On the Inclusion of Analytic Symplectic Maps in Analytic Hamiltonian Flows and Its Applications

PROBING THE LOCAL DYNAMICS OF PERIODIC ORBITS BY THE GENERALIZED ALIGNMENT INDEX (GALI) METHOD

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412502185 ◽

2012 ◽

Vol 22 (09) ◽

pp. 1250218 ◽

Cited By ~ 24

Author(s):

T. MANOS ◽

CH. SKOKOS ◽

CH. ANTONOPOULOS

Keyword(s):

Periodic Orbits ◽

Nonlinear Dynamical Systems ◽

Power Laws ◽

Oscillatory Behavior ◽

Unstable Periodic Orbits ◽

Symplectic Maps ◽

Local Dynamics ◽

Hamiltonian Flows ◽

Homoclinic Tangle ◽

Stable Periodic

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper, we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits, the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around nonzero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using the components of deviation vectors orthogonal to the direction of motion for the computation of GALIs, the indices of stable periodic orbits behave for flows as they do for maps.

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Transport Barriers in Symplectic Maps

Brazilian Journal of Physics ◽

10.1007/s13538-021-00894-8 ◽

2021 ◽

Vol 51 (3) ◽

pp. 899-909

Author(s):

R. L. Viana ◽

I. L. Caldas ◽

J. D. Szezech ◽

A. M. Batista ◽

C. V. Abud ◽

...

Keyword(s):

Symplectic Maps ◽

Transport Barriers

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Gaussian Model for Chaotic Instability of Hamiltonian Flows

Physical Review Letters ◽

10.1103/physrevlett.74.375 ◽

1995 ◽

Vol 74 (3) ◽

pp. 375-378 ◽

Cited By ~ 88

Author(s):

Lapo Casetti ◽

Roberto Livi ◽

Marco Pettini

Keyword(s):

Gaussian Model ◽

Hamiltonian Flows

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Internal Arnold diffusion and chaos thresholds in coupled symplectic maps

Journal of Physics A General Physics ◽

10.1088/0305-4470/21/3/003 ◽

1988 ◽

Vol 21 (3) ◽

pp. L127-L133 ◽

Cited By ~ 31

Author(s):

H Kantz ◽

P Grassberger

Keyword(s):

Arnold Diffusion ◽

Symplectic Maps

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A Direct Proof of the Nekhoroshev Theorem for Nearly Integrable Symplectic Maps

Annales Henri Poincaré ◽

10.1007/s00023-004-0188-2 ◽

2004 ◽

Vol 5 (6) ◽

pp. 1013-1039 ◽

Cited By ~ 18

Author(s):

Massimiliano Guzzo

Keyword(s):

Direct Proof ◽

Symplectic Maps ◽

Nekhoroshev Theorem

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Analysis of resonances in action space for symplectic maps

Physical Review E ◽

10.1103/physreve.57.1178 ◽

1998 ◽

Vol 57 (1) ◽

pp. 1178-1180 ◽

Cited By ~ 4

Author(s):

A. Bazzani ◽

L. Bongini ◽

G. Turchetti

Keyword(s):

Action Space ◽

Symplectic Maps

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Chain recurrence, growth rates and ergodic limits

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385707000363 ◽

2007 ◽

Vol 27 (5) ◽

pp. 1509-1524 ◽

Cited By ~ 5

Author(s):

FRITZ COLONIUS ◽

ROBERTA FABBRI ◽

RUSSELL JOHNSON

Keyword(s):

Lyapunov Exponents ◽

Growth Rates ◽

General Linear ◽

Chain Recurrence ◽

Hamiltonian Flows ◽

Rotation Numbers ◽

Linear Flows

AbstractAverages of functionals along trajectories are studied by evaluating the averages along chains. This yields results for the possible limits and, in particular, for ergodic limits. Applications to Lyapunov exponents and to concepts of rotation numbers of linear Hamiltonian flows and of general linear flows are given.

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