On Local Lyapunov Exponents of Chaotic Hamiltonian Systems
Faculty of Computer Science, Mathematics and Natural Sciences
PF 30 11 66, 04251 Leipzig, Germany
∗E-mail: jochen.merker@htwk-leipzig.de
Received:
Received: 31 December 2017; revised: 27 March 2018; accepted: 05 April 2018; published online: 30 May 2018
DOI: 10.12921/cmst.2017.0000053
Abstract:
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.
Key words:
chaos, Hamiltonian systems, Iwasawa decomposition, Lyapunov exponents, symplectic
References:
[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 Ex-
ponents 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 Lya-
punov 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 Covari-
ant 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 decom-
position, 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), Inter-
national 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), Inter-
national Journal of Bifurcation and Chaos 20(11), 3391–3441
(2010).
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.
Key words:
chaos, Hamiltonian systems, Iwasawa decomposition, Lyapunov exponents, symplectic
References:
[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 Ex-
ponents 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 Lya-
punov 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 Covari-
ant 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 decom-
position, 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), Inter-
national 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), Inter-
national Journal of Bifurcation and Chaos 20(11), 3391–3441
(2010).
