TY - JOUR

T1 - Geometrical properties of local dynamics in Hamiltonian systems

T2 - The Generalized Alignment Index (GALI) method

AU - Skokos, Ch

AU - Bountis, T. C.

AU - Antonopoulos, Ch

N1 - Funding Information:
This work was partially supported by the European Social Fund (ESF), Operational Program for Educational and Vocational Training II (EPEAEK II) and particularly the Programs HERAKLEITOS, providing a Ph. D. scholarship for the third author (C. A.) and the Program PYTHAGORAS II, partially supporting the first author (Ch. S.). Ch. S. was also supported by the Marie Curie Intra-European Fellowship No MEIF–CT–2006–025678. The second author (T. B.) wishes to express his gratitude to the beautiful Centro Internacional de Ciencias of the Universidad Autonoma de Mexico for its excellent hospitality during his visit in January–February 2006, when some of this work was completed. In particular, T. B. wants to thank the main researchers of this Center, Dr. Christof Jung and Thomas Seligman, for numerous conversations on the stability of multi-dimensional Hamiltonian systems. Finally, we would like to thank the referees for very useful comments which helped us improve the clarity of the paper.

PY - 2007/7/1

Y1 - 2007/7/1

N2 - We investigate the detailed dynamics of multi-dimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on N-dimensional tori. More specifically we introduce the Generalized Alignment Index of order k (GALIk) as the volume of a generalized parallelepiped, whose edges are k initially linearly independent unit deviation vectors with respect to the orbit studied whose magnitude is normalized to unity at every time step. We show analytically and verify numerically on particular examples of N-degree-of-freedom Hamiltonian systems that, for chaotic orbits, GALIk tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, GALIk fluctuates around non-zero values for 2 ≤ k ≤ N and goes to zero for N < k ≤ 2 N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus: ∝ t- 2 (k - N) + m if 0 ≤ m < k - N, and ∝ t- (k - N) if m ≥ k - N. The GALIk is a generalization of the Smaller Alignment Index (SALI) (GALI2 ∝ SALI). However, GALIk provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.

AB - We investigate the detailed dynamics of multi-dimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on N-dimensional tori. More specifically we introduce the Generalized Alignment Index of order k (GALIk) as the volume of a generalized parallelepiped, whose edges are k initially linearly independent unit deviation vectors with respect to the orbit studied whose magnitude is normalized to unity at every time step. We show analytically and verify numerically on particular examples of N-degree-of-freedom Hamiltonian systems that, for chaotic orbits, GALIk tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, GALIk fluctuates around non-zero values for 2 ≤ k ≤ N and goes to zero for N < k ≤ 2 N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus: ∝ t- 2 (k - N) + m if 0 ≤ m < k - N, and ∝ t- (k - N) if m ≥ k - N. The GALIk is a generalization of the Smaller Alignment Index (SALI) (GALI2 ∝ SALI). However, GALIk provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.

KW - Chaos detection methods

KW - Chaotic motion

KW - Hamiltonian systems

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U2 - 10.1016/j.physd.2007.04.004

DO - 10.1016/j.physd.2007.04.004

M3 - Article

AN - SCOPUS:34250003484

VL - 231

SP - 30

EP - 54

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1

ER -