On Local Lyapunov Exponents of Chaotic Hamiltonian Systems

• Autorzy: Hofmann, T. , Merker, J.
• Języki publikacji: EN

Abstrakty

EN

Chaos in conservative systems, particularly in Hamiltonian systems, is different from chaos in dissipative systems. For example, not only the eigenvalues of the symmetric Jacobian, but also the global Lyapunov exponents of Hamiltonian systems occur in pairs (λ;-λ). In this article, we even show that appropriately defined local Lyapunov exponents occur in pairs, and in turn this allows to give a new and easily accessible proof of the pairing property for global Lyapunov exponents. As examples of low dimensional chaotic Hamiltonian systems, we discuss the classical Hénon-Heiles system and a sixth order generalisation. For the latter, there is numerical evidence of two disjoint chaotic seas.

Słowa kluczowe

EN

Lyapunov exponents; Hamiltonian systems; chaos; symplectic; Iwasawa decomposition

• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2018
• Tom: Vol. 24, No. 2
• Strony: 97--111
• Opis fizyczny: Bibliogr. 16 poz., rys.

Twórcy

autor

Hofmann, T.

• Faculty of Computer Science, Mathematics and Natural Sciences PF 30 11 66, 04251 Leipzig, Germany

autor

Merker, J.

• Faculty of Computer Science, Mathematics and Natural Sciences PF 30 11 66, 04251 Leipzig, Germany,

Bibliografia

• [1] V.I. Arnold , Mathematical Methods of Classical Mechanics,2nd Ed., Springer, New York 1980.
• [2] G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Part 1: Theory, Meccanica 15(1), 9–20 (1980).
• [3] L. Barreira, Ya. Pesin, Lectures on Lyapunov exponents and smooth ergodic theory, in: Proceedings of symposia in pure mathematics, AMS, Providence, RI, 3–90 (2001).
• [4] M. Benzi, N. Razouk, On the Iwasawa decomposition of a symplectic matrix, Applied Mathematics Letters 20, 260–265 (2007).
• [5] L.A. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos 11(4), 802–808 (2001).
• [6] Farouk Cherif, Theoretical Computation of Lyapunov Exponents for Almost Periodic Hamiltonian Systems, IAENG International Journal of Applied Mathematics 41:1, IJAM_41_1_02 (2011).
• [7] J.P. Eckmann, D. Ruelle, Ergodic theory of chaos and strange attractors, Reviews of Modern Physics 57, 617–656 (1985).
• [8] F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, A. Politi, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett. 99, 130601 (2007).
• [9] Wm.G. Hoover, C.G. Hoover, Instantaneous Pairing of Lyapunov Exponents in Chaotic Hamiltonian Dynamics and the 2017 Ian Snook Prizes, CMST 23(1), 73–79 (2017).
• [10] J.E. Marsden, T.S. Ratiu, Introduction to mechanics and symmetry Springer, 1994.
• [11] H.A. Posch, Symmetry Properties of Orthogonal and Covariant Lyapunov Vectors and Their Exponents, Journal of Physics A 46, 254006 (2013).
• [12] P. Sawyer, Computing the Iwasawa decomposition of the classical Lie groups of noncompact type using the QR decomposition, Linear Algebra and its Applications 493, 573–579 (2016).
• [13] I.I. Shevchenko, A.V. Mel’nikov, Lyapunov Exponents in the Hénon-Heiles Problem, JETP Letters 77(12), 642–646 (2003).
• [14] W. Li, S. Shi, Non-integrability of Hénon-Heiles system, Celest. Mech. Dyn. Astr. 109, 1–12 (2011).
• [15] L. Zachilas, A review study of the 3-particle Toda lattice and higher-order truncations: The odd-order cases (part I), International Journal of Bifurcation and Chaos 20(10), 3007–3064 (2010).
• [16] L. Zachilas, A review study of the 3-particle Toda lattice and higher-order truncations: The even-order cases (part II), International Journal of Bifurcation and Chaos 20(11), 3391–3441 (2010).

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).

Identyfikatory

DOI

10.12921/cmst.2017.0000053