In this work, we investigate different timescales of chaotic dynamics in a multi-parametric 4D symplectic map. We compute the Lyapunov time and a macroscopic timescale, the instability time, for a wide range of values of the system’s parameters and many different ensembles of initial conditions in resonant domains. The instability time is obtained by plain numerical simulations and by its estimates from the diffusion time, which we derive in three different ways: through a normal and an anomalous diffusion law and by the Shannon entropy, whose formulation is briefly revisited. A discussion about which of the four approaches provide reliable values of the timescale for a macroscopic instability is addressed. The relationship between the Lyapunov time and the instability time is revisited and studied for this particular system where in some cases, an exponential or polynomial law has been observed. The main conclusion of the present research is that only when the dynamical system behaves as a nearly ergodic one such relationship arises and the Lyapunov and instability times are global timescales, independent of the position in phase space. When stability regions prevent the free diffusion, no correlations between both timescales are observed, they are local and depend on both the position in phase space and the perturbation strength. In any case, the instability time largely exceeds the Lyapunov time. Thus, when the system is far from nearly ergodic, the timescale for predictable dynamics is given by the instability time, being the Lyapunov time its lower bound.

1.
B. V.
Chirikov
, “
A universal instability of many-dimensional oscillator systems
,”
Phys. Rep.
52
,
263
(
1979
).
2.
J. D.
Meiss
, “
Symplectic maps, variational principles, and transport
,”
Rev. Mod. Phys.
64
,
795
(
1992
).
3.
A. J.
Lichtenberg
and
M. A.
Lieberman
,
Regular and Chaotic Dynamics
(
Springer-Verlag
,
New York
,
1992
).
4.
I. I.
Shevchenko
,
Dynamical Chaos in Planetary Systems
(
Springer Nature
,
2020
).
5.
P. M.
Cincotta
,
C. M.
Giordano
,
R.
Alves Silva
, and
C.
Beuagé
, “
The Shannon entropy: An efficient indicator of dynamical stability
,”
Physica D
417
,
132816
(
2021
).
6.
P. M.
Cincotta
,
C. M.
Giordano
,
R.
Alves Silva
, and
C.
Beuagé
, “
Shannon entropy diffusion estimates: Sensitivity on the parameters of the method
,”
Celest. Mech. Dyn. Astron.
133
,
7
(
2021
).
7.
P. M.
Cincotta
and
C. M.
Giordano
, “
Estimation of diffusion time with the Shannon entropy approach
,”
Phys. Rev. E
107
,
064101
(
2023
).
8.
P. M.
Cincotta
and
C. M.
Giordano
, “
Estimation of the diffusion time in a triaxial galactic potential
,”
Mon. Not. R. Astron. Soc.
526
,
895
(
2023
).
9.
C.
Efthymiopoulos
and
M.
Harsoula
, “
The speed of Arnold diffusion
,”
Physica D
251
,
19
(
2013
).
10.
C.
Froeschlé
,
M.
Guzzo
, and
E.
Lega
, “
Local and global diffusion along resonant lines in discrete quasi-integrable dynamical systems
,”
Celest. Mech. Dyn. Astron.
92
,
243
(
2005
).
11.
C.
Froeschlé
,
E.
Lega
, and
M.
Guzzo
, “
Analysis of the chaotic behaviour of orbits diffusing along the Arnold web
,”
Celest. Mech. Dyn. Astron.
95
,
141
(
2006
).
12.
N.
Guillery
and
J. D.
Meiss
, “
Diffusion and drift in volume-preserving maps
,”
Regul. Chaotic Dyn.
22
,
700
(
2017
).
13.
C.
Froechlé
,
M.
Guzzo
, and
E.
Lega
, “
First numerical evidence of global Arnold diffusion in quasi-integrable systems
,”
Discrete Contin. Dyn. Syst. B
5
(3),
687–698
(
2005
).
14.
E.
Lega
,
C.
Froeschlé
, and
M.
Guzzo
, “
Diffusion in Hamiltonian quasi-integrable systems
,” in
Topics in Gravitational Dynamics
, Lecture Notes in Physics Vol. 729, edited by D. Benest, C. Froeschle, and E. Lega (Springer, Berlin, Heidelberg, 2007), pp. 29–65.
15.
E.
Lega
,
M.
Guzzo
, and
C.
Froeschlé
, “
Detection of Arnold diffusion in Hamiltonian systems
,”
Physica D
182
,
179
(
2003
).
16.
J. D.
Meiss
,
N.
Miguel
,
C.
Simó
, and
A.
Vieiro
, “
Accelerator modes and anomalous diffusion in 3D volume-preserving maps
,”
Nonlinearity
31
,
5615
(
2018
).
17.
P. M.
Cincotta
,
C. M.
Giordano
, and
I. I.
Shevchenko
, “
Revisiting the relation between the Lyapunov time and the instability time
,”
Physica D
430
,
133101
(
2022
).
18.
A.
Morbidelli
and
C.
Froeschelé
, “
On the relationship between the Lyapunov times and macroscopic instability times
,”
Celest. Mech. Dyn. Astron.
63
,
227
(
1996
).
19.
A.
Milani
and
A. M.
Nobili
, “
An example of stable chaos in the solar system
,”
Nature
357
,
569
(
1992
).
20.
R. S.
Mackay
,
J. D.
Meiss
, and
I. C.
Percival
, “
Stochasticity and transport in Hamiltonian systems
,”
Phys. Rev. Lett.
52
,
697
(
1984
).
21.
E.
Efthymiopoulos
,
G.
Contopoulos
,
N.
Voglis
, and
R.
Dvorak
, “
Stickiness and cantori
,”
J. Phys. A
30
,
8167
(
1997
).
22.
G.
Contopoulos
and
M.
Harsoula
, “
Stickiness in chaos
,”
J. Bifurcat. Chaos
18
,
2929
(
2008
).
23.
G.
Contopoulos
and
M.
Harsoula
, “
Stickiness effects in conservative systems
,”
Int. J. Bifurcat. Chaos
20
(7), 2005 (2010).
24.
N.
Miguel
,
C.
Simó
, and
A.
Vieiro
, “
Escape times across the golden Cantorus of the standard map
,”
Regul. Chaotic Dyn.
27
,
281
(
2022
).
25.
K. D. N. T.
Lam
and
J.
Kurchan
, “
Stochastic perturbation of integrable systems: A window to weakly chaotic systems
,”
J. Stat. Phys.
156
,
619
(
2014
).
26.
T.
Goldfriend
and
J.
Kurchan
, “
Equilibration of quasi-integrable systems
,”
Phys. Rev. E
99
,
022146
(
2019
).
27.
F.
Mogavero
,
N. H.
Hoang
, and
J.
Laskar
, “
Timescales of chaos in the inner solar system: Lyapunov spectrum and quasi-integrals of motion
,”
Phys. Rev. X
13
,
021018
(
2023
).
28.
V.
Gelfreich
,
C.
Simó
, and
A.
Vieiro
, “
Dynamics of 4D symplectic maps near a double resonance
,”
Physica D
243
,
92
(
2013
).
29.
P. M.
Cincotta
,
C. M.
Giordano
, and
I. I.
Shevchenko
, “
Diffusion and Lyapunov timescales in the Arnold model
,”
Phys. Rev. E
106
,
044205
(
2022
).
30.
V. I.
Arnold
, “
Instability of dynamical systems with many degrees of freedom
,”
Dokl. Akad. Nauk SSSR
156
(1),
9–12
(
1964
), https://www.mathnet.ru/eng/dan29524.
31.
P. M.
Cincotta
,
C. M.
Giordano
,
J. G.
Martí
, and
C.
Beaugé
, “
On the chaotic diffusion in multidimensional Hamiltonian systems
,”
Celest. Mech. Dyn. Astron.
130
,
Article ID 7, 23 pp.
(
2018
).
32.
C. M.
Giordano
and
P. M.
Cincotta
, “
The Shannon entropy as a measure of diffusion in multidimensional dynamical systems
,”
Celest. Mech. Dyn. Astron.
130
, Article ID. N ° 35 (
2018
).
33.
E.
Kövári
,
B.
Érdi
, and
Z.
Sándor
, “
Application of the Shannon entropy in the planar (non-restricted) four-body problem: The long-term stability of the Kepler-60 exoplanetary system
,”
Mon. Not. R. Astron. Soc.
509
(1),
884–893
(
2022
).
34.
R.
Alves Silva
,
C.
Beuagé
,
S.
Ferraz-Mello
,
P. M.
Cincotta
, and
C. M.
Giordano
, “
Instability times in the HD 181433 exoplanetary system
,”
Astron. Astrophys.
652
,
A112
(
2021
).
35.
P. M.
Cincotta
,
C. M.
Giordano
, and
C.
Simó
, “
Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits
,”
Physica D
182
,
151
(
2003
).
36.
P. M.
Cincotta
and
C. M.
Giordano
, “
Theory and applications of the mean exponential growth factor of nearby orbits (MEGNO) method
,” in
Chaos Detection and Predictability
, Lecture Notes in Physics Vol. 915, edited by C. Skokos, G. Gottwald, and J. Laskar (Springer, Berlin, Heidelberg, 2016), p. 93.
37.
P. M.
Cincotta
and
C.
Simó
, “
Global dynamics and diffusion in the rational standard map
,”
Physica D
413
,
132661
(
2020
).
38.
P. M.
Cincotta
,
C. M.
Giordano
, and
C.
Simó
, “
Numerical and theoretical studies on the rational standard map at moderate-to-large values of the amplitude parameter
,”
Regul. Chaotic Dyn.
28
,
265
(
2023
).
39.
P. M.
Cincotta
and
C.
Simó
, “
Simple tools to study global dynamics in non-axisymmetric galactic potentials -I
,”
Astron. Astrophys. Suppl. Ser.
147
,
205
(
2000
).
40.
B.
Benettin
,
L.
Galgani
,
A.
Giorgilli
, and
J. M.
Strelcyn
, “
Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 2: Numerical application
,”
Meccanica
15
,
21
(
1980
).
41.
H.
Kook
and
J. D.
Meiss
, “
Diffusion in symplectic maps
,”
Phys. Rev. A
41
,
4143
(
1990
).
42.
M.
Guzzo
and
E.
Lega
, “
The numerical detection of the Arnold web and its use for long-term diffusion studies in conservative and weakly dissipative systems
,”
Chaos
23
,
023124
(
2013
).
43.
N.
Miguel
,
C.
Simó
, and
A.
Vieiro
, “
On the effect of islands in the diffusive properties of the standard map, for large parameter values
,”
Found. Comput. Math.
15
,
89
(
2014
).
44.
B. V.
Chirikov
, “
Anomalous diffusion in a microtron and critical structure at the chaos boundary
,”
J. Exp. Theor. Phys.
83
,
646
(
1996
).
45.
T.
Manos
and
M.
Robnik
, “
Survey on the role of accelerator modes for anomalous diffusion: The case of the standard map
,”
Phys. Rev. E
89
,
022905
(
2014
).
46.
G. M.
Zaslavsky
and
S. S.
Abdullaev
, “
Scaling properties and anomalous transport of particles inside the stochastic layer
,”
Phys. Rev. E
51
,
3901
(
1995
).
47.
G. M.
Zaslavsky
, “
Chaos, fractional kinetics, and anomalous transport
,”
Phys. Rep.
371
,
461
(
2002
).
48.
C.
Shannon
and
W.
Weaver
,
The Mathematical Theory of Communication
(
University of Illinois Press
,
Urbana
,
1949
).
49.
V.
Arnold
and
A.
Avez
,
Ergodic Problems of Classical Mechanics
, 2nd ed. (
Addison-Wesley
,
New York
,
1989
).
50.
A.
Katz
,
Principles of Statistical Mechanics, The Information Theory Approach
(
W. H. Freeman
,
San Francisco
,
1967
).
51.
A.
Lesne
, “
Shannon entropy: A rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics
,”
Math. Struct. Comput. Sci.
24
,
e240311
(
2014
).
52.
P. M.
Cincotta
and
I. I.
Shevchenko
, “
Correlations in area preserving maps: A Shannon entropy approach
,”
Physica D
402
,
132235
(
2020
).
You do not currently have access to this content.