Richness of chaotic dynamics in nonholonomic models of a Celtic stone. (English) Zbl 1417.37222
Summary: We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.
MSC:
| 37J60 | Nonholonomic dynamical systems |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37G35 | Dynamical aspects of attractors and their bifurcations |
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
Keywords:
Celtic stone; nonholonomic model; strange attractor; discrete Lorenz attractor; Shilnikov-like spiral attractor; mixed dynamicsReferences:
| [1] | Astapov, I. S., On Rotation Stability of Celtic Stone, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1980, no. 2, pp. 97–100 (Russian). · Zbl 0445.70003 |
| [2] | Karapetyan, A.V., On permanent rotations of heavy rigid body on the absolutely rough horizontal plane, Prikl. Mat. Mekh., 1981, vol. 45, no. 5, pp. 808–814 [J. Appl. Math. Mech., 1981, vol. 45, no. 5, pp. 808–814]. |
| [3] | Markeev, A.P., The Dynamics of a Rigid Body on an Absolutely Rough Plane, Prikl. Mat. Mekh., 1983, vol. 47, no. 4, pp. 575–582 [J. Appl. Math. Mech., 1983, vol. 47, no. 4, pp. 473–478]. |
| [4] | Markeev, A.P., Dynamics of a Body Touching a Rigid Surface, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2011 (Russian). |
| [5] | Kazakov, A. O., Chaotic Dynamics Phenomena in the Rubber Rock-n-Roller on a Plane Problem, Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 2, pp. 309–325 (Russian). |
| [6] | Kazakov, A. O., Strange Attractors and Mixed Dynamics in the Unbalanced Rubber Ball on a Plane Problem, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 508–520. · Zbl 1417.37223 · doi:10.1134/S1560354713050043 |
| [7] | Borisov, A.V. and Mamaev, I. S., Strange Attractors in the Rattleback Dynamics, in Nonholonomic Dynamical Systems: Integrability, Chaos, Strange Attractors, A. V. Borisov, I. S. Mamaev (Eds.), Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2002, pp. 296–326 (Russian). · Zbl 1097.37002 |
| [8] | Borisov, A.V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 408–418 [Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393–403]. |
| [9] | Shilnikov, L.P., Bifurcation Theory and Turbulence, in Methods of Qualitative Theory of Differential Equations, E.A. Leontovich (Ed.), Gorky: Gorky Gos. Univ., 1986, pp. 150–165, 215 (Russian). |
| [10] | Gonchenko, S. V., Turaev, D. V., and Shilnikov, L.P., On Newhouse Domains of Two-Dimensional Diffeomorphisms That Are Close To a Diffeomorphism with a Structurally Unstable Heteroclinic Contour, in Dynamical Systems and Related Topics: Collection of Articles to the 60th Anniversary of Academician D.V.Anosov, Tr. Mat. Inst. Steklova, vol. 216, Moscow: Nauka, 1997, pp. 76–125 [Proc. Steklov Inst. Math., 1997, vol. 216, pp. 70–118]. · Zbl 0931.37006 |
| [11] | Gonchenko, S. V., Shilnikov, L.P., and Stenkin, O.V., On Newhouse Regions with Infinitely Many Stable and Unstable Invariant Tori, in Proc. of Intern. Conf. ”Progress in Nonlinear Science”: Dedicated to 100th Anniversary of A.A.Andronov, N.Novgorod, 2002, pp. 80–102. |
| [12] | Lamb, J. S.W. and Stenkin, O. V., Newhouse Regions for Reversible Systems with Infinitely Many Stable, Unstable and Elliptic Periodic Orbits, Nonlinearity, 2004, vol. 17, no. 4, pp. 1217–1244. · Zbl 1061.37030 · doi:10.1088/0951-7715/17/4/005 |
| [13] | Delshams, A., Gonchenko, S. V., Gonchenko, A. S., Lázaro, J. T., and Sten’kin, O., Abundance of Attracting, Repelling and Elliptic Periodic Orbits in Two-Dimensional Reversible Maps, Nonlinearity, 2013, vol. 26, no. 1, pp. 1–33. · Zbl 1277.37044 · doi:10.1088/0951-7715/26/1/1 |
| [14] | Newhouse, S.E., The Abundance of Wild Hyperbolic Sets and Non-Smooth Stable Sets for Diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 1979, vol. 50, no. 1, pp. 101–151. · Zbl 0445.58022 · doi:10.1007/BF02684771 |
| [15] | Gonchenko, S.V., Turaev, D.V., and Shilnikov, L.P., On the Existence of Newhouse Regions Near Systems with Non-Rough Poincaré Homoclinic Curve (Multidimensional Case), Dokl. Ross. Akad. Nauk, 1993, vol. 329, no. 4, pp. 404–407 [Russian Acad. Sci. Dokl. Math., 1993, vol. 47, no. 2, pp. 268–283]. |
| [16] | Palis, J. and Viana, M., High Dimension Diffeomorphisms Displaying Infinitely Many Sinks, Ann. of Math. (2), 1994, vol. 140, no. 1, pp. 91–136. · Zbl 0817.58004 · doi:10.2307/2118546 |
| [17] | Romero, N., Persistence of Homoclinic Tangencies in Higher Dimensions, Ergodic Theory Dynam. Systems, 1995, vol. 15, no. 4, pp. 735–757. · Zbl 0833.58020 · doi:10.1017/S0143385700008634 |
| [18] | Gonchenko, S., Shilnikov, L., and Turaev, D., Homoclinic Tangencies of Arbitrarily High Orders in Conservative and Dissipative Two-Dimensional Maps, Nonlinearity, 2007, vol. 20, no. 2, pp. 241–275. · Zbl 1132.37023 · doi:10.1088/0951-7715/20/2/002 |
| [19] | Newhouse, S.E., Diffeomorphisms with Infinitely Many Sinks, Topology, 1974, vol. 13, pp. 9–18. · Zbl 0275.58016 · doi:10.1016/0040-9383(74)90034-2 |
| [20] | Turaev, D.V. and Shil’nikov, L.P., An Example of a Wild Strange Attractor, Mat. Sb., 1998, vol. 189, no. 2, pp. 137–160 [Sb. Math., 1998, vol. 189, nos. 1–2, pp. 291–314]. · Zbl 0927.37017 · doi:10.4213/sm300 |
| [21] | Turaev, D.V. and Shil’nikov, L.P., Pseudohyperbolicity and the Problem on Periodic Perturbations of Lorenz-Type Attractors, Dokl. Akad. Nauk, 2008, vol. 418, no. 1, pp. 23–27 [Russian Dokl. Math., 2008, vol. 77, no. 1, pp. 17–21]. |
| [22] | Gonchenko, S. V., Ovsyannikov, I. I., Simó, C., and Turaev, D., Three-Dimensional Hénon-Like Maps and Wild Lorenz-Like Attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2005, vol. 15, no. 11, pp. 3493–3508. · Zbl 1097.37023 · doi:10.1142/S0218127405014180 |
| [23] | Gonchenko, A. S. and Gonchenko, S.V., On Existence of Lorenz-Like Attractors in a Nonholonomic Model of Celtic Stones, Rus. J. Nonlin. Dyn., 2012, vol. 9, no. 1, pp. 77–89 (Russian). |
| [24] | Gonchenko, A. S., On Lorenz-Like Attractors in Model of Celtic Stone, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2013, no. 2, pp. 3–11 (Russian). · Zbl 1299.37066 |
| [25] | Borisov, A.V. and Mamaev, I. S., Dynamics of a Rigid Body, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2001 (Russian). · Zbl 1004.70002 |
| [26] | Borisov, A.V., Jalnine, A.Yu., Kuznetsov, S.P., Sataev, I.R., and Sedova, J.V., Dynamical Phenomena Occurring Due To Phase Volume Compression in Nonholonomic Model of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512–532. · Zbl 1263.74021 · doi:10.1134/S1560354712060044 |
| [27] | Karapetyan, A.V., Hopf Bifurcation in a Problem of Rigid Body Moving on a Rough Plane, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1985, no. 2, pp. 19–24 (Russian). |
| [28] | Shilnikov, L.P., Existence of a Countable Set of Periodic Motions in a Neighborhood of a Homoclinic Curve, Dokl. Akad. Nauk SSSR, 1967, vol. 172, no. 2, pp. 298–301 (Russian). |
| [29] | Anishchenko, V. S., Complicated Oscillations in Simple Systems, Moscow: Nauka, 1990 (Russian). · Zbl 0746.58006 |
| [30] | Vitolo, R., Bifurcations of Attractors in 3D Diffeomorphisms, PhD Thesis, Groningen Univ. Press, 2003. |
| [31] | Shilnikov, L.P., The Bifurcation Theory and the Lorenz Model, in Bifurcation of the Cycle and Its Applications, Moscow: Mir, 1980, pp. 317–335 (Russian). |
| [32] | Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., On Some New Aspects of Celtic Stone Chaotic Dynamics, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 3, pp. 507–518 (Russian). |
| [33] | Kuznetsov, S.P., Jalnine, A.Y., Sataev, I.R., and Sedova, J. V., Phenomena of Nonlinear Dynamics of Dissipative Systems in Nonholonomic Mechanics of the Rattleback, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 4, pp. 735–762 (Russian). |
| [34] | Gonchenko, A. S., Gonchenko, S.V., and Shilnikov, L.P., Towards Scenarios of Chaos Appearance in Three-Dimensional Maps, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 1, pp. 3–28 (Russian). |
| [35] | Anosov, D.V., Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature, Trudy Mat. Inst. Steklov, 1967, vol. 90, pp. 3–209 [Proc. Steklov. Inst. Math., Providence, R. I.: AMS, 1969]. |
| [36] | Afraimovich, V. S. and Shil’nikov, L.P., Strange Attractors and Quasiattractors, in Nonlinear Dynamics and Turbulence, G. I. Barenblatt, G. Iooss, D. D. Joseph (Eds.), Interaction Mech. Math. Ser., Boston, MA: Pitman, 1983, pp. 1–34. |
| [37] | Anosov, D.V. and Solodov, V. V., Hyperbolic Sets, in Dynamical Systems – 9, Encyclopaedia Math. Sci., vol. 66, Berlin: Springer, 1995, pp 10–92. |
| [38] | Afraimovich, V. S., Bykov, V.V., and Shil’nikov, L.P., On Attracting Structurally Unstable Limit Sets of Lorenz Attractor Type, Trudy Moskov. Mat. Obshch., 1982, vol. 44, pp. 150–212 [Trans. Mosc. Math. Soc., 1982, vol. 44, pp. 153–216]. · Zbl 0506.58023 |
| [39] | Ruelle, D., Small Random Perturbations of Dynamical Systems and the Definition of Attractors, Comm. Math. Phys., 1981, vol. 82, pp. 137–151. · Zbl 0482.58017 · doi:10.1007/BF01206949 |
| [40] | Auslander, J. and Seibert, P., Prolongations and Stability in Dynamical Systems, Ann. Inst. Fourier (Grenoble), 1964, vol. 14, fasc. 2, pp. 237–267. · Zbl 0128.31303 · doi:10.5802/aif.179 |
| [41] | Gonchenko, A. S., Gonchenko, S.V., Ovsyannikov, I. I., and Turaev, D., Lorenz-Like Attractors in Three-Dimensional Hénon Maps, Math. Model. Nat. Phenom., 2013, vol. 8, no. 5, pp. 80–92. · Zbl 1331.37040 |
| [42] | Afraimovich, V. S. and Shil’nikov, L.P., On invariant two-dimensional tori, their breakdown and stochasticity,Methods of the Qualitative Theory of Differential Equations (Gorky), 1983, pp. 2–26. [English translation in: Amer. Math. Soc. Transl., 149 (1991), pp. 201–212]. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
