[1] |
Afraimovich, V. S., Bykov, V., and Shilnikov, L. P., “On the origin and structure of the Lorenz attractor,” in Akademiia Nauk SSSR Doklady (Nauka, 1977), Vol. 234, pp. 336-339. · Zbl 0451.76052 |
[2] |
Afraimovich, V.; Bykov, V.; Shil’nikov, L., On structurally unstable attracting limit sets of Lorenz attractor type, Trans. Moscow Math. Soc., 44, 153-216 (1983) · Zbl 0527.58024 |
[3] |
Kuznetsov, S.; Kruglov, V.; Sataev, I., Smale-Williams solenoids in autonomous system with saddle equilibrium, Chaos, 31, 013140 (2021) · Zbl 1465.37035 · doi:10.1063/5.0028921 |
[4] |
Lyubimov, D.; Zaks, M., Two mechanisms of the transition to chaos in finite-dimensional models of convection, Phys. D, 9, 52-64 (1983) · Zbl 0598.58029 · doi:10.1016/0167-2789(83)90291-9 |
[5] |
Rovella, A., The dynamics of perturbations of the contracting Lorenz attractor, Braz. Math. Soc., 24, 233-259 (1993) · Zbl 0797.58051 · doi:10.1007/BF01237679 |
[6] |
Kazakov, A., On bifurcations of Lorenz attractors in the Lyubimov-Zaks model, Chaos, 31, 093118 (2021) · Zbl 07866711 · doi:10.1063/5.0058585 |
[7] |
Kuznetsov, S. P., Example of a physical system with a hyperbolic attractor of the Smale-Williams type, Phys. Rev. Lett., 95, 144101 (2005) · doi:10.1103/PhysRevLett.95.144101 |
[8] |
Kuznetsov, S. P.; Pikovsky, A., Autonomous coupled oscillators with hyperbolic strange attractors, Phys. D, 232, 87-102 (2007) · Zbl 1128.34024 · doi:10.1016/j.physd.2007.05.008 |
[9] |
Kruglov, V. P.; Kuznetsov, S. P.; Pikovsky, A., Attractor of Smale-Williams type in an autonomous distributed system, Regul. Chaotic Dyn., 19, 483-494 (2014) · Zbl 1335.37014 · doi:10.1134/S1560354714040042 |
[10] |
Kuznetsov, S.; Kruglov, V.; Sedova, Y., Mechanical systems with hyperbolic chaotic attractors based on Froude pendulums, Russ. J. Nonlinear Dyn., 16, 51-58 (2020) · Zbl 1447.37051 · doi:10.20537/nd200105 |
[11] |
Kuznetsov, S. P., Hyperbolic Chaos: A Physicist’s View (2012), Higher Education Press/Springer: Higher Education Press/Springer, Beijing · Zbl 1239.37002 |
[12] |
Vladimirov, A.; Toronov, V. Y.; Derbov, V., Properties of the phase space and bifurcations in the complex Lorenz model, Tech. Phys., 43, 877-884 (1998) · doi:10.1134/1.1259094 |
[13] |
Vladimirov, A.; Toronov, V. Y.; Derbov, V. L., The complex Lorenz model: Geometric structure, homoclinic bifurcation and one-dimensional map, Int. J. Bifurc. Chaos, 8, 723-729 (1998) · Zbl 1109.37303 · doi:10.1142/S0218127498000516 |
[14] |
Ning, C.-Z.; Haken, H., Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations, Phys. Rev. A, 41, 3826 (1990) · doi:10.1103/PhysRevA.41.3826 |
[15] |
Gibbon, J.; McGuinness, M., The real and complex Lorenz equations in rotating fluids and lasers, Phys. D, 5, 108-122 (1982) · Zbl 1194.76280 · doi:10.1016/0167-2789(82)90053-7 |
[16] |
Fowler, A.; Gibbon, J.; McGuinness, M., The complex Lorenz equations, Phys. D, 4, 139-163 (1982) · Zbl 1194.37039 · doi:10.1016/0167-2789(82)90057-4 |
[17] |
Gibbon, J.; McGuinness, M., A derivation of the Lorenz equations for some unstable dispersive physical systems, Phys. Lett. A, 77, 295-299 (1980) · doi:10.1016/0375-9601(80)90700-8 |
[18] |
Fowler, A.; Gibbon, J.; McGuinness, M., The real and complex Lorenz equations and their relevance to physical systems, Physica D, 7, 126-134 (1983) · Zbl 1194.76087 · doi:10.1016/0167-2789(83)90123-9 |
[19] |
Lorenz, E. N., Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141 (1963) · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |
[20] |
Shimizu, T.; Morioka, N., On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model, Phys. Lett. A, 76, 201-204 (1980) · doi:10.1016/0375-9601(80)90466-1 |
[21] |
Shil’nikov, A. L., On bifurcations of the Lorenz attractor in the Shimizu-Morioka model, Phys. D, 62, 338-346 (1993) · Zbl 0783.58052 · doi:10.1016/0167-2789(93)90292-9 |
[22] |
Shil’Nikov, A.; Shil’Nikov, L.; Turaev, D., Normal forms and Lorenz attractors, Int. J. Bifurc. Chaos, 3, 1123-1139 (1993) · Zbl 0885.58080 · doi:10.1142/S0218127493000933 |
[23] |
Anosov, D. V., “Dynamical systems in the 1960s: The hyperbolic revolution,” in Mathematical Events of the Twentieth Century (Springer, 2006), pp. 1-17. · Zbl 1096.01006 |
[24] |
Smale, S., Differentiable dynamical systems, Bull. Am. Math. Soc., 73, 747-817 (1967) · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1 |
[25] |
Anosov, D.; Gould, G.; Aranson, S.; Grines, V.; Plykin, R.; Safonov, A.; Sataev, E.; Shlyachkov, S.; Solodov, V.; Starkov, A., Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour (1995), Springer |
[26] |
Katok, A.; Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems (1997), Cambridge University Press · Zbl 0878.58019 |
[27] |
Shilnikov, L. P., Mathematical problems of nonlinear dynamics: A tutorial, Int. J. Bifurc. Chaos, 7, 1953-2001 (1997) · Zbl 0909.58008 · doi:10.1142/S0218127497001527 |
[28] |
Kuznetsov, S. P., Dynamical chaos and uniformly hyperbolic attractors: From mathematics to physics, Phys.-Usp., 54, 119 (2011) · doi:10.3367/UFNe.0181.201102a.0121 |
[29] |
Turaev, D. V.; Shil’nikov, L. P., An example of a wild strange attractor, Sb.: Math., 189, 291 (1998) · Zbl 0927.37017 · doi:10.1070/SM1998v189n02ABEH000300 |
[30] |
Gonchenko, A. S.; Gonchenko, S. V.; Kazakov, A. O.; Kozlov, A., Elements of contemporary theory of dynamical chaos: A tutorial. Part I. Pseudohyperbolic attractors, Int. J. Bifurc. Chaos, 28, 1830036 (2018) · Zbl 1468.37002 · doi:10.1142/S0218127418300367 |
[31] |
Turaev, D. and Shil’nikov, L., “Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors,” in Doklady Mathematics (SP MAIK Nauka/Interperiodica, 2008), Vol. 77, pp. 17-21. · Zbl 1157.37022 |
[32] |
Malkin, M., Topological conjugacy of discontinuous maps of a closed interval, Ukr. Math. J., 32, 398-403 (1980) · Zbl 0488.58020 · doi:10.1007/BF01091562 |
[33] |
Malkin, M., “Rotation intervals and dynamics of Lorenz-like maps,” in Methods of Qualitative Theory of Differential Equations (American Mathematical Society, 1985), pp. 122-139. |
[34] |
Gonchenko, S.; Kazakov, A.; Turaev, D., Wild pseudohyperbolic attractor in a four-dimensional Lorenz system, Nonlinearity, 34, 2018 (2021) · Zbl 1472.34106 · doi:10.1088/1361-6544/abc794 |
[35] |
Afraimovich, V. S. and P, S. L., “Strange attractors and quasi-attractors,” in Nonlinear Dynamics and Turbulence, edited by G. I. Barenblatt, G. Iooss, and D. D. Joseph (Pitman, New York, 1983), pp. 1-28. · Zbl 0532.58018 |
[36] |
Williams, R. F., Expanding attractors, Publ. Math. l’Inst. Haut. Étud. Sci., 43, 169-203 (1974) · Zbl 0279.58013 · doi:10.1007/BF02684369 |
[37] |
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, Phys. Lett. A, 365, 97-104 (2007) · Zbl 1203.37043 · doi:10.1016/j.physleta.2006.12.071 |
[38] |
Wilczak, D., Uniformly hyperbolic attractor of the Smale-Williams type for a Poincaré map in the Kuznetsov system, SIAM J. Appl. Dyn. Syst., 9, 1263-1283 (2010) · Zbl 1213.37046 · doi:10.1137/100795176 |
[39] |
Shilnikov, L. P., On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Mat. Sb., 119, 461-472 (1968) · Zbl 0188.15303 · doi:10.1070/SM1968v006n03ABEH001069 |
[40] |
Chua, L. O.; Shilnikov, L. P.; Shilnikov, A. L.; Turaev, D. V., Methods Of Qualitative Theory in Nonlinear Dynamics (Part II) (2001), World Scientific · Zbl 1046.34003 |
[41] |
Tigan, G.; Turaev, D., Analytical search for homoclinic bifurcations in the Shimizu-Morioka model, Phys. D, 240, 985-989 (2011) · Zbl 1218.34046 · doi:10.1016/j.physd.2011.02.013 |
[42] |
Capiński, M. J.; Turaev, D.; Zgliczyński, P., Computer assisted proof of the existence of the Lorenz attractor in the Shimizu-Morioka system, Nonlinearity, 31, 5410 (2018) · Zbl 1405.34048 · doi:10.1088/1361-6544/aae032 |
[43] |
Barrio, R.; Shilnikov, A.; Shilnikov, L., Kneadings, symbolic dynamics and painting Lorenz chaos, Int. J. Bifurc. Chaos, 22, 1230016 (2012) · Zbl 1258.37035 · doi:10.1142/S0218127412300169 |
[44] |
Xing, T.; Barrio, R.; Shilnikov, A., Symbolic quest into homoclinic chaos, Int. J. Bifurc. Chaos, 24, 1440004 (2014) · Zbl 1300.34101 · doi:10.1142/S0218127414400045 |
[45] |
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. Part 1: Theory, Meccanica, 15, 9-20 (1980) · Zbl 0488.70015 · doi:10.1007/BF02128236 |
[46] |
Shimada, I.; Nagashima, T., A numerical approach to ergodic problem of dissipative dynamical systems, Prog. Theor. Phys., 61, 1605-1616 (1979) · Zbl 1171.34327 · doi:10.1143/PTP.61.1605 |
[47] |
Pikovsky, A.; Politi, A., Lyapunov Exponents: A Tool to Explore Complex Dynamics (2016), Cambridge University Press · Zbl 1419.37002 |
[48] |
Kuptsov, P. V., Fast numerical test of hyperbolic chaos, Phys. Rev. E, 85, 015203 (2012) · doi:10.1103/PhysRevE.85.015203 |
[49] |
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, 908-932 (2018) · Zbl 1412.37045 · doi:10.1134/S1560354718070079 |
[50] |
Frederickson, P.; Kaplan, J. L.; Yorke, E. D.; Yorke, J. A., The Liapunov dimension of strange attractors, J. Differ. Equ., 49, 185-207 (1983) · Zbl 0515.34040 · doi:10.1016/0022-0396(83)90011-6 |
[51] |
Lai, Y.-C.; Grebogi, C.; Yorke, J.; Kan, I., How often are chaotic saddles nonhyperbolic?, Nonlinearity, 6, 779 (1993) · Zbl 0785.58035 · doi:10.1088/0951-7715/6/5/007 |
[52] |
Anishchenko, V. S.; Kopeikin, A.; Kurths, J.; Vadivasova, T.; Strelkova, G., Studying hyperbolicity in chaotic systems, Phys. Lett. A, 270, 301-307 (2000) · Zbl 1115.37364 · doi:10.1016/S0375-9601(00)00338-8 |
[53] |
Kuptsov, P. V.; Parlitz, U., Theory and computation of covariant Lyapunov vectors, J. Nonlinear Sci., 22, 727-762 (2012) · Zbl 1301.37065 · doi:10.1007/s00332-012-9126-5 |
[54] |
Belykh, V. N.; Barabash, N. V.; Belykh, I. V., A Lorenz-type attractor in a piecewise-smooth system: Rigorous results, Chaos, 29, 103108 (2019) · Zbl 1425.37023 · doi:10.1063/1.5115789 |
[55] |
Belykh, V. N.; Barabash, N. V.; Belykh, I. V., Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs, Chaos, 31, 043117 (2021) · Zbl 1473.37060 · doi:10.1063/5.0044731 |