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

For citation:

Kruglov V. P., Kuptsov P. V. Theoretical models of physical systems with rough chaos. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 1, pp. 35-77. DOI: 10.18500/0869-6632-2021-29-1-35-77

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 320)
Article type: 

Theoretical models of physical systems with rough chaos

Kruglov Vjacheslav Pavlovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Kuptsov Pavel Vladimirovich, Yuri Gagarin State Technical University of Saratov

The purpose of this review is to present in a unified manner the latest results on mathematical modeling of rough hyperbolic chaos in systems of various physical nature. Main research Methods are the numerical solution of systems of differential equations and partial differential equations, numerical extraction of the phase of oscillatory processes or spatial patterns, calculating of Lyapunov exponents, and studying the mutual arrangement of the stable and unstable manifolds of chaotic trajectories, the calculation of Gaussian curvature of surfaces. These procedures allow to reveal the typical attributes of rough hyperbolic chaos. Results reproduce the studies of already known phenomena, however, their qualitative explanation and quantitative confirmation are given in in a more detailed form, in accordance with the development of ideas about them. Conclusion. Methodologically the proposed review article may be interesting for undergraduate and graduate students in terms of studying the principles of construction and analysis of systems with chaotic behavior.

Работа В.П. Круглова (Раздел 2, Модели пространственно распределенных систем с аттрактором типа Смейла–Вильямса) поддержана грантом РНФ № 19-11-00280. Работа П.В. Купцова выполнена в рамках государственного задания Института радиотехники и электроники им. В.А. Котельникова РАН.
  1. Andronov AA, Pontryagin LS. Rough Systems. Proceedings of the Academy of Sciences of the USSR. 1937;14(5):247–250. In Russian.
  2. Andronov AA, Vitt AA, Khaikin SE. Theory of Oscillations. Elsevier; 1966.
  3. Kuznetsov SP. Hyperbolic Chaos: A Physicist’s View. Berlin, Heidelberg: Higher Education Press: Bijing and Springer-Verlag; 2012.
  4. Kuznetsov SP. Dynamical chaos and hyperbolic attractors: from mathematics to physics. Izhevsk, Moscow: Institute of computer research; 2013. In Russian.
  5. Turaev DV, Shil’nikov LP. An example of a wild strange attractor. Sb Math. 1998;189(2):291– 314.
  6. Turaev DV, Shil’nikov LP. Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors. Doklady Mathematics. 2008;77(1):17–21.
  7. Gonchenko AS, Gonchenko SV, Kazakov AO, Kozlov AD. Mathematical theory of dynamical chaos and its applications: Review. Part 1. Pseudohyperbolic attractors. Izvestiya VUZ Applied Nonlinear Dynamics. 2017;25(2):4–36. In Russian.
  8. Gonchenko AS, Gonchenko SV. Variety of strange pseudohyperbolic attractors in threedimensional generalized Henon maps. Physica D: Nonlinear Phenomena. 2016;337:43–57. ´
  9. Kuznetsov SP. Example of a physical system with a hyperbolic attractor of the Smale-Williams type. Physical Review Letters. 2005;95(14):144101.
  10. Smale S. Differentiable dynamical systems. Bulletin of the American mathematical Society. 1967;73(6):747–817.
  11. Williams RF. Expanding attractors. Publications Mathematiques de l’IH ´ ES. 1974;43:169–203. ´
  12. Guckenheimer J, Holmes PJ. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. vol. 42 of Applied Mathematical Sciences. New York: Springer-Verlag; 1983.
  13. Anosov DV. Geodesic flows on closed Riemannian manifolds of negative curvature. Proceedings of the Mathematical Institute of the Academy of Sciences of the USSR. 1967;90(5):3–210. In Russian.
  14. Anosov DV, Aranson DK, Grines VZ, Plykin RV, Sataev EA, Safonov AV, et al. Dynamical systems with hyperbolic behavior. vol. 66 of Encyclopaedia of Mathematical Sciences. Berlin, Heidelberg: Springer-Verlag; 1995. p. 236.
  15. Balazs NL, Voros A. Chaos on the pseudosphere. Physics Reports. 1986;143(3):109–240.
  16. Wilczak D. Uniformly hyperbolic attractor of the Smale–Williams type for a Poincare map in the ´ Kuznetsov system. SIAM Journal on Applied Dynamical Systems. 2010;9(4):1263–1283.
  17. Kuznetsov SP, Seleznev EP. A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. JETP. 2006;102(2):355–364.
  18. Belyakin ST, A SS. Model fibrillation as an analogue of the hyperbolic the Smale-Williams attractor. American Journal of Biomedical Science & Research. 2019;2(5):197–201.
  19. Loskutov AY. Fascination of chaos. Phys Usp. 2010;53(12):1257–1280.
  20. Zeraoulia E, Sprott JC. Robust Chaos and Its Applications. vol. 79. World Scientific; 2011.
  21. 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(6):718–733.
  22. Kuznetsov SP. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics. Phys Usp. 2011;54(2):119–144.
  23. Kuznetsov SP. Self-oscillating system generating rough hyperbolic chaos. Izvestiya VUZ Applied Nonlinear Dynamics. 2019;27(6):39–62. In Russian.
  24. Anosov DV. Rough Systems. Proceedings of the Mathematical Institute of the Academy of Sciences of the USSR. 1985;169:59–93. In Russian.
  25. Poznyak EG, Shikin EV. Differential Geometry: First Exposure. Moscow Univerity Press; 1990. In Russian.
  26. Dubrovin BA, Novikov SP, Fomenko AT. Modern Geometry–Methods and Applications. Graduate Texts in Mathematics. New York: Springer-Verlag; 1990.
  27. Gantmacher FR. The Theory of Matrices. AMS Chelsea Publishing: Reprinted by American Mathematical Society; 2000.
  28. Meeks III WH, Ros A, Rosenberg H. The Global Theory of Minimal Surfaces in Flat Spaces. vol. 1775 of Lecture Notes in Mathematics. Berlin, Heidelberg: Springer-Verlag; 2002.
  29. Meeks III WH, Perez J. A survey on Classical Minimal Surface Theory. vol. 60 of University ´ Lecture Series. American Mathematical Society; 2012.
  30. Donnay V, Visscher D. A new proof of the existence of embedded surfaces with Anosov geodesic flow. Regular and Chaotic Dynamics. 2018;23(6):685–694.
  31. Borisovich UG, Bliznyakov NM, Izrailevich YA, Fomenko TN. Introduction to Topology: A Tutorial. Moscow: Science, Fizmatlit; 1995. In Russian.
  32. Kozlov VV. Topological obstructions to the integrability of natural mechanical systems. 1979;249(6):1299–1302. In Russian.
  33. Thurston WP, Weeks JR. The mathematics of three-dimensional manifolds. Scientific American. 1984;251(1):108–121.
  34. Hunt TJ, MacKay RS. Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity. 2003;16(4):1499–1510.
  35. Kuznetsov SP. Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: testing absence of tangencies of stable and unstable manifolds for phase trajectories. Regular and Chaotic Dynamics. 2015;20(6):649–666.
  36. Benettin G, Galgani L, Giorgilli A, Strelcyn JM. All Lyapunov characteristic numbers are effectively computable. C R Acad Sc Paris, Ser A. 1978;286:431–433. ´
  37. Benettin G, Galgani L, Giorgilli A, Strelcyn JM. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part I: Theory. Part II: Numerical application. Meccanica. 1980;15:9–30.
  38. Shimada I, Nagashima T. A numerical approach to ergodic problem of dissipative dynamical systems. Prog Theor Phys. 1979;61(6):1605–1616.
  39. Kuznetsov SP. Chaos in the system of three coupled rotators: from Anosov dynamics to hyperbolic attractor. Izv Saratov Univ (N S), Ser Physics. 2015;15(2):5–17. In Russian.
  40. Kuznetsov SP. From Anosov’s dynamics on a surface of negative curvature to electronic generator of robust chaos. Izv Saratov Univ (N S), Ser Physics. 2016;16(3):132–144. In Russian.
  41. Kuznetsov SP, Kruglov VP. On some simple examples of mechanical systems with hyperbolic chaos. vol. 297. Springer; 2017. p. 208–234.
  42. Anosov DV, Sinai YG. Some smooth ergodic systems. Russ Math Surv. 1967;22(5):103–167.
  43. Kuptsov PV. Fast numerical test of hyperbolic chaos. Physical Review E. 2012;85:015203.
  44. Kuznetsov SP. Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: examples and numerical study. The Bulletin of Udmurt University Mathematics Mechanics Computer Science. 2018;28(4):565–581. In Russian.
  45. Kuznetsov SP. From geodesic flow on a surface of negative curvature to electronic generator of robust chaos. International Journal of Bifurcation and Chaos. 2016;26(14):1650232.
  46. Kuznetsov SP, Pikovsky AS. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D: Nonlinear Phenomena. 2007;232(2):87–102.
  47. Kuznetsov AP, Kuznetsov SP, Pikovsky AS, Turukina LV. Chaotic dynamics in the systems of coupling nonautonomous oscillators with resonance and nonresonance communicator of the signal. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(6):75–85. In Russian.
  48. Kuznetsov SP. On the feasibility of a parametric generator of hyperbolic chaos. JETP. 2008;106(2):380–387.
  49. Tyuryukina LV, Pikovsky AS. Hyperbolic chaos in a system of nonlinear coupled Landau-Stuart oscillators. Izvestiya VUZ Applied Nonlinear Dynamics. 2009;17(2):99–113. In Russian.
  50. Kruglov VP. Circular non-autonomous generator of hyperbolic chaos. Izvestiya VUZ Applied Nonlinear Dynamics. 2010;18(5):132–147. In Russian.
  51. Kuznetsov SP. Example of blue sky catastrophe accompanied by a birth of Smale-Williams attractor. Regular and Chaotic Dynamics. 2010;15(2–3):348–353.
  52. Kuznetsov SP. Some mechanical systems manifesting robust chaos. Nonlinear Dynamics & Mobile Robotics. 2013;1(1):3–22. In Russian.
  53. Doroshenko VM, Kruglov VP, Kuznetsov SP. Smale–Williams Solenoids in a System of Coupled Bonhoeffer–van der Pol Oscillators. Russian Journal of Nonlinear Dynamics. 2018;14(4): 435–451.
  54. Kuznetsov SP, Kruglov VP. Hyperbolic chaos in a system of two Froude pendulums with alternating periodic braking. Communications in Nonlinear Science and Numerical Simulation. 2019;67:152–161.
  55. Kuznetsov SP, Sedova YV. Hyperbolic chaos in the Bonhoeffer–van der Pol oscillator with additional delayed feedback and periodically modulated excitation parameter. Izvestiya VUZ Applied Nonlinear Dynamics. 2019;27(1):77–95. In Russian.
  56. Kuznetsov SP, Seleznev EP. A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. JETP. 2006;102(2):355–364.
  57. Kuznetsov SP, Ponomarenko VI. Realization of a strange attractor of the Smale-Williams type in a radiotechnical delay-feedback oscillator. Technical Physics Letters. 2008;34(9):771–773.
  58. Kuznetsov SP. Hyperbolic strange attractors of physically realizable systems. Izvestiya VUZ Applied Nonlinear Dynamics. 2009;17(4):5–34. In Russian.
  59. Kuptsov PV, Kuznetsov SP, Pikovsky AS. Hyperbolic chaos of Turing patterns. Physical Review Letters. 2012;108(19):194101.
  60. Cross MC, Hohenberg PC. Pattern formation outside of equilibrium. Reviews of Modern Physics. 1993;65(3):851–1112.
  61. Swift J, Hohenberg PC. Hydrodynamic fluctuations at the convective instability. Physical Review A. 1977;15(1):319–329.
  62. Kuznetsov SP. Some lattice models with hyperbolic chaotic attractors. Russian Journal of Nonlinear Dynamics. 2020;16(1):13–21.
  63. Kruglov VP, Kuznetsov SP, Pikovsky AS. Attractor of Smale-Williams type in an autonomous distributed system. Regular and Chaotic Dynamics. 2014;19(4):483–494.
  64. Kruglov VP, Khadzhieva LB. Uniformly hyperbolic attractor in a system based on coupled oscillators with «figure-eight» separatrix. Izvestiya VUZ Applied Nonlinear Dynamics. 2016;24(6):54–64. In Russian.
  65. Kuznetsov SP. Generation of robust hyperbolic chaos in CNN. Russian Journal of Nonlinear Dynamics. 2019;15(2):109–124.
  66. Glansdorff P, Prigogine IR. Thermodynamic theory of structure, stability and fluctuations. Wiley– Interscience; 1971.
  67. Kruglov VP. Technique and results of numerical test for hyperbolic nature of attractors for reduced models of distributed systems. Izvestiya VUZ Applied Nonlinear Dynamics. 2014;22(6):79–93. In Russian.
  68. Isaeva OB, Kuznetsov AS, Kuznetsov SP. Hyperbolic chaos of standing wave patterns generated parametrically by a modulated pump source. Physical Review E. 2013;87(4):040901.
  69. Kruglov VP, Kuznetsov SP, Kuznetsov AS. Hyperbolic chaos in systems with parametrically excited patterns of standing waves. Russian Journal of Nonlinear Dynamics. 2014;10(3):265– 277. In Russian.
  70. Kuznetsov SP. Chaotic dynamics of pendulum ring chain with vibrating suspension. Izvestiya VUZ Applied Nonlinear Dynamics. 2019;27(4):99–113. In Russian.
  71. Kuznetsov SP, Pikovsky AS, Rosenblum M. Collective phase chaos in the dynamics of interacting oscillator ensembles. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2010;20(4):043134.
  72. Kuptsov PV, Kuznetsov SP, Pikovsky AS. Hyperbolic chaos at blinking coupling of noisy oscillators. Physical Review E. 2013;87(3):032912.