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Lyapunov analysis of strange pseudohyperbolic attractors: angles between tangent subspaces, local volume expansion and contraction. (English) Zbl 1412.37045

Summary: Pseudohyperbolic attractors are genuine strange chaotic attractors. They do not contain stable periodic orbits and are robust in the sense that such orbits do not appear under variations. The tangent space of these attractors is split into a direct sum of volume expanding and contracting subspaces and these subspaces never have tangencies with each other. Any contraction in the first subspace, if it occurs, is weaker than contractions in the second one. In this paper we analyze the local structure of several chaotic attractors recently suggested in the literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also, we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well-known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite-time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite-time Lyapunov exponents, we demonstrate that average expanding and contracting properties specific to pseudohyperbolicity are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37D30 Partially hyperbolic systems and dominated splittings
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure

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[1] Kuptsov, P.V. and Politi, A., Large-Deviation Approach to Space-Time Chaos, Phys. Rev. Lett., 2011, vol. 107, no. 11, 114101, 5 pp. · doi:10.1103/PhysRevLett.107.114101
[2] Kuptsov, P. V. and Parlitz, U., Strict and Fussy Mode Splitting in the Tangent Space of the Ginzburg-Landau Equation, Phys. Rev. E, 2010, vol. 81, no. 3, 036214, 6 pp. · doi:10.1103/PhysRevE.81.036214
[3] Kuptsov, P. V. and Kuznetsov, S.P., Numerical Test for Hyperbolicity in Chaotic Systems with Multiple Time Delays, Commun. Nonlinear Sci. Numer. Simul., 2018, vol. 56, Suppl.C, pp. 227-239. · Zbl 1510.65131 · doi:10.1016/j.cnsns.2017.08.016
[4] Yang, H. L., Takeuchi, K.A., Ginelli, F., Chaté, H., and Radons, G., Hyperbolicity and the Effective Dimension of Spatially Extended Dissipative Systems, Phys. Rev. Lett., 2009, vol. 102, no. 7, 074102, 4 pp. · doi:10.1103/PhysRevLett.102.074102
[5] Takeuchi, K.A., Yang, H. L., Ginelli, F., Radons, G., and Chaté, H., Hyperbolic Decoupling of Tangent Space and Effective Dimension of Dissipative Systems, Phys. Rev. E, 2011, vol. 84, no. 4, 046214, 19 pp. · doi:10.1103/PhysRevE.84.046214
[6] Sprott, J. C., Elegant Chaos: Algebraically Simple Chaotic Flows, Singapore: World Sci., 2010. · Zbl 1222.37005 · doi:10.1142/7183
[7] Bonatti, Ch., Díaz, L. J., and Viana, M., Dynamics beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia Math. Sci., vol. 102, Berlin: Springer-Verlag, 2005. · Zbl 1060.37020
[8] Gonchenko, A. S. and Gonchenko, S.V., Variety of Strange Pseudohyperbolic Attractors in Three-Dimensional Generalized Hénon Maps, Phys. D, 2016, vol. 337, pp. 43-57. · Zbl 1376.37047 · doi:10.1016/j.physd.2016.07.006
[9] Benettin, G., Galgani, L., Giorgilli, A., and 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, 1980, vol. 15, no. 1, pp. 9-20. · Zbl 0488.70015 · doi:10.1007/BF02128236
[10] Ginelli, F., Poggi, P., Turchi, A., Chaté, H., Livi, R., and Politi, A., Characterizing Dynamics with Covariant Lyapunov Vectors, Phys. Rev. Lett., 2007, vol. 99, no. 13, 130601, 4 pp. · doi:10.1103/PhysRevLett.99.130601
[11] Golub, G.H. and Van Loan, Ch. F., Matrix Computations, 4th ed., Baltimore,Md.: Johns Hopkins Univ., 2013. · Zbl 1268.65037
[12] Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., vol. 54, Cambridge: Cambridge Univ. Press, 1995. · Zbl 0878.58020 · doi:10.1017/CBO9780511809187
[13] Kuptsov, P.V., Fast Numerical Test of Hyperbolic Chaos, Phys. Rev. E, 2012, vol. 85, no. 1, 015203(R), 4 pp.
[14] Kuptsov, P.V. and Kuznetsov, S.P., Numerical Test for Hyperbolicity of Chaotic Dynamics in Time-Delay Systems, Phys. Rev. E, 2016, vol. 94, no. 1, 010201(R), 7 pp.
[15] Kuptsov, P. V. and Parlitz, U., Theory and Computation of Covariant Lyapunov Vectors, J. Nonlinear Sci., 2012, vol. 22, no. 5, pp. 727-762. · Zbl 1301.37065 · doi:10.1007/s00332-012-9126-5
[16] Kuznetsov, S.P., Hyperbolic Chaos: A Physicist’s View, Berlin: Springer, 2012. · Zbl 1239.37002 · doi:10.1007/978-3-642-23666-2
[17] Kuznetsov, S.P., Dynamical Chaos and Uniformly Hyperbolic Attractors: From Mathematics to Physics, Phys. Uspekhi, 2011, vol. 54, no. 2, pp. 119-144; see also: Uspekhi Fiz. Nauk, 2011, vol. 181, no. 2, pp. 121-149. · doi:10.3367/UFNe.0181.201102a.0121
[18] Legras, B.; Vautard, R.; Palmer, T. (ed.), A Guide to Lyapunov Vectors, 143-156 (1996), Reading, UK
[19] Shimada, I. and Nagashima, T., A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems, Progr. Theoret. Phys., 1979, vol. 61, no. 6, pp. 1605-1616. · Zbl 1171.34327 · doi:10.1143/PTP.61.1605
[20] Hogben, L., Handbook of Linear Algebra, 2nd ed., Boca Raton, Fla.: CRC, 2017. · Zbl 1284.15001
[21] Pesin, Ya.B., Lectures on Partial Hyperbolicity and Stable Ergodicity, Zur. Lect. Adv. Math., Zürich: EMS, 2004. · Zbl 1098.37024 · doi:10.4171/003
[22] Pikovsky, A. and Politi, A., Lyapunov Exponents: A Tool to Explore Complex Dynamics, Cambridge: Cambridge Univ. Press, 2016. · Zbl 1419.37002 · doi:10.1017/CBO9781139343473
[23] Wolfe, Ch. L. and Samelson, R.M., An Efficient Method for Recovering Lyapunov Vectors from Singular Vectors, Tellus A, 2007, vol. 59, no. 3, pp. 355-366. · doi:10.1111/j.1600-0870.2007.00234.x
[24] Gonchenko, S.V., Gonchenko, A. S., Kazakov, A.O., and Kozlov, A.D., Elements of Contemporary Mathematical Theory of Dynamical Chaos: Part 1. Pseudohyperbolic Attractors, arXiv:1712.04032 (2017). · Zbl 1468.37002
[25] Turaev, D.V. and Shil’nikov, L.P., An Example of a Wild Strange Attractor, Sb. Math., 1998, vol. 189, nos. 1-2, pp. 291-314; see also: Mat. Sb., 1998, vol. 189, no. 2, pp. 137-160. · Zbl 0927.37017 · doi:10.1070/SM1998v189n02ABEH000300
[26] Dmitriev, A. S., Efremova, E.V., Maksimov, N.A., and Panas, A. I., Generation of Chaos, Moscow: Tekhnosfera, 2012 (Russian).
[27] Turaev, D.V. and Shil’nikov, L.P., Pseudohyperbolicity and the Problem of the Periodic Perturbation of Lorenz-Type Attractors, Dokl. Math., 2008, vol. 77, no. 1, pp. 17-21; see also: Dokl. Akad. Nauk, 2008, vol. 418, no. 1, pp. 23-27. · Zbl 1157.37022 · doi:10.1134/S1064562408010055
[28] Gonchenko, A. S., Gonchenko, S.V., Kazakov, A.O., and Kozlov, A.D., Mathematical Theory of Dynamical Chaos and Its Applications: Review. Part 1. Pseudohyperbolic Attractors, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2017, vol. 25, no. 2, pp. 4-36 (Russian).
[29] Lorenz, E.N., Deterministic Nonperiodic Flow, J. Atmospheric Sci., 1963, vol. 20, no. 2, pp. 130-141. · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[30] Sparrow, C., The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, New York: Springer, 1982. · Zbl 0504.58001 · doi:10.1007/978-1-4612-5767-7
[31] Schuster, H.G. and Just, W., Deterministic Chaos: An Introduction, Weinheim: Wiley-VCH, 2005. · Zbl 1094.37001 · doi:10.1002/3527604804
[32] Frøyland, J. and Alfsen, K. H., Lyapunov-Exponent Spectra for the Lorenz Model, Phys. Rev. A, 1984, vol. 29, no. 5, pp. 2928-2931. · doi:10.1103/PhysRevA.29.2928
[33] Kuznetsov, S.P., Dynamical Chaos, 2nd ed., Moscow: Fizmatlit, 2006 (Russian).
[34] Bykov, V. V. and Shil’nikov, L.P., On the Boundaries of the Domain of Existence of the Lorenz Attractor, Selecta Math. Soviet., 1992, vol. 11, no. 4, pp. 375-382.
[35] Rössler, O.E., An Equation for Continuous Chaos, Phys. Lett. A, 1976, vol. 57, no. 5, pp. 397-398. · Zbl 1371.37062 · doi:10.1016/0375-9601(76)90101-8
[36] 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
[37] Hunter, J.D., Matplotlib: A 2D Graphics Environment, Comput. Sci. Eng., 2007, vol. 9, no. 3, pp. 90-95. · doi:10.1109/MCSE.2007.55
[38] Smale, S., Differentiable Dynamical Systems, Bull. Amer. Math. Soc., 1967, vol. 73, no. 6, pp. 747-817. · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1
[39] Dynamical Systems 9: Dynamical Systems with Hyperbolic Behaviour, D.V.Anosov (Ed.), Encyclopaedia Math. Sci., vol. 66, Berlin: Springer, 1995.
[40] Shilnikov, L.P., Shilnikov, A. L., Turaev, D., and Chua, L.O., Methods of Qualitative Theory in Nonlinear Dynamics: Part 2, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 5, River Edge,N.J.: World Sci., 2001. · Zbl 1046.34003
[41] Gonchenko, S.V., Kazakov, A.O., and Turaev, D., Wild Pseudohyperbolic Attractors in a Four-Dimensional Lorenz System, in preparation (2018).
[42] Borisov, A.V., Kazakov, A.O., and Sataev, I.R., Spiral Chaos in the NonholonomicModel of a Chaplygin Top, Regul. Chaotic Dyn., 2016, vol. 21, nos. 7-8, pp. 939-954. · Zbl 1378.37112 · doi:10.1134/S1560354716070157
[43] Aston, Ph. J. and Laing, C. R., Symmetry and Chaos in the Complex Ginzburg-Landau Equation: 1. Reflectional Symmetries, Dynam. Stabil. Syst., 1999, vol. 14, no. 3, pp. 233-253. · Zbl 0936.35025 · doi:10.1080/026811199281985
[44] Aston, Ph. J. and Laing, C. R., Symmetry and Chaos in the Complex Ginzburg-Landau Equation: 2. Translational Symmetries, Phys. D, 2000, vol. 135, no. 1, pp. 79-97. · Zbl 0941.35102 · doi:10.1016/S0167-2789(99)00046-9
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