Numerical study of discrete Lorenz-like attractors. (English) Zbl 07851472
Summary: We consider a homotopic to the identity family of maps, obtained as a discretization of the Lorenz system, such that the dynamics of the last is recovered as a limit dynamics when the discretization parameter tends to zero. We investigate the structure of the discrete Lorenz-like attractors that the map shows for different values of parameters. In particular, we check the pseudohyperbolicity of the observed discrete attractors and show how to use interpolating vector fields to compute kneading diagrams for near-identity maps. For larger discretization parameter values, the map exhibits what appears to be genuinely-discrete Lorenz-like attractors, that is, discrete chaotic pseudohyperbolic attractors with a negative second Lyapunov exponent. The numerical methods used are general enough to be adapted for arbitrary near-identity discrete systems with similar phase space structure.
MSC:
| 37M22 | Computational methods for attractors of dynamical systems |
| 37G35 | Dynamical aspects of attractors and their bifurcations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
Software:
MATCONTReferences:
| [1] | Afraimovich, V. S.; Bykov, V. V.; Shilnikov, L. P., The Origin and Structure of the Lorenz Attractor, Dokl. Akad. Nauk SSSR, 234, 2, 336-339, 1977 |
| [2] | Afraimovich, V. S.; Bykov, V. V.; Shil’nikov, L. P., On Attracting Structurally Unstable Limit Sets of Lorenz Attractor Type, Trans. Mosc. Math. Soc., 44, 153-216, 1982 · Zbl 0527.58024 |
| [3] | Barrio, R.; Blesa, F.; Serrano, S.; Shilnikov, A., Global Organization of Spiral Structures in Biparameter Space of Dissipative Systems with Shilnikov Saddle-Foci, Phys. Rev. E, 84, 3, 2011 · doi:10.1103/PhysRevE.84.035201 |
| [4] | Barrio, R.; Shilnikov, A.; Shilnikov, L., Kneadings, Symbolic Dynamics and Painting Lorenz Chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22, 4, 2012 · Zbl 1258.37035 · doi:10.1142/S0218127412300169 |
| [5] | Barros, D., Bonatti, Ch., and Pacifico, M. J., Up, Down, Two-Sided Lorenz Attractor, Collisions, Merging and Switching, arXiv:2101.07391 (2021). |
| [6] | Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.-M., Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1: Theory, Meccanica, 15, 1, 9-20, 1980 · Zbl 0488.70015 · doi:10.1007/BF02128236 |
| [7] | Bykov, V. V., On the Generation of a Non-Trivial Hyperbolic Set from a Contour Formed by Separatrices of Saddles, Methods of Qualitative Theory of Differential Equations, 22-32, 1988, Gorky: Gorky Gos. Univ., Gorky |
| [8] | Bykov, V. V., The Bifurcations of Separatrix Contours and Chaos, Phys. D, 62, 1-4, 290-299, 1993 · Zbl 0799.58054 · doi:10.1016/0167-2789(93)90288-C |
| [9] | Bykov, V. V., The Generation of Periodic Motions from the Separatrix Contour of a Three-Dimensional System, Uspekhi Mat. Nauk, 32, 6-198, 213-214, 1977 · Zbl 0376.34026 |
| [10] | Bykov, V. V.; Shilnikov, A. L.; al., L. P. Shil’nikov et, On the Boundaries of the Domain of Existence of the Lorenz Attractor, Methods of Qualitative Theory and Theory of Bifurcations, 151-159, 1989, Gorky: Gorky Gos. Univ., Gorky · Zbl 0803.34049 |
| [11] | Bykov, V. V.; Shilnikov, A. L., On the Boundaries of the Domain of Existence of the Lorenz Attractor, Selecta Math. Soviet., 11, 4, 375-382, 1992 · Zbl 0789.58052 |
| [12] | Capiński, M. J.; Turaev, D.; Zgliczyński, P., Computer Assisted Proof of the Existence of the Lorenz Attractor in the Shimizu - Morioka System, Nonlinearity, 31, 12, 5410-5440, 2018 · Zbl 1405.34048 · doi:10.1088/1361-6544/aae032 |
| [13] | Creaser, J. L.; Krauskopf, B.; Osinga, H. M., Finding First Foliation Tangencies in the Lorenz System, SIAM J. Appl. Dyn. Syst., 16, 4, 2127-2164, 2017 · Zbl 1381.37097 · doi:10.1137/17M1112716 |
| [14] | Dhooge, A.; Govaerts, W.; Kuznetsov, Yu. A.; Meijer, H. G. E.; Sautois, B., New Features of the Software MatCont for Bifurcation Analysis of Dynamical Systems, Math. Comput. Model. Dyn. Syst., 14, 2, 147-175, 2008 · Zbl 1158.34302 · doi:10.1080/13873950701742754 |
| [15] | Eilertsen, J.; Magnan, J., On the Chaotic Dynamics Associated with the Center Manifold Equations of Double-Diffusive Convection near a Codimension-Four Bifurcation Point at Moderate Thermal Rayleigh Number, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28, 8, 2018 · Zbl 1395.37023 · doi:10.1142/S0218127418500943 |
| [16] | Eilertsen, J. S.; Magnan, J. F., Asymptotically Exact Codimension-Four Dynamics and Bifurcations in Two-Dimensional Thermosolutal Convection at High Thermal Rayleigh Number: Chaos from a Quasi-Periodic Homoclinic Explosion and Quasi-Periodic Intermittency, Phys. D, 382/383, 1-21, 2018 · Zbl 1415.37046 · doi:10.1016/j.physd.2018.06.004 |
| [17] | Gavrilov, N. K.; Shilnikov, L. P., On Three-Dimensional Dynamical Systems Close to Systems with a Structurally Unstable Homoclinic Curve: 1, Math. USSR-Sb., 17, 4, 467-485, 1972 · Zbl 0255.58006 · doi:10.1070/SM1972v017n04ABEH001597 |
| [18] | Gavrilov, N. K.; Shilnikov, L. P., On Three-Dimensional Dynamical Systems Close to Systems with a Structurally Unstable Homoclinic Curve: 2, Math. USSR-Sb., 19, 1, 139-156, 1973 · Zbl 0273.58009 · doi:10.1070/SM1973v019n01ABEH001741 |
| [19] | Gelfreich, V.; Vieiro, A., Interpolating Vector Fields for Near Identity Maps and Averaging, Nonlinearity, 31, 9, 4263-4289, 2018 · Zbl 1406.37059 · doi:10.1088/1361-6544/aacb8e |
| [20] | Glendinning, P.; Sparrow, C., \(T\)-Points: A Codimension Two Heteroclinic Bifurcation, J. Statist. Phys., 43, 3-4, 479-488, 1986 · Zbl 0635.58031 · doi:10.1007/BF01020649 |
| [21] | Glendinning, P.; Sparrow, C., Prime and Renormalisable Kneading Invariants and the Dynamics of Expanding Lorenz Maps, Phys. D, 62, 1-4, 22-50, 1993 · Zbl 0783.58046 · doi:10.1016/0167-2789(93)90270-B |
| [22] | Gonchenko, A. S.; Gonchenko, S. V.; Kazakov, A. O., Richness of Chaotic Dynamics in the Nonholonomic Model of Celtic Stone, Regul. Chaotic Dyn., 18, 5, 521-538, 2013 · Zbl 1417.37222 · doi:10.1134/S1560354713050055 |
| [23] | Gonchenko, A. S.; Gonchenko, S. V.; Kazakov, A. O.; Kozlov, A. D., Elements of Contemporary Theory of Dynamical Chaos: A Tutorial: Part 1. Pseudohyperbolic Attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28, 11, 2018 · Zbl 1468.37002 · doi:10.1142/S0218127418300367 |
| [24] | Gonchenko, A. S.; Samylina, E. A., On the Region of Existence of a Discrete Lorenz Attractor in the Nonholonomic Model of a Celtic Stone, Radiophys. Quantum El., 62, 5, 369-384, 2019 · doi:10.1007/s11141-019-09984-9 |
| [25] | Gonchenko, S. V.; Kazakov, A. O.; Turaev, D. V.; Kaynov, M. N., On Methods for Verification of the Pseudohyperbolicity of Strange Attractors, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 29, 1, 160-185, 2021 |
| [26] | Gonchenko, S. V.; Shilnikov, L. P.; Turaev, D. V., Dynamical Phenomena in Multidimensional Systems with a Structurally Unstable Homoclinic Poincaré Curve, Russian Acad. Sci. Dokl. Math., 47, 3, 410-415, 1993 · Zbl 0864.58043 |
| [27] | Gonchenko, S. V.; Turaev, D. V.; Shil’nikov, L. P., On the Existence of Newhouse Regions in a Neighborhood of Systems with a Structurally Unstable Homoclinic Poincaré Curve (the Multidimensional Case), Dokl. Math., 47, 2, 268-273, 1993 · Zbl 0823.58030 |
| [28] | Gonchenko, S.; Gonchenko, A., On Discrete Lorenz-Like Attractors in Three-Dimensional Maps with Axial Symmetry, Chaos, 33, 12, 2023 · Zbl 07863409 · doi:10.1063/5.0172243 |
| [29] | Gonchenko, S.; Gonchenko, A.; Kazakov, A.; Samylina, E., On Discrete Lorenz-Like Attractors, Chaos, 31, 2, 2021 · Zbl 1465.37047 · doi:10.1063/5.0037621 |
| [30] | Gonchenko, S.; Karatetskaia, E.; Kazakov, A.; Kruglov, V., Conjoined Lorenz Twins: A New Pseudohyperbolic Attractor in Three-Dimensional Maps and Flows, Chaos, 32, 12, 2022 · Zbl 07880293 · doi:10.1063/5.0123426 |
| [31] | Gonchenko, S. V.; Kazakov, A. O.; Turaev, D. V., Wild Pseudohyperbolic Attractor in a Four-Dimensional Lorenz System, Nonlinearity, 34, 4, 2018-2047, 2021 · Zbl 1472.34106 · doi:10.1088/1361-6544/abc794 |
| [32] | Gonchenko, S. V.; Ovsyannikov, I. I.; Simó, C.; Turaev, D., Three-Dimensional Hénon-Like Maps and Wild Lorenz-Like Attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15, 11, 3493-3508, 2005 · Zbl 1097.37023 · doi:10.1142/S0218127405014180 |
| [33] | Gonchenko, S. V.; Shilnikov, L. P.; Turaev, D. V., On Dynamical Properties of Multidimensional Diffeomorphisms from Newhouse Regions: 1, Nonlinearity, 21, 5, 923-972, 2008 · Zbl 1160.37019 · doi:10.1088/0951-7715/21/5/003 |
| [34] | Gonchenko, S. V.; Gonchenko, A. S.; Ovsyannikov, I. I.; Turaev, D. V., Examples of Lorenz-Like Attractors in Hénon-Like Maps, Math. Model. Nat. Phenom., 8, 5, 48-70, 2013 · Zbl 1331.37040 · doi:10.1051/mmnp/20138504 |
| [35] | Gonchenko, S. V.; Shil’nikov, L. P.; Turaev, D. V., Dynamical Phenomena in Systems with Structurally Unstable Poincaré Homoclinic Orbits, Chaos, 6, 1, 15-31, 1996 · Zbl 1055.37578 · doi:10.1063/1.166154 |
| [36] | Guckenheimer, J.; Marsden, J. E.; McCracken, M., A Strange, Strange Attractor, The Hopf Bifurcation and Its Applications, 368-381, 1976, New York: Springer, New York · Zbl 0346.58007 · doi:10.1007/978-1-4612-6374-6_25 |
| [37] | Guckenheimer, J.; Williams, R. F., Structural Stability of Lorenz Attractors, Inst. Hautes Études Sci. Publ. Math., 50, 59-72, 1979 · Zbl 0436.58018 · doi:10.1007/BF02684769 |
| [38] | Kuptsov, P. V., Fast Numerical Test of Hyperbolic Chaos, Phys. Rev. E, 85, 1, 2012 · doi:10.1103/PhysRevE.85.015203 |
| [39] | Kuptsov, P. V.; Kuznetsov, S. P., Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles between Tangent Subspaces, Local Volume Expansion and Contraction, Regul. Chaotic Dyn., 23, 7-8, 908-932, 2018 · Zbl 1412.37045 · doi:10.1134/S1560354718070079 |
| [40] | Lorenz, E. N., Deterministic Nonperiodic Flow, J. Atmos. Sci., 20, 2, 130-141, 1963 · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |
| [41] | Luzzatto, S.; Tucker, W., Non-Uniformly Expanding Dynamics in Maps with Singularities and Criticalities, Inst. Hautes Études Sci. Publ. Math., 89, 179-226, 1999 · Zbl 0978.37029 · doi:10.1007/BF02698857 |
| [42] | Luzzatto, S.; Viana, M.; Flexor, M.; Sentenac, P.; Yoccoz, J.-Ch., Positive Lyapunov Exponents for Lorenz-Like Families with Criticalities, Géométrie complexe et systèmes dynamiques: Colloque en l’honneur d’Adrien Douady (Orsay, 1995), 201-237, 2000, Paris: Soc. Math. France, Paris · Zbl 0944.37025 |
| [43] | Lyubimov, D. V.; Zaks, M. A., Two Mechanisms of the Transition to Chaos in Finite-Dimensional Models of Convection, Phys. D, 9, 1-2, 52-64, 1983 · Zbl 0598.58029 · doi:10.1016/0167-2789(83)90291-9 |
| [44] | Malkin, M. I., Periodic Orbits, Entropy, and Rotation Sets of Continuous Mappings of the Circle, Ukr. Math. J., 35, 3, 280-285, 1983 · Zbl 0559.54035 · doi:10.1007/BF01092176 |
| [45] | Malkin, M. I., Rotation Intervals and the Dynamics of Lorenz Type Mappings, Methods of the Qualitative Theory of Differential Equations, 122-139, 1986, Gorky: GGU, Gorky |
| [46] | Malkin, M. I., Rotation Intervals and the Dynamics of Lorenz Type Mappings, Selecta Math. Soviet., 10, 3, 265-275, 1991 · Zbl 0732.58025 |
| [47] | Milnor, J. W.; Thurston, W. P., On Iterated Maps of the Interval, 1977, Prinseton: Prinseton Univ. Press, Prinseton |
| [48] | Milnor, J.; Thurston, W., On Iterated Maps of the Interval, Dynamical Systems (College Park, Md., 1986/87), 465-563, 1988, Berlin: Springer, Berlin · Zbl 0664.58015 |
| [49] | Morales, C. A.; Pacifico, M. J.; Pujals, E. R., Robust Transitive Singular Sets for \(3\)-Flows Are Partially Hyperbolic Attractors or Repellers, Ann. of Math. (2), 160, 2, 375-432, 2004 · Zbl 1071.37022 · doi:10.4007/annals.2004.160.375 |
| [50] | Newhouse, Sh. E., The Abundance of Wild Hyperbolic Sets and Nonsmooth Stable Sets for Diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 50, 101-151, 1979 · Zbl 0445.58022 · doi:10.1007/BF02684771 |
| [51] | Ovsyannikov, I. I.; Turaev, D. V., Analytic Proof of the Existence of the Lorenz Attractor in the Extended Lorenz Model, Nonlinearity, 30, 1, 115-137, 2017 · Zbl 1367.34051 · doi:10.1088/1361-6544/30/1/115 |
| [52] | Pusuluri, K.; Meijer, H. G. E.; Shilnikov, A. L., Homoclinic Puzzles and Chaos in a Nonlinear Laser Model, Commun. Nonlinear Sci. Numer. Simul., 93, 2021 · Zbl 1466.78019 · doi:10.1016/j.cnsns.2020.105503 |
| [53] | Pusuluri, K.; Shilnikov, A., Homoclinic Chaos and Its Organization in a Nonlinear Optics Model, Phys. Rev. E, 98, 4, 2018 · doi:10.1103/PhysRevE.98.040202 |
| [54] | Rand, D., The Topological Classification of Lorenz Attractors, Math. Proc. Cambridge Philos. Soc., 83, 3, 451-460, 1978 · Zbl 0375.58015 · doi:10.1017/S0305004100054736 |
| [55] | Rucklidge, A. M., Chaos in Models of Double Convection, J. Fluid Mech., 237, 209-229, 1992 · Zbl 0747.76089 · doi:10.1017/S0022112092003392 |
| [56] | Shilnikov, A. L., Bifurcation and Chaos in the Morioka - Shimizu System, Selecta Math. Soviet., 10, 2, 105-117, 1991 · Zbl 0745.58036 |
| [57] | Shilnikov, A. L.; al., E. A. Leontovich et, Bifurcation and Chaos in the Morioka - Shimizu System, Methods of Qualitative Theory of Differential Equations, 180-193, 1986, Gorky: Gorky Gos. Univ., Gorky · Zbl 0723.58001 |
| [58] | Shil’nikov, A. L.; Shil’nikov, L. P.; Turaev, D. V., Normal Forms and Lorenz Attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 3, 5, 1123-1139, 1993 · Zbl 0885.58080 · doi:10.1142/S0218127493000933 |
| [59] | Shilnikov, A. L., On Bifurcations of the Lorenz Attractor in the Shimizu - Morioka Model, Phys. D, 62, 1-4, 338-346, 1993 · Zbl 0783.58052 · doi:10.1016/0167-2789(93)90292-9 |
| [60] | Shilnikov, L. P., Bifurcation Theory and the Lorenz Model, J. Marsden and M. McCraken Bifurcation of Cycle Birth and Its Applications, 317-335, 1980, Moscow: Mir, Moscow |
| [61] | Tatjer, J. C., Three-Dimensional Dissipative Diffeomorphisms with Homoclinic Tangencies, Ergodic Theory Dynam. Systems, 21, 1, 249-302, 2001 · Zbl 0972.37013 · doi:10.1017/S0143385701001146 |
| [62] | Tucker, W., The Lorenz Attractor Exists, C. R. Acad. Sci. Paris Sér. 1 Math., 328, 12, 1197-1202, 1999 · Zbl 0935.34050 · doi:10.1016/S0764-4442(99)80439-X |
| [63] | Turaev, D. V.; Shil’nikov, L. P., An Example of a Wild Strange Attractor, Sb. Math., 189, 1-2, 291-314, 1998 · Zbl 0927.37017 · doi:10.1070/SM1998v189n02ABEH000300 |
| [64] | Turaev, D. V.; Shil’nikov, L. P., Pseudohyperbolicity and the Problem of the Periodic Perturbation of Lorenz-Type Attractors, Dokl. Math., 77, 1, 17-21, 2008 · Zbl 1157.37022 · doi:10.1134/S1064562408010055 |
| [65] | Williams, R., The Structure of Lorenz Attractors, Inst. Hautes Études Sci. Publ. Math., 50, 73-99, 1979 · Zbl 0484.58021 · doi:10.1007/BF02684770 |
| [66] | Xing, T.; Barrio, R.; Shilnikov, A., Symbolic Quest into Homoclinic Chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24, 8, 2014 · Zbl 1300.34101 · doi:10.1142/S0218127414400045 |
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.
