678. DOI: 0.1088/0951-7715/26/3/621.
  55. Gonchenko SV. On a two parameter family of systems close to a system with a nontransversal
    Poincare homoclinic curve. I. Selecta Math. Sovietica. 1990, 10.
  56. Henon M. A two-dimensional mapping with a strange attractor // Commun. Math. Phys. 1976. Vol. 50. P. 69–77. DOI: 10.1007/BF01608556.
  57. Benediks M., Carleson L. Dynamics of the Henon Map // Ann. Math. 1991. Vol. 133. P. 73–169. DOI: 10.2307/2944326.
  58. Gonchenko SV, Turaev DV, Shilnikov LP. On the existence of Newhouse domains in a neighborhood of systems with a structurally unstable Poincare homoclinic curve (the higher-dimensional ´
    case). Dokl. Math. 1993;47(2):268–273.
  59. Gonchenko S. V., Gonchenko V. S, Shilnikov L. P. On homoclinic origin of Henon-like maps // Regular and Chaotic Dynamics. 2010. Vol. 15, no. 4–5. P. 462–481. DOI: 10.1134/S1560354710040052.
  60. Arnold VI. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer;
    1988.
  61. Biragov V. S. Bifurcations in a two-parameter family of conservative mappings that are close to the Henon mapping // Methods of the qualitative theory of differential equations. 1987. P. 10–24. 
  62. Simo C., Vieiro A. Resonant zones, inner and outer splitting in generic and low order resonances of area preserving maps // Nonlinearity. 2009. Vol. 22, no. 5. P. 1191–1245. DOI: 10.1088/0951-7715/22/5/012.
  63. Milnor J. On the concept of attractor // Comm. Math. Phys. 1985. Vol. 99. P. 177–195. DOI: 10.1007/BF01212280.
  64. Minkov SS. Thick attractors and skew products. PhD thesis math. and phys., 2016. 61 p.
  65. Turaev DV, Shilnikov LP. Pseudohyperbolicity and the problem on periodic perturbation of
    Lorenz-like attractors. Dokl. Math. RAN. 2008;418(1).
  66. Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Kozlov A. D. Elements of contemporary theory of dynamical chaos: A tutorial. Part I. Pseudohyperbolic attractors // International Journal of Bifurcation and Chaos. 2018. Vol. 28, no. 11. P. 1830036. DOI: 10.1142/S0218127418300367.
  67. Gonchenko SV, Lamb JSW, Rios I, Turaev D. Attractors and repellers near elliptic points of
    reversible systems. Dokl. Math. RAN. 2014;454:375–378. DOI: 10.7868/S0869565214040045.
  68. Newhouse S. Nondensity of axiom A(α) on S2 // Global Analysis, Proc. Sympos. Pure Math. 1970. Vol. 14. C. 191–202.
  69. Newhouse S. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms // Publ. Math. IHES. 1979. Vol. 50. P. 101–152. DOI: 10.1007/BF02684771.
  70. Guckenheimer J., Williams R. F. Structural stability of Lorenz attractors // Inst. Hautes Etudes Sci. Publ. Math. 1979. Vol. 50. P. 59–72. DOI: 10.1007/BF02684769.
  71. Malkin M. I. Rotation intervals and dynamics of Lorenz-like maps // In: Methods of qualitative theory of diff. eq. Gorki, 1985. C. 122–139.
  72. Gonchenko SV, Turaev DV, Shilnikov LP. On models with non-rough Poincare homoclinic curves.
    Sov. Math. Docl. 1991;320(2):269–272.
  73. Gonchenko S. V., Shil’nikov L. P., Turaev D. V. On models with non-rough Poincare homoclinic curves // Physica D. 1993. Vol. 62, no. 1–4. P. 1–14. DOI: 10.1016/0167-2789(93)90268-6.
  74. Gonchenko SV, Turaev DV, Shilnikov LP. Dynamical phenomena in multidimensional systems
    with non-rough Poincare homoclinic curv. Docl. RAN. 1993;330(2):144–147.
  75. Gonchenko S. V., Shilnikov L. P., Turaev D. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions // Nonlinearity. 2008. Vol. 21, no. 5. P. 923–972. DOI: 10.1088/0951-7715/21/5/003.
  76. Gonchenko SV, Shilnikov LP. On invariants of Ω-conjugacy of diffeomorphisms with a nontransversal homoclinic orbit. Ukr. Math. J. 1990;42(2):153–159.
  77. Gonchenko SV, Shilnikov LP. On moduli of systems with a non-rough Poincare homoclinic curve.
    Izv. RAN, ser. math. 1992;56(6):1165–1197.
  78. Gonchenko SV, Turaev DV, Shilnikov LP. Homoclinic tangencies of an arbitrary order in Newhouse
    domains. J. Math. Sci. 2001;105:1738–1778. DOI: 10.1023/A:1011359428672.
  79. Palis J., Viana M. High dimension diffeomorphisms displaying infinitely many sinks // Ann. Math. 1994. Vol. 140. P. 91–136.
  80. Romero N. Persistence of homoclinic tangencies in higher dimensions // Ergod. Th. and Dyn. Sys. 1995. Vol. 15. P. 735–757.
  81. Gonchenko S. V., Shilnikov L. P., Turaev D. V. Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits // Chaos. 1996. Vol. 6, No. 1. P. 15–31. DOI: 10.1063/1.166154.
  82. Gonchenko SV, Gonchenko VS. On bifurcations of birth of closed invariant curves in the
    case of two-dimensional diffeomorphisms with homoclinic tangencies. Proc. Steklov Math. Inst.
    2004;244:87–114.
  83. Gavrilov NK. On three-dimensional dynamical systems with a non-rough homoclinic contour.
    Mat. Notes. 1973;14(5):953–957. DOI: 10.1007/BF01462256.
  84. Gonchenko SV. Moduli of Ω-conjugacyes for two-dimensional diffeomorphisms with non-rough
    heteroclinic contour. Math. Sb. 1996;187(9):1261–1281.
  85. Gonchenko S. V., Shilnikov L. P., Stenkin O. V. On Newhouse regions with infinitely many stable and unstable invariant tori // Proc. Int. Conf. Progress in Nonlinear Science dedicated to 100th Anniversary of A. A. Andronov, July 2–6; “Mathematical Problems of Nonlinear Dynamics”, vol. 1. Nizhni Novgorod, 2002. P. 80–102.
  86. Gonchenko SV. Dynamical systems with homoclinic tangency, omega-moduli and bifurcations.
    Thesis of Doctor of Phys. and Math. Sci., N. Novgorod, 2004. 300 p.
  87. Gonchenko SV, Stenkin OV., Shilnikov LP. On the existence of a countable set of stable and
    unstable invariant tori for systems from Newhouse domains with heteroclinic tangencies. Rus.
    Nonlinear Dynamics. 2006;2(1):3–25.
  88. Dmitriev A. S., Komlev Yu. A., Turaev D. V. Bifurcation phenomena in the 1:1 resonant horn for the forced van Der Pol-Duffing equation // Int. J. Bifurcation Chaos. 1992. Vol. 2, no. 1. P. 93–100. DOI: 10.1142/S0218127492000094.
  89. Lamb J. S. W., Roberts J. A. G. Time-reversal symmetry in dynamical systems: A survey // Physica D. 1998. Vol. 112. P. 1–39.
  90. Rom-Kedar V., Wiggins S. Transport in two-dimensional maps // Arch. Ration. Mech. Anal. 1990. Vol. 109, no. 3. P. 239–298.
  91. Haragus M., Iooss G. Reversible bifurcations // In: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. London: Universitext, Springer, 2011.
  92. Lerman L. M., Turaev D. V. Breakdown of symmetry in reversible systems // Reg. Chaotic Dyn. 2012. Vol. 17, no. 3–4. P. 318–336. DOI: 10.1134/S1560354712030082.
  93. Lamb J. S. W., Stenkin O. V. Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits // Nonlinearity. 2004. Vol. 17, no. 4. P. 1217–1244. DOI: 10.1088/0951-7715/17/4/005.
  94. Borisov AV, Mamaev IS. Strange attractors in dynamics of Celtic stones. Advances in Physical
    Sciences. 2003;173(4):407–418. DOI: 10.1070/pu2003v046n04abeh001306.
  95. Kuznetsov SP, Zhalnin AYu, Sataev IR, Sedova YuV. Phenomena of nonlinear dynamics of
    dissipative systems in the nonholonomic mechanics of Celtic stone. Rus. Nonlinear dynamics.
    2012;84:735–762.
  96. Kozlov V. V. On the theory of integration of the equations of nonholonomic mechanics // Uspekhi Mekh. 1985. Vol. 8, no. 3. P. 85–107. DOI: 10.1070/RD2002v007n02ABEH000203.
  97. Bizyaev I. A., Borisov A. V., Kazakov A. O. Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors // Regular and Chaotic Dynamics. 2015. Vol. 20, no. 5. P. 605–626.
Received: 
15.07.2024
Accepted: 
04.08.2024
Available online: 
26.11.2024
Published: 
29.11.2024
  • 247 reads