#### For citation:

Bakhanova Y. V., Kazakov A. O., Karatetskaia E. Y., Kozlov A. D., Safonov K. A. On homoclinic attractors of three-dimensional flows. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2020, vol. 28, iss. 3, pp. 231-258. DOI: 10.18500/0869-6632-2020-28-3-231-258

# On homoclinic attractors of three-dimensional flows

The main goal is to construct a classification of such attractors and to distinguish among them the classes of pseudohyperbolic attractors which chaotic dynamics is preserved under perturbations of the system. The main research method is a qualitative method of saddle charts, which consists of constructing an extended bifurcation diagram on the plane of the system parameters in the form $\dot x=y+g_1(x,y,z), \dot y=z+g_2(x,y,z), \dot z=Ax+By+Cz+g_3(x,y,z), \;\; g_i(0,0,0) = (g_i)^\prime_x(0,0,0) = (g_i)^\prime_y(0,0,0) = (g_i)^\prime_z(0,0,0) = 0, \; i = 1, 2, 3$, the linearization matrix of which is represented in the Frobenius form, and the eigenvalues that determine the type of equilibrium state are expressed only through the coefficients A, B, C. The pseudohyperbolicity of the attractors under consideration is verified by means of a numerical method which helps to check the continuity of the subspaces of strong contractions and volume expansion on the attractor. The homoclinic nature of attractors is established using the numerical method of calculating the distance from an attractor to a saddle equilibrium. Results. An extended bifurcation diagram is constructed on the parameter plane (A,B), on which the stability region of the equilibrium state is highlighted, as well as six regions corresponding to two different types of spiral figure-eight attractors, Shilnikov attractor, Lorenz-like attractor, Shilnikov saddle attractor, and Lyubimov–Zaks–Rovella attractor. The pseudohyperbolicity of the Lorenz-like attractor is confirmed numerically. For the attractors of Lyubimov– Zaks–Rovella, it is shown that despite the continuity of strong contracting and volume-expanding subspaces such attractors cannot be pseudohyperbolic. The paper discusses that in three-dimensional flows, in addition to Lorenz-like attractors, only Shilnikov saddle attractors containing a saddle equilibrium state with a two-dimensional unstable manifold can be pseudohyperbolic. However, we currently do not know examples of such attractors.

1. Turaev D.V., Shilnikov L.P. An example of a wild strange attractor. Sb. Math., 1998, vol. 189, pp. 291–314.

2. Gonchenko A.S., Gonchenko S.V. Variety of strange pseudohyperbolic attractors in three-dimensional generalized Henon maps // Physica D: Nonlinear Phenomena, 2016. Vol. 337. P. 43–57.

3. Afraimovich V.S., Bykov V.V., Shilnikov L.P. On the origin and structure of the Lorenz attractor. Akademiia Nauk SSSR Doklady, 1977, vol. 234, pp. 336–339.

4. 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.

5. Hayashi S. Hyperbolicity, stability, and the creation of homoclinic points // Documenta Mathematica, Extra Volume ICM, 1998. T. 2. C. 789–796.

6. Shilnikov L.P. The theory of bifurcations and turbulence. Selecta Mathematica Sovietica, 1991, vol. 10, no. 1, pp. 43–53.

7. Gonchenko S.V., Turaev D.V., Gaspard P. and Nicolis G. Complexity in the bifurcation structure of homoclinic loops to a saddle-focus // Nonlinearity, 1997. Vol. 10, no. 2. P. 409.

8. Shilnikov L.P. Some cases of generation of periodic motions in an n-dimensional space. Soviet Math. Dokl., 1962, vol. 3, pp. 394–397.

9. Shilnikov L.P. Some cases of generation of period motions from singular trajectories. Matematicheskii Sbornik, 1963, vol. 103, no. 4, pp. 443–466

10. Lorenz E. Deterministic nonperiodic flow // Journal of the Atmospheric Sciences. 1963. Vol. 20, no. 2. P. 130–141.

11. Shilnikov A.L. Bifurcation and chaos in the Morioka–Shimizu system. Selecta Math. Soviet., 1991, vol. 10, no. 2, pp. 105–117.

12. Shilnikov A.L. On bifurcations of the Lorenz attractor in the Shimuizu–Morioka model // Physica D. 1993. Vol. 62. P. 338–346.

13. Chua L.O., Komuro M., Matsumoto T. The double scroll family // IEEE Transactions on Circuits and Systems. 1986. Vol. 33, no. 11. P. 1072–1118.

14. Rossler O.E. ¨ An equation for continuous chaos // Physics Letters A. 1976. Vol. 57, № 5. P. 397–398.

15. 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, no. 4. P. 467–485.

16. 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, no. 1. P. 139–156.

17. Aframovich V.S., Shilnikov L.P. Strange Attractors and Quasiattractors. Nonlinear Dynamics and Turbulence, G.I. Barenblatt, G. Iooss, D.D. Joseph (Eds.). Boston: Pitmen, 1983.

18. Gonchenko S.V., Shilnikov L.P., Turaev D.V. Quasiattractors and homoclinic tangencies // Computers and Mathematics with Applications. 1997. Vol. 34, no. 2–4. P. 195–227.

19. Shilnikov L.P. A case of the existence of a denumerable set of periodic motions. Dokl. Akad. Nauk SSSR, 1965, vol. 160, no. 3, pp. 558–561.

20. 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).

21. 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. 291–314

22. Kuznetsov S.P. Dynamic Chaos and Hyperbolic Attractors: From Mathematics to Physics. M.-Izhevsk: Izhevsk Institute for Computer Research, 2013, 488 p. (in Russian).

23. Kuznetsov S.P. Dynamical chaos and uniformly hyperbolic attractors: From mathematics to physics. Phys. Usp., 2011, vol. 54, no. 2, pp. 119–144.

24. Grines V.Z., Zhuzhoma E.V., Pochinka O.V. Rough diffeomorphisms with basic sets of codimension one // Journal of Mathematical Sciences. 2017. Vol. 225. P. 195–219.

25. Kuznetsov S.P. Example of a physical system with a hyperbolic attractor of the Smale–Williams type // Physical Review Letters. 2005. Т. 95, no. 14. 144101.

26. Kuznetsov S.P. and Seleznev E.P. Strange attractor of Smale–Williams type in the chaotic dynamics of a physical system. J. Exp. Theor. Phys., 2006, vol. 102, no. 2. pp. 355–364.

27. Kuznetsov S.P., Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors // Physica D: Nonlinear Phenomena. 2007. Т. 232, no. 2. С. 87–102

28. Kuznetsov S.P. Hyperbolic strange attractors of physically realizable systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, no. 4, pp. 5–34 (in Russian).

29. Kruglov V.P., Kuznetsov S.P. An autonomous system with attractor of Smale–Williams type with resonance transfer of excitation in a ring array of van der Pol oscillators // Communications in Nonlinear Science and Numerical Simulation. 2011. Vol. 16. P. 3219–3223.

30. Jalnine A.Yu. Hyperbolic and non-hyperbolic chaos in a pair of coupled alternately excited FitzHughNagumo systems // Communications in Nonlinear Science and Numerical Simulation. 2015. Vol. 23, no. 1–3. P. 202–208.

31. Kuznetsov S.P., Sataev I.R. Hyperbolic attractor in a system of coupled non-autonomous van der Pol oscillators: Numerical test for expanding and contracting cones // Physics Letters. 2007. Vol. A365. P. 97–104.

32. Kuptsov P.V. Fast numerical test of hyperbolic chaos // Phys. Rev. E. 2012. Vol. 85. 015203(R).

33. Kruglov V.P. Technique and results of numerical test for hyperbolic nature of attractors for reduced models of distributed systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, no. 6, pp. 79–93 (in Russian).

34. Kuznetsov S.P., Kruglov V.P. Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics // Regular and Chaotic Dynamics. 2016. Vol. 21, no. 2. P. 160–174.

35. Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity in chaotic systems with multiple time delays // Communications in Nonlinear Science and Numerical Simulation. 2018. Vol. 56. P. 227–239.

36. 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.

37. Tucker W. The Lorenz attractor exists // Comptes Rendus de l’Academie des Sciences-Series ´ I-Mathematics. 1999. Vol. 328, no. 12. P. 1197–1202.

38. Gonchenko S.V., Kazakov A.O., Turaev D. Wild pseudohyperbolic attractors in a four-dimensional Lorenz system // arXiv preprint arXiv:1809.07250. 2018.

39. Kuptsov P.V., Kuznetsov S.P. Lyapunov analysis of strange pseudohyperbolic attractors: Angles between tangent subspaces, local volume expansion and contraction // Regular and Chaotic Dynamics. 2018. Vol. 23, no. 7–8. P. 908–932

40. Shilnikov A.L., Shilnikov L.P. On the nonsymmetrical Lorenz model // International Journal of Bifurcation and Chaos. 1991. Vol. 1, no. 4. P. 773–776.

41. Kazakov A.O, Kozlov A.D. The asymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems. J. SVMO, 2018, vol. 20, no. 2, pp. 187–198.

42. 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.

43. Gonchenko A.S., Gonchenko S.V., Shilnikov L.P. Towards scenarios of chaos appearance in three-dimensional maps. Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 3–28 (in Russian).

44. Gonchenko A., Gonchenko S., Kazakov A., Turaev D. Simple scenarios of onset of chaos in three-dimensional maps // International Journal of Bifurcation and Chaos. 2014. Vol. 24, № 8. P. 1440005.

45. Gonchenko A.S., Kozlov A.D. On scenaria of chaos appearance in three-dimension nonorientable maps. J. SVMO, 2016. vol. 18, no. 4, pp. 17–29.

46. Kozlov A.D. Examples of strange attractors in three-dimentional nonoriented maps. Zhurnal SVMO, 2017, vol. 19, no. 2, pp. 62–75.

47. Gantmacher F. R. The Theory of Matrices. AMS Chelsea Publishing: Reprinted by American Mathematical Society; 2000, 660 p.

48. Guckenheimer J., Holmes P. Local bifurcations. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York, 1983. P. 117–165.

49. Lyubimov D.V., Zaks M.A. Two mechanisms of the transition to chaos in finite-dimensional models of convection // Physica D: Nonlinear Phenomena. 1983. Vol. 9, no. 1–2. P. 52–64.

50. Rovella A. The dynamics of perturbations of the contracting Lorenz attractor // Boletim da Sociedade Brasileira de Matematica-Bulletin/Brazilian Mathematical Society. 1993. Vol. 24, ´ no. 2. P. 233–259.

51. Ovsyannikov I. M., Shilnikov L. P. On systems with a saddle-focus homoclinic curve. Matematicheskii Sbornik, 1986, vol. 172, no. 4, pp. 552–570.

52. 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.

53. Coullet P., Tresser C., Arneodo A. Possible new strange attractors with spiral structure // Communications in Mathematical Physics. 1981. Vol. 79, no. 4. P. 573–579.

54. 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.

55. 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.

56. Hastings A., Powell T. Chaos in a three-species food chain // Ecology. 1991. Vol. 72, no. 3. P. 896–903.

57. 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.

58. Kuznetsov Y.A., Rinaldi S. Remarks on food chain dynamics // Mathematical biosciences. 1996, vol. 134, no. 1. P. 1–33.

59. 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.

60. 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 // The European Physical Journal Special Topics. 2018. Vol. 227, no. 7–9. P. 959–970.

61. 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.

62. 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. 035201.

63. Gonchenko S.V., Turaev D.V., Shilnikov L.P. Dynamical phenomena in multidimensional systems with non-rough Poincare homoclinic curve. Doklady Mathematics, 1993. Vol. 330, no. 2, pp. 144–147.

64. Gonchenko S.V., Turaev D.V., Shilnikov L.P. On an existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case). Doklady Akademii Nauk, 1993. Vol. 329, No. 4, pp. 404–407.

- 1270 reads