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

For citation:

Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Kozlov A. D., Bakhanova Y. V. Mathematical theory of dynamical chaos and its applications: Review Part 2. Spiral chaos of three-dimensional flows. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 5, pp. 7-52. DOI: 10.18500/0869-6632-2019-27-5-7-52

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):
(downloads: 412)
Article type: 
517.925 + 517.93

Mathematical theory of dynamical chaos and its applications: Review Part 2. Spiral chaos of three-dimensional flows

Gonchenko Aleksandr Sergeevich, Lobachevsky State University of Nizhny Novgorod
Gonchenko Sergey Vladimirovich, Lobachevsky State University of Nizhny Novgorod
Kazakov Aleksej Olegovich, National Research University "Higher School of Economics"
Kozlov Aleksandr Dmitrievich, Lobachevsky State University of Nizhny Novgorod
Bakhanova Yulia Viktorovna, Lobachevsky State University of Nizhny Novgorod

The main goal of the present paper is an explanation of topical issues of the theory of spiral chaos of three-dimensional flows, i.e. the theory of strange attractors associated with the existence of homoclinic loops to the equilibrium of saddle-focus type, based on the combination of its two fundamental principles, Shilnikov’s theory and universal scenarios of spiral chaos, i.e. those elements of the theory that remain valid for any models, regardless of their origin. The mathematical foundations of this theory were laid in the 60th in the famous works of L.P. Shilnikov, and on this subject to date, a lot of important and interesting results have been accumulated. However, these results, for the most part, were related to applications, and, perhaps for this reason, the theory of spiral chaos lacked internal unity – until recently it seemed to consist of separate parts. As it seems for us, the main results of our review allow to fill this gap. So, in the paper we present a fairly complete and illustrative proof of the famous theorem of Shilnikov (1965), describe the main elements of the phenomenological theory of universal scenarios for the emergence of spiral chaos, and also, from a unified point of view, consider a number of three-dimensional models which demonstrate this chaos. They are both the classical models (the systems of Rossler and Arneodo–Coullet–Tresser) and several models known from applications. We discuss advantages of such a new approach to the study of problems of dynamical chaos (including the spiral one), and our recent works devoted to the study of chaotic dynamics of four-dimensional flows and three-dimensional maps show that it is also quite effective. In particular, the next, third, part of the review will be devoted to these results.


  1. Gonchenko A.S., Gonchenko S.V., Kazakov A O., Kozlov A D. Mathematical theory of dynamical chaos and its applications: review part 1. Pseudohyperbolic attractors. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, no. 2, pp. 4–36 (in Russian). 
  2. Shilnikov L.P. A case of the existence of a denumerable set of periodic motions. Doklady Akademii Nauk. Russian Academy of Sciences, 1965, vol. 160, no. 3, pp. 558–561.
  3. Shilnikov L.P. A case of the existence of a denumerable set of periodic motions. Inter-University Symposium on KTDU, 1964, 1 p.(in Russian).
  4. Shilnikov L.P. Leontovich–Andronova Evgeniya Aleksandrovna. Sb. Personality in Science. Women scientists of Nizhny Novgorod, Izdat. NNSU, 1999, pp. 83–102 (in Russian).
  5. Shilnikov L.P. Some cases of generation of periodic motions in an n-dimensional space. Soviet Math. Dokl., 1962, vol. 3, pp. 394–397.
  6. Shilnikov L.P. Some cases of generation of period motions from singular trajectories. Matematicheskii Sbornik, 1963, vol. 103, no. 4, pp. 443–466.
  7. Shilnikov L.P. Existence of a countable set of periodic motions in a neighborhood of a homoclinic curve. Doklady Akademii Nauk, 1967, vol. 172, no. 2, pp. 298–301.
  8. Shilnikov L.P. A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type. Matematicheskii Sbornik, 1970, vol. 123, no. 1, pp. 92–103.
  9. Lorenz E. Deterministic nonperiodic flow // Journal of the Atmospheric Sciences. 1963. Vol. 20, no. 2. P. 130–141.
  10. Sharkovsky O.M. Coexistence of the cycles of a continuous mapping of the line into itself. Ukrainskij matematicheskij zhurnal, 1964, vol. 16, no. 01, pp. 61–71.
  11. Smale S. Differentiable dynamical systems // Bull. AMS. 1963. Vol. 73. P. 747–817.
  12. Chua L.O., Komuro M., Matsumoto T. The double scroll family // Circuits and Systems. IEEE Transactions on. 1986. Vol. 33, no. 11. P. 1072–1118.
  13. Anishchenko V.S. Complex oscillations in simple systems, M.: Nauka, 1990 (in Russian). 
  14. Arecchi F.T., Meucci R., Gadomski W. Laser dynamics with competing instabilities // Physical Review Letters. 1987. Vol. 58, no. 21. P. 2205.
  15. Arecchi F.T., Lapucci A., Meucci R., Roversi J.A., Coullet P.H. Experimental characterization of Shil’nikov chaos by statistics of return times // EPL (Europhysics Letters). 1988. Vol. 6, no. 8. P. 677.
  16. Pisarchik A.N., Meucci R., Arecchi F.T. Theoretical and experimental study of discrete behavior of Shilnikov chaos in a CO2 laser // The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics. 2001. Vol. 13, no. 3. P. 385–391. 
  17. Zhou C.S., Kurths J., Allaria E., Boccaletti S., Meucci R., Arecchi F.T. Constructive effects of noise in homoclinic chaotic systems // Physical Review E. 2003. Vol. 67, no. 6. P. 066220.
  18. Argoul F., Arneodo A., Richetti P. Experimental evidence for homoclinic chaos in the BelousovZhabotinskii reaction // Physics Letters A. 1987. Vol. 120, no. 6. P. 269–275.
  19. Arneodo A., Argoula F., Elezgarayab J., Richettia P. Homoclinic chaos in chemical systems // Physica D: Nonlinear Phenomena. 1993. Vol. 62, no. 1. P. 134–169.
  20. Feudel U., Neiman A., Pei X., Wojtenek W., Braun H., Huber M., Moss F. Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2000. Vol. 10, no. 1. P. 231–239.
  21. Parthimos D., Edwards D.H., Griffith T.M. Shilnikov homoclinic chaos is intimately related to type-III intermittency in isolated rabbit arteries: role of nitric oxide // Physical Review E. 2003. Vol. 67, no. 5. P. 051922.
  22. Koper M.T.M., Gaspard P., Sluyters J.H. Mixed–mode oscillations and incomplete homoclinic scenarios to a saddle focus in the indium / thiocyanate electrochemical oscillator // Journal of Chemical Physics. 1992. Vol. 97, no. 11. P. 8250–8260.
  23. Chedjou J.C., Woafo P., Domngang S. Shilnikov chaos and dynamics of a self-sustained electromechanical transducer // Journal of vibration and acoustics. 2001. Vol. 123, no. 2. P. 170–174.
  24. Bassett M.R., Hudson J.L. Shilnikov chaos during copper electrodissolution // Journal of Physical Chemistry. 1988. Vol. 92, no. 24. P. 6963–6966.
  25. Noh T. Shilnikov chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode // Electrochimica Acta. 2009. Vol. 54, no. 13. P. 3657–3661.
  26. Rucklidge A.M. Chaos in a low-order model of magnetoconvection // Physica D: Nonlinear Phenomena. 1993. Vol. 62, no. 1. P. 323–337.
  27. Henderson M.E., Levi M., Odeh F. The geometry and computation of the dynamics of coupled pendula // International Journal of Bifurcation and Chaos. 1991. Vol. 1, no. 01. P. 27–50.
  28. Gonchenko A.S., Gonchenko S.V., Kazakov A.O. On some new aspects of Celtic stone chaotic dynamics. Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics], 2012, vol. 8, no. 3, pp. 507–518 (in Russian). 
  29. 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. Pp. 521–538.
  30. Vano, J.A., Wildenberg J.C., Anderson M.B., Noel J.K., Sprott J.C. Chaos in low-dimensional Lotka–Volterra models of competition // Nonlinearity. 2006. Vol. 19, no. 10. P. 2391.
  31. Ovsyannikov I.M., Shilnikov L.P. Systems with a homoclinic curve of multidimensional saddlefocus, and spiral chaos. Matematicheskii Sbornik, 1991, vol. 182, no. 7, pp. 1043–1073. 
  32. Afraimovich V.S., Shilnikov L.P. Strange attractors and quasiattractors // Nonlinear Dynamics and Turbulence / Eds G.I.Barenblatt, G. Iooss, D.D. Joseph (Boston, Pitmen), 1983.
  33. Gonchenko S.V., Shilnikov L.P., Turaev D.V. Quasi-attractors and homoclinic tangencies // Computers Math. Applic. 1997. Vol. 34, no. 2–4. P. 195–227.
  34. Gavrilov N.K., Shilnikov L.P. On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. Part 1. Math. USSR Sb., 1972, vol. 17, pp. 467–485.
  35. Gavrilov N.K., Shilnikov L.P. On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. Part 2. Math. USSR Sb, 1973, vol. 19, pp. 139–156.
  36. Ovsyannikov I. M., Shilnikov L. P. On systems with a saddle-focus homoclinic curve. Matematicheskii Sbornik, 1986, vol. 172, no. 4, pp. 552–570. 
  37. Gonchenko S.V. On stable periodic motions in systems close to systems with nonrough homoclinic curve. Math. Notes, 1983, vol. 33, no. 5, pp. 745–755.
  38. Turaev D.V., Shilnikov L.P. An example of a wild strange attractor. Sb. Math., 1998, vol. 189, pp. 291–314.
  39. Gonchenko S.S., Kazakov A.O., Turaev D. Wild pseudohyperbolic attractors in a four-dimensional Lorenz system // arXiv preprint arXiv:1809.07250, 2018.
  40. Gonchenko A.S., Gonchenko S.V., Shilnikov L.P. Towards Scenarios of Chaos Appearance in Three-Dimensional Maps. Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics], 2012, vol. 8, no. 1, pp. 3–28 (in Russian).
  41. Gonchenko A.S., Gonchenko S.V., Kazakov A.O. and Turaev D. Simple scenarios of onset of chaos in three-dimensional maps // Int. J. Bif. and Chaos. 2014. Vol. 24, no. 8. 25 p.
  42. Gonchenko A.S., Gonchenko S.V. Variety of strange pseudohyperbolic attractors in three-dimensional generalized Henon maps // Physica D. 2016. Vol. 337. P. 43–57.
  43. Shilnikov L.P. The theory of bifurcations and turbulence. Selecta Mathematica Sovietica, 1991, vol. 10, no. 1, pp. 43–53.
  44. Shilnikov A.L., Shilnikov L.P., Turaev D.V., Chua L.O. Methods of Qualitative Theory in Nonlinear Dynamics. World Scientific; Part 1, 1998, 412 p.
  45. Shilnikov A.L., Shilnikov L.P., Turaev D.V., Chua L.O. Methods of Qualitative Theory in Nonlinear Dynamics. World Scientific; Part 2, 2001, 577 p.
  46. Belyakov L.A. A case of the generation of a periodic motion with homoclinic curves. Mathematical Notes, 1974, vol. 15, no. 4, pp. 336–341.
  47. Belyakov L.A. Bifurcation set in a system with homoclinic saddle curve. Mathematical Notes, 1980, vol. 28, no. 6, pp. 910–916.
  48. Belyakov L.A. Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero. Mathematical Notes, 1984, vol. 36, no. 5, pp. 838–843. 
  49. Biragov V., Shilnikov L. On the bifurcation of a saddle-focus separatrix loop in a three-dimensional conservative dynamical system. Selecta Math. Soviet. 1992, vol. 11, no. 4, pp. 333–340.
  50. Gonchenko V.S., Shilnikov L.P. On bifurcations of a homoclinic loop to a saddle-focus of index 1/2. Rus. Math. Docl., 2007, vol. 417, no. 6. 
  51. Gonchenko S.V., Turaev D.V., Shilnikov L.P. On models with nonrough Poincare homoclinic curves // Physica D. 1993. Vol. 62. P. 1–14.
  52. Gonchenko S. V., Turaev D. V., Shilnikov L. P. On models with a structurally unstable homoclinic Poincare curve. ´ Sov. Math. Dokl., 1992, vol. 44, no. 2., pp. 422–426.
  53. Gaspard P., Gonchenko S.V., Nicolis G., Turaev D.V. Complexity in the bifurcation structure of homoclinic loops to a saddle-focus. Nonlinearity. 1997. Vol. 10. P. 409–423.
  54. Ruelle D., Takens F. On the nature of turbulence // Communications in Mathematical Physics. 1971. Vol. 20, no. 3. P. 167–192.
  55. Feigenbaum M.J. Quantitative universality for a class of nonlinear transformations // Journal of statistical physics. 1978. Vol. 19, no. 1. P. 25–52.
  56. Pomeau Y., Manneville P. Intermittent transition to turbulence in dissipative dynamical systems // Comm. Math. Phys. 1980. Vol. 74, no. 2. P. 189–197.
  57. Luk’yanov V.I., Shilnikov L.P. On some bifurcations of dynamical systems with homoclinic structures. Doklady Akademii Nauk, 1978, vol. 243, no. 1, pp. 26–29
  58. Afraimovich V.S., Shilnikov L.P. Invariant two-dimensional tori, their breakdown and stochasticity. Amer. Math. Soc. Transl., 1991, vol. 149, no. 2, pp. 201–212.
  59. Afraimovich V.S., Shilnikov L.P. On some global bifurcations connected with the disappearance of a fixed point of saddle-node type. Doklady Akademii Nauk, 1974, vol. 219. no. 6, pp. 1281–1284.
  60. Afraimovich V.S., Shilnikov L.P. The ring principle in problems of interaction between two self-oscillating systems. Prikladnaia Matematika i Mekhanika. 1977, vol. 41, pp. 618–627. 
  61. Aronson D.G., Chory M.A., Hall G.R., McGehee R.P. Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study // Communications in Mathematical Physics. 1982. Т. 83, no. 3. P. 303–354.
  62. Newhouse S.E., Palis J., Takens F. Bifurcations and stability of families of diffeomorphisms // Publications Mathematiques IHES. 1983. Vol. 57. P. 5–71.
  63. Turaev D.V., Shilnikov L.P. in book «Mathematical mechanisms of turbulence». Kiev, 1986, pp. 113–121 (in Russian).
  64. Coullet P., Tresser C., Arneodo A. Transition to stochasticity for a class of forced oscillators // Physics letters A. 1979. Vol. 72, no. 4–5. P. 268–270.
  65. Arneodo A., Coullet P., Tresser C. Possible New Strange Attractors With Spiral Structure // Commun. Math. Phys. 1981. Vol. 79. P. 573–579
  66. Shilnikov L.P. Bifurcation theory and the Lorenz model. Appendix to Russian edition of «The Hopf Bifurcation and Its Applications». Eds. J. Marsden and M. McCraken, 1980, pp. 317–335 (in Russian).
  67. Afraimovich V.S., Bykov V.V., Shilnikov L.P. Attractive nonrough limit sets of Lorenz-attractor type. Trudy Moskovskoe Matematicheskoe Obshchestvo, 1982, vol. 44, pp. 150–212.
  68. Bykov V.V. Ob Bifurcations of dynamical systems close to systems with a separatrix contour containing a saddle-focus. Methods of Qualitative Theory of Differential Equations. Gorki, 1980, pp. 44–72 (in Russian).
  69. Bykov V.V. The bifurcations of separatrix contours and chaos // Physica D: Nonlinear Phenomena. 1993. Vol. 62, no. 1–4. P. 290–299.
  70. Bykov V.V. On systems with separatrix contour containing two saddle-foci // Journal of Mathematical Sciences. 1999. Vol. 95, no. 5. P. 2513–2522.
  71. Rossler O.E. Continuous chaos-four prototype equations // Annals of the New York Academy of Sciences 1979. Vol. 316, no. 1. P. 376–392.
  72. Rossler O.E. An equation for continuous chaos // Physics Letters A. 1976. Vol. 57, no. 5. P. 397–398.
  73. Arneodo A., Coullet P., Tresser C. Oscillators with chaotic behavior: an illustration of a theorem by Shilnikov // Journal of Statistical Physics. 1982. Vol. 27, no. 1. P. 171–182.
  74. Gaspard P., Kapral R., Nicolis G. Bifurcation phenomena near homoclinic systems: a twoparameter analysis // Journal of Statistical Physics. 1984. Vol. 35, no. 5. P. 697–727.
  75. Vitolo R., Glendinning P., Gallas J.A.C. Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows // Physical Review E. 2011. Vol. 84, no. 1. P. 016216.
  76. Barrio R., Blesa F., Serrano S., Shilnikov A. Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci // Physical Review E. 2011. Vol. 84, no. 3. P. 035201. 
  77. Letellier C., Dutertre P., Maheu B. Unstable periodic orbits and templates of the Rossler system: ¨ toward a systematic topological characterization // Chaos: An Interdisciplinary Journal of Nonlinear Science. 1995. Vol. 5, no. 1. P. 271–282.
  78. Arneodo A., Coullet P., Tresser C. Occurence of strange attractors in three-dimensional Volterra equations // Physics Letters A. 1980. Vol. 79, no. 4. P. 259–263.
  79. Arneodo A., Coullet P., Spiegel E., Tresser C. Asymptotic chaos // Physica D: Nonlinear Phenomena. 1985. Vol. 14, no. 3. P. 327–347.
  80. Shilnikov L.P., Turaev D.V. Super-homoclinic orbits and multi-pulse homoclinic loops in Hamiltonian systems with discrete symmetries // Regul. Khaoticheskaya Din. 1997. Vol. 2, no. 3–4. P. 126–138.
  81. Rosenzweig M.L. Exploitation in three trophic levels // The American Naturalist. 1973. Vol. 107, no. 954. P. 275–294.
  82. Hastings A., Powell T. Chaos in a three-species food chain // Ecology. 1991. Vol. 72, no. 3. P. 896–903.
  83. Rai V., Sreenivasan R. Period-doubling bifurcations leading to chaos in a model food chain // Ecological modelling. 1993. Vol. 69, no. 1–2. P. 63–77.
  84. Kuznetsov Y.A., Rinaldi S. Remarks on food chain dynamics // Mathematical biosciences. 1996. Vol. 134, no. 1. P. 1–33.
  85. Deng B. Hines G. Food chain chaos due to Shilnikov’s orbit // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2002. Vol. 12, no. 3. P. 533–538.
  86. Kuznetsov Y.A., De Feo O., Rinaldi S. Belyakov homoclinic bifurcations in a tritrophic food chain model // SIAM Journal on Applied Mathematics. 2001. Vol. 62, no. 2. P. 462–487.
  87. Bakhanova Y.V., Kazakov A.O., Korotkov A.G., Levanova T.A., Osipov G.V. Spiral attractors as the root of a new type of «bursting activity» in the Rosenzweig–MacArthur model // European Physical Journal. Special Topics. 2018. Vol. 227, no. 7–9. P. 959–970.
  88. Gaspard P., Nicolis G. What can we learn from homoclinic orbits in chaotic dynamics? // Journal of statistical physics. 1983. Vol. 31, no. 3. P. 499–518.
  89. Gallas J.A.C. The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows // International Journal of Bifurcation and Chaos. 2010. Vol. 20, no. 02. P. 197–211.
  90. Afraimovich V.S., Gonchenko S.V., Lerman L.M., Shilnikov A.L., Turaev D.V. Scientific heritage of L.P. Shilnikov // Regul. Chaot. Dyn. 2014. Vol. 19, no. 4. P. 435–460.
  91. Afraimovich V.S., Belyakov L.A., Gonchenko S.V., Lerman L.M., Morozov A.D., Turaev D.V., Shilnikov A.L. «L.P. Shilnikov. Selected Works». UNN press, 2017, p. 429 (in Russian).
  92. Korotkov A.G., Levanova T.A., Kazakov A.O. Effects of memristor-based coupling in the ensemble of FitzHugh–Nagumo elements // Eur. Phys. J. Special Topics. 2019. Vol. 228. P. 2325–2337.
  93. 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.
  94. Borisov A.V., Kazakov A.O., Sataev I.R. Spiral chaos in the nonholonomic model of a Chaplygin top // Regular and Chaotic Dynamics. 2016. Vol. 21, no. 7–8. P. 939–954. 
  95. Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Samylina E.A, Chaotic dynamics and multistability in a nonholonomic model of Celtic stone. Izvestiya VUZ. Radiophysics, 2018, vol. 61, no. 10, pp. 867–882.
  96. Stankevich N., Kuznetsov A., Popova E., Seleznev E. Chaos and hyperchaos after secondary Neimark–Sacker bifurcation in a model of radio-physical generator // Nonlinear dynamics. 2019. Vol. 97, no. 4. P. 2355–2370.
  97. Garashchuk I.R., Sinelshchikov D.I., Kazakov A.O., Kudryashov N.A. Hyperchaos and multistability in the model of two interacting microbubble contrast agents. Chaos: Interdisciplinary Journal of Nonlinear Science, 2019, vol. 29, no. 6, pp. 063131.
Short text (in English):
(downloads: 164)