Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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168
S.-N. Liang and
B.
L.
Lan
3. Results
In the example given here, the parameters
of
the bouncing ball system
are: g
= 981 cmls2, C = 2.998x103
c
m/s
(we have to use an
artificially smaller c value for accurate numerical calculation
of
the
general relativistic map), table's frequency
(OJ/2;r)
= 60 Hz, table's
amplitude
A = 0.012 cm, and coefficient
of
restitution a = 0.5 . The
initial conditions are
8.17001 cmls for the
ball's
velocity and 0.12001
for the normalized impact phase (i.e., impact phase divided by 2n.
The Newtonian and general relativistic trajectories are plotted in Figure
1. The figure shows that the two trajectories are close to each other for
a while but they are completely different after
21
impacts although the
ball's
speed and table's speed remained low (about
1O-
3
c) and the
gravitational field is
weak (gravitational potential gy is about
1O-
6
c
2
).
We have carefully checked that the breakdown
of
agreement between
the two trajectories is not due to numerical errors.
4.
Conclusion
For a slow-moving dynamical system where gravity is weak, we have
shown, contrary to expectation (see, for example, [1,9]), that the
trajectories predicted by Newtonian mechanics and general relativistic
mechanics from the same parameters and initial conditions could
rapidly disagree completely. The bouncing ball system could be
realized experimentally
[10-12] to test which
of
the two completely
different trajectory predictions is physically correct.
References
1.
A. Einstein, Relativity: the special and the general theory, Random
House, New York,
1961.
2.
J.
Ford and G. Mantica, Am.
1.
Phys. 60, 1086-1098 (1992).
3. W. D. McComb, Dynamics and relativity, Oxford University
Press, Oxford, 1999.
4.
J.
B. Hartle, Gravity:
An
Introduction to Einstein's General
Relativity, Addison-Wesley, San Francisco, 2003.
5.
B. L. Lan, Chaos 16,033107 (2006).
6.
B. L. Lan, in Topics on chaotic systems: selected papers from
Chaos2008 international conference, eds. C. H. Skiadas, I.
Dimotikalis, and C. Skiadas, World Scientific, 2009, pp. 199-203.
7.
B. L. Lan, Chaos, Solitons and Fractals 42, 534-537 (2009).
Dynamics
of
a Bouncing Ball
169
8.
P.
Davies, in The new physics, ed.
P.
Davies, Cambridge
University Press, 1992, pp. 1-6.
9.
I.
R.
Ladipus, Am.
1.
Phys. 40, 1509-15lO (1972).
lO.
N.
B.
Tufillaro,
T.
Abbott,
J.
Reilly,
An
experimental approach to
nonlinear dynamics
and
chaos, Addison-Wesley, California, 1992.
11.
N.
B.
Tufillaro,
T.
M.
Mello,
Y.
M.
Choi and A.
M.
Albano,
1.
Physique 47,1477-1482 (1986).
12.
N.
B.
Tufillaro and A.
M.
Albano, Am.
1.
Phys. 54,939-944, 1986.
II.)
'"
0.8
~
0.6
Po
...
<J
~
0.4
.§
10
.~
8
<J
o
Q)
7
>
. '
impact
. '
' .
.'
I'
I
I
,
Il
t:.
~~
'.
'.
"
.'
I
I
,
I
III
11
I
'.
' .
A
'.
'.
.'
t
I
I
I
11
4
L-
______________
-4
________________________
~
o
10
20 30 40 50
impact
Figure 1. Comparison
of
the Newtonian and general relativistic trajectories: normalized
impact phases (top) and velocities (bottom). Newtonian (general relativistic) values are
plotted with triangles (diamonds).
170
Symmetry-Break in a Minimal Lorenz-Like System
Valerio Lucarini I and Klaus Fraedrich2
1 Department
of
Mathematics, University
of
Reading
Whiteknights,
PO Box 220, Reading, RG6 6AX, UK
Department
of
Meteorology, University
of
Reading
Earley Gate, PO Box 243, Reading, RG6 6BB, UK
Department
of
Physics, University
of
Bologna
Viale Berti-Pichat 6/2
40127 Bologna, Italy
(e-mail:
v.lucarini@reading.ac.uk)
2 Meteorologisches Institut, KlimaCampus, University
of
Hamburg
Grindelberg
5,20144
Hamburg, Germany
Abstract:
Starting from the classical Saltzman 2D convection equations, we derive via
spectral truncation a minimal
10
ODE system which includes the thermal effect
of
viscous
dissipation. Neglecting this process leads to a dynamical system which includes a decoupled,
generalized Lorenz system. The consideration
of
this process breaks an important symmetry, couples
the dynamics
of
fast and slow variables, and modifies the structural properties
of
the attractor. When
the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time
scale
of
0(Ec··
1
)
is introduced in the system. Moreover, the system is ergodic and hyperbolic, the
slow variables feature long term memory with
11/
312
power spectra, and the fast variables feature
amplitude modulation. Increasing the strength
of
the thermal-viscous feedback has a stabilizing
effect,
as
both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically
with Ec. The analyzed system features very rich dynamics: it overcomes some
of
the limitations
of
the Lorenz system and might have prototypical value
in
relevant processes in complex systems
dynamics.
1.
Introduction
The Lorenz system [1] has a central role in modern science
as
it has provided
the first example
of
low-dimensional chaos [2], and has literally paved the way
for new scientific paradigms. The Lorenz system can be derived with a minimal
truncation
of
the Fourier-modes projection
of
the 2D Boussinesq convection
equations introduced by Saltzman [3], where a specific selection
of
the spatial
symmetry
of
the fields
is
considered. Extensions
of
the Lorenz system taking
into account higher-order spectral truncations in the 2D case have been
presented, see e.g. [4,5,6], whereas in [7] the standard procedure has been
extended to the 3D case. The mathematical properties
of
the Lorenz system have
been the subject
of
an intense analysis; For several classic results, see [8].
Recently, at more theoretical level, the investigation
of
Lorenz-like systems has
stimulated the introduction
of
the family
of
singular hyperbolic systems as
extension
of
the family
of
hyperbolic systems [9]. Moreover, moving from the
theory developed for non-equilibrium systems [10-12], in a recent paper a
careful verification on the Lorenz system
of
the Kramers-Kronig dispersion
relations and sum rules has been performed [13].
In spite
of
its immense value, the Lorenz system does not provide an
efficient representation
of
several crucial phenomena typically associated to
Symmetry-Break in a Minimal Lorenz-Like System
171
complex, chaotic systems. When looking at finite time predictability properties,
the Lorenz system features an unrealistic return-of-skill in the forecast,
i.e. there
are regions in the attractor within which all infinitesimal uncertainties decrease
with time [14]. Additionally, the Lorenz system does not feature an interplay
between fast and slow variables, so that it cannot mimic the coupling between
systems with different internal time scales.
Palmer [15] introduced artificially
ad
hoc
components to the Lorenz system in order to derive a toy-model able to
generate low-frequency variability. Moreover, as discussed in
[16], the Lorenz
system does not allow for a closed-form computation
of
the entropy production.
In this work, we wish to propose a minimal dynamical system, possibly
with some paradigmatic value, able to overcome some
of
the limitation
of
the
Lorenz system, and endowed with a much richer dynamics. Starting from the 2D
convection equations, we derive with a truncation
a la Lorenz a minimal 10
d.oJ.
ODEs system which includes the description
of
the thermal effect
of
viscous dissipation. We discuss how, while neglecting this process lead
us
to a
dynamical system which includes a decoupled (generalized) Lorenz system, its
consideration breaks an important symmetry
of
the systems, couples the
dynamics
of
fast and slow variable, with the ensuing modifications to the
structural properties
of
the attractor and in the spectral features
of
the system.
2. Rayleigh-Benard convective
system
We
consider a thermodynamic fluid having kinematic viscosity
V,
thermal
conductivity
k,
thermal capacity at constant volume
Cl"
and linearized equation
of
state P =
Po
(1-cff). The 2D
(x,
z)
Rayleigh-Benard top-heavy convective
system confined within
z E
[0,
H]
can be completely described, when adopting
the Boussinesq approximation, by the following system
of
PDEs:
d,V2V+J(V,V2V)= gad/J+
vV
4
V
d/J+J(V,e)=
I1T
dxV+kV2e+~diiVdijV
H
C,
..
where g is the gravity
acceleration,
(la-lb)
T =
To
- zl1T / H + e , with H
uniform depth
of
the fluid and
I1T
imposed temperature difference. The suitable boundary conditions in
the case
of
free-slip system are e(x, z = 0) = e(x, z = H) = 0 and
V(x, z =
0)
= V(x, z = H) = V
2
V(x, Z =
0)
= V
2
V(x, Z = H) = 0 for all values
of
x.
These PDEs can be non-dimensionalized
by
applying the linear transformations
x=Hx, z=HZ,
t=H
2
/kt,
v=k¥/,and
e=kv/(gaH
3
)B:
172
V.
Lucarini
and
K.
Fraedrich
a
i
V
2
v/
+
'(v/,
V
2
v/)=
ciJ/)+
cfi4v/
ai)+
'(Iii,
B)
=
Ra.v/+
V2B+
aEca,v/a
,
v/'
t r x
IJ
I)
(2a-2b)
where
fJ=v/k
is the Prandl number,
R=gaH
3
f:,.T/(kv)
is
the Rayleigh
number, and
Ec
= e /(C/y..TH
2
)
is the Eckert number, and the
suitable boundary conditions are
B(x,
z =
0)
=
B(x,
z =
1)
= 0 and
v/(x,
z =
0)
=
v/(x,
z =
1)
= V
2
v/(x,
z =
0)
= V
2
v/(x,
z =
1)
=
O.
The Eckert number
in
Eq. 2b quantifies the impact
of
viscous dissipation on the thermal balance
of
the system. As this number is usually rather small in actual fluids, the
corresponding term in the 2previous set
of
PDEs is typically discarded. We
perform a truncated Fourier expansion
of
v/
and
B,
assuming that they are
periodic along the x and z with periodicity
of
2/
a and
2,
respectively. When
boundary conditions are considered,
we
then derive a set
of
ODEs describing the
temporal evolution
of
the corresponding (complex valued) modes
'P
m
.
1I
and
Om
II
' which are associated to the wave vector R =
(J/Zln,mn).
See
[3]
for a
detailed derivation in the case
Ec
=
0,
considering that the sign
of
the term
proportional
to
R
is
wrong in both Eqs. 34 and 35. In the case
Ec
=
0,
the
seminal Lorenz system can be derived
by
severely truncating the system,
considering the evolution equation for the real part
of
'Pl.l and for the imaginary
parts
of
°
1
,1
and °
0
,2' and performing suitable rescaling (see below). While
°
0
,2
has
no
real part because
of
the boundary conditions [3], neglecting the
imaginary (real) part
of
'Pl.l
(°
1
,1)
amounts to an arbitrary selection
of
the
phase
of
the waves in the system. An entire hierarchy
of
generalized Lorenz
models, all obeying to this constraint, can be derived with lengthy but
straightforward calculations. See, e.g., [4-6] for a detailed discussion.
3. Symmetries
of
the extended Lorenz system
In this work
we
include the modes 'PI,I' 'P
2
,
2'
01.1'
02,2
in our truncation, and
retain both the real and the imaginary parts, whereas the considered horizontally
symmetric modes
00,2
and
00
.4
are, as mentioned above, imaginary. Finally,
we
assume, in general, a non vanishing value for
Ec.
If
we define
'PI
, I = a(X
I
+iXJ,
O1,1
=
f3(~
+iyJ, 'P
2
,2
=
a(AI
+iAJ, 0
2.2
=
f3(B
I
+iBJ,
0
0
,2=iJi.lI'
and 0
0
,4=iJi.l2'
with a={a
2
+1)/{2.J2a),
f3
=
JZ'3
(a
2
+
1)
j
{2.J2a
2
),
r =
JZ'
3
(a
2
+ 1
JI{2a
2
)'
set a =
I/.J2
[1]
,
we
derive the
following system
of
real ODEs:
Symmetry-Break
in
a Minimal Lorenz-Like System 173
XI =
O"Y
2
-(JX
I
X
2
=
-(J~
-(JX
2
~
= X
2
Z
1
-rX
2
-~
+s/(.J2;r)(JEc(A
I
X
I
+~X2)
1'2
= -
XIZ
I
+ rX I -
Y2
+ s/(.J2;r )oEc(A
2
X
I
-
AI
X
2)
AI
= (J/2
B2
-
4(JA
I
,12
=
-(J/2B
I
-4(JA
2
BI
= 4A
2
Z
2
- 2rA
2
-4BI
-1/(2.J2;r
)oEC(XI2
- xi)
B2
=
-4A
I
Z
2
+ 2rA
I
-
4B2
-1/(.J2;r
)(JEcX
I
X 2
ZI = X
I
Y
2
-
X2~
-bZ
I
Z2
=
4AIB2
-4A2BI
-4bZ
2
(3a-3j)
where the dot indicates the derivative with respect to
r=;r2(a
2
+1}=3/2;r2
t
,
r =
R/
Rc
= R/(27;r4
/4)
is the relative Rayleigh number, and b is a geometric
factor. Note that
if
we
exclude the faster varying modes
tp2,2
and O
2
,2
in our
truncation (as in the Lorenz case) the Eckert number
is
immaterial in the
equations
of
motion, the reason being that viscous dissipation acts on small
spatial scales. Therefore, the ODEs (3a-31) provide the minimal system which
includes the feedback due to the thermal effect
of
viscosity.
If
we
retain all the
variables in the system (3a-3j) and Ec is set to 0, the following symmetry is
obeyed. Let
Set,
So)
= (X
(t),
y(t),
A(t),
B(t},
z(t)Y -with -
(t)
=
(-I
(t}'-2
(t))
-be a
solution
of
the system with initial conditions
So
= (X(O),Y(O),A(O),B(O),Z(O)Y.
We then have that:
y-I
(m,¢
)S(t,
T(m,
¢
)So)
=
S(t,
So)
(4)
for all
(m,¢)E 9\2,
where
T(m,¢)
is
a
real, linear, orthogonal
(y-I
(m,
¢) =
TT
(m,¢))
transformation:
RaJ
0
0 0 0
X
0
RaJ
0 0 0
y
T(m,¢)S
=
0 0
R¢
0
0
A
(S)
0 0 0
RI/J
0
B
0 0 0
0 I Z
174
V.
Lucarini and
K.
Fraedrich
with I being the 2X2 identity matrix and
R;;
being the 2X2 rotation matrix
of
angle
~.
The
T(OJ,¢)
-matrix shifts the phases
of
both the \f'1,1 and 0
1
,1
waves
by
of
the same angle
OJ
and, independently, the phases
of
both the
\f'2,2
and
02.2
waves by the same angle
¢.
The symmetry (4-5) gives a solid framework
to and generalizes the results given in [17].
4. Statistical properties
of
the system - symmetric case
In
the
Ec
= 0 case, we expect the presence
of
degeneracies in the dynamics,
leading to non-ergodicity
of
the system. Since, from (4), we have that
S(t,T(OJ,¢)SeJ=T(OJ,¢)S(t,So), we readily obtain that the statistical properties
of
the flow depend on the initial conditions, and that by suitably choosing the
matrix
T(OJ,
¢)
, we can, e.g., exchange the statistical properties
of
XI and
r;
with those
of
X 2 and
Y2
by setting
OJ
=
;r/2
(and similarly for the A and B
variables by setting ¢ =
;r/2).
Whereas, in actual integration, numerical noise
coupled with sensitive dependence on initial conditions (see below) breaks the
instantaneous identity
S(t,T(OJ,
¢
)So)
=
T(OJ,¢
)S(t,
So)
after sufficiently long
time, the statistical properties
of
the observables transform according to the
symmetry defined
in Eq. (5).
The dynamics
of
the variables (Xi' X
2'
r;,
Y
2
,
ZJ
defines an extended
Lorenz system [17]. The classical three-component system results from a
specific selection
of
the phase
of
the waves
of
the system, which is obtained by
setting vanishing initial conditions for X
2 and
r;.
With such as choice, we have
that X
2 =
r;
= 0 at all times, whereas the x and y variables
of
the Lorenz system
are
Xl
and Y
2
,
respectively. As a consequence
of
the symmetry given in Eqs.
(4-5), the statistical properties
of
ZI do not depend on the initial conditions, and
agree with those
of
the z variable
of
the classical Lorenz system. Moreover, the
statistical properties
of
the quadratic quantities
X2
==
X
I
2
+
xi,
y2
==
r;2
+ y
2
2
,
and
XY
==
X
I
Y
2
-
X
2r;
do not depend on the initial conditions (whereas those
of
each term in the previous sums do!), and agree (for all values
of
r,
a,
b
!)
with
those
of
x
2
and
l,
and
xy
of
the classical Lorenz system, respectively.
Analogously, since the symmetry defined
in Eq. (5) is obeyed, the statistical
properties
of
AI' A
2
, B
I
,
B2
depend on the initial conditions, whereas the
statistical properties
of
the quadratic quantities A
2
==
AI2
+
Ai
B2
==
Bl2
+
Bi
'
AB
==
AIB2
-
A2BI
' and
of
Z2
are well defined. The symmetry (4-5) implies that
the attractor
of
the system (3a-3j) can be expressed as a Cartesian product
of
a 2-
torus times a
"fundamental" attractor. The initial conditions will define where on
the torus the system is located.
Symmetry-Break
in
a Minimal Lorenz-Like System
175
From a physical point
of
view, the symmetry (4-5)
is
related
to
the fact
that the system does not preferentially select streamfunction and temperature
waves
of
a specific phase, and, does not mix phases. Therefore, in a statistical
sense, the relative strength
of
waves with the same periodicity but different
phase can be arbitrarily chosen by suitably selecting the initial conditions.
Instead, the statistical properties
of
the space-averaged convective heat
transport, which is determined by a linear combination
of
Z, and Z2' as well as
those
of
the total kinetic energy (determined by a linear combination
of
X 2 and
A
2)
and available potential energy (determined by a linear combination
of
y2
and
B2)
of
the fluid [3] must not depend on the initial conditions - but only on
the system's parameters - as they are robust properties
of
the flow.
In general, since the dynamics
of
the variables
(X"X2'~'Y2'Z,)
is
entirely decoupled from that
of
the variables {A" A
2
,
B"
B
2
,
ZJ,
the attractor
of
the system
(3a-31)
will be strange at least for the same values
of
the parameters
r,
a,
b providing the classical Lorenz
system
with a chaotic dynamics.
Additionally, strange attractors could result from choices
of
the values
of
r,
a,
b determining a chaotic dynamics for the (A" A
2
,
B"
B
2
, Z2) variables and a
trivial or periodic behavior for the
(X"
X
2'
y,
'
Y2
,Z,) variables.
With the
"classical" Lorenz parameter values r =
28,
a = lO, b = 8/3 ,
the variables
(X"X2'~'Y2'Z,)
obviously have
an
erratic behavior, whereas the
variables
(Ai' A
2
,
B"
B
2
, Z2)'
which describe the faster spatially varying wave
components, do not feature any time variability when asymptotic dynamics
is
considered, as they converge to fixed values
(A,=,
A;
,B,=,
B;,
Z;
) . As discussed
above, the
(A,
=, A;,
B,
=, B;) values depend on the initial conditions, while, after
some manipulations on Eqs (3a-3j),
we
obtain
(A2
r = b/8{r/2-8),
(B2
r = 8b{r/2-8) ,
Z;
=
r/2-8,
and (ABt =
b{r/2-8).
In general, since the fundamental properties
of
the system do not depend
on where in the 2-torus the system converges, despite the fact that ergodicity
is
not obeyed, it makes sense to compute the Lyapunov exponents [2,18]
of
the
system (3a-3j). In fact,
if
we
compute the Lyapunov exponents
(,11'
...
'..1
10
)
using
the algorithm by Benettin et al. [19] with the classical Lorenz parameter values,
we
obtain the same results independently
of
the initial conditions. The sum
of
the Lyapunov exponents
is
given by the trace
of
the jacobian J
of
the system
(3a-3j) and yelds
Tr(J) =
LAj
=-5{2+2a+b)"" -123.333. In particular,
we
obtain one positive exponent
(AI
""
0.905
),
which gives the metric entropy h
of
the system
[2]
and matches exactly the same value
as
the positive Lyapunov
exponent
in
the classical Lorenz system, and three vanishing exponents
Az
=
~
=
,14
=
o.
The non-hyperbolicity
of
the system,
as
described by the
176
V.
Lucarini
and
K.
Fraedrich
presence
of
two additional vanishing exponents, is related to the existence
of
two neutral directions due to the symmetry property (4-5). Moreover, we have 6
negative Lyapunov exponents, one
of
which agrees with the negative exponent
of
the Lorenz system
(As
""
-14.572).
Given the very structure (and derivation)
of
the model discussed above, the fact that we recover the classical results
of
the
3-component Lorenz system is quite reinsuring. The Kaplan-Yorke dimension
[20]
of
the system is:
(6)
where k is such that the sum
of
the first k (4, in our case) Lyapunov exponents is
positive and the sum
of
the first
k+
1 Lyapunov exponents is negative.
5.
Symmetry-break and phase mixing
Considering a non-vanishing value for Ec amounts to including an additional
coupling between the temperature and the streamfunction waves. Such a
coupling breaks the symmetry (4-5), so that a dramatic impact on the attractor
properties is expected. Using standard multi scale procedure,
we
can emphasize
the emergence
of
the additional time scale
Ec-
1
•
We introduce a slow time
variable r =
£t
with c = Ec . Then
we
take the expansion
Vi(t)=Voi(t,r)+c~j(t,r)+
..
+cl7v,/(t,r)+
...
where
Vi,j=l,
...
,lO
is a generic
variable
of
the ODEs system considered,
we
redefine the time derivative
d/
dt
as
d/
dt
==
d/dt + liJ/dr ,
we
plug these new definitions in Eqs. (3a-3j), and group
together terms with the same power
of
£.
At zero order - the r variable
is
frozen - we obtain that the variables V
0'
obey the Eqs. (3a-3j) with Ec = 0 .
Therefore, for time scales shorter than
0(c-
1
),
the symmetries (4-5) described in
the previous section are approximately obeyed, which confirms that our
multiscale expansion is well-defined.
We hereby consider the specific case where
we
select the classical values
for the other 3 parameters
of
the system, so that r =
28,
(j
=
10,
b = 8/3 . When
large values
of
Ec are considered
(Ec
> 0.045), the system loses its chaotic
nature as no positive Lyapunov exponents are detected (not shown), whereas
periodic motion is realized. The analysis
of
this transition and
of
this regime
is
beyond the scopes
of
this paper,
as
we
confine ourselves to studying the
properties
of
the system when chaotic motion is realized, thus focusing on the
Ec
~
0 limit. We consider Ec E [0,0.02].
In physical terms, the coupling allows for a mixing
of
the phases
of
the
thermal and streamfunction waves: after a
0(c-
1
)
time the system "realizes"
that the symmetry (4-5)
is
broken, so that the previously described degeneracies
Symmetry-Break
in
a Minimal Lorenz-Like System
177
are destroyed and ergodicity
is
established
in
the system. Note that, since also in
the case
of
Ec
> 0 the convection does not preferentially act on waves
of
a
specific phase, we have that, pairwise, the statistical properties
of
XI and X 2 '
and
of
~
and -Y2 are identical and do not depend on the initial conditions. The
same applies for the
A and B pairs
of
variables, respectively.
If,
in particular,
we
consider the long term averages
of
the observables
(X
2),
(y2),
(Xy)
,
(A2),
(B2),
and
(AB),
we
obtain that in all cases the two addends give the
same contribution, e.g.
(Xn
=
(xi)
=
1/2(X
2
).
In this case, long is considered
with respect to
[,-I
=
Ec
-
I
:
these results can be obtained by first and second
order
£ -terms in the multi scale expansion envisioned above. As
an
example, the
(Xn
=
(Xi)
= 1/2 ( X
2)
identity is obtained by taking the long term average
of
the first order £ -term
of
Eq. (3g). Whereas having
Ec
> 0
is
crucial for mixing
the phases
of
waves, the impact
of
the symmetry break on the long term
averages
of
the physically sensitive observables 0 = X2 ,
y2,
A2, B2,
ZI'
and
Z2
is relatively weak
in
the considered
Ec
-values range.
6. Symmetry-break
and
hyperbolicity
We hereby want to take a different point
of
view
on
the process
of
symmetry
break
of
the dynamical system and on its sensitivity with respect to
Ec,
by
analyzing the
Ec
-dependence
of
the spectrum
of
the Lyapunov exponents.
Again, the classical values for
r,
(J,
b are considered. The first, crucial result
system
is
that the system
is
hyperbolic for any finite positive value
of
Ec:
two
of
the vanishing Lyapunov exponents branch off from 0 with a distinct linear
dependence
on
Ec,
so that ~
'"
5.IEc and
,14
'"
-6.0Ec
for Ec < 0.008,
whereas ~ = 0 corresponds to the direction
of
the flow. The largest Lyapunov
exponent also decreases linearly
as
~
'"
0.905-9.0Ec.
Time scale separations
emerges again, as the two globally unstable directions define two characteristic
times
II
~
and
1/
~
, which are
0(1)
and
O(£c-
I
),
respectively. All the other
Lyapunov exponents feature a negligible dependence on
Ec,
except
As
'"
-14.572
+ 1O.0Ec, which ensures that the sum
of
Lyapunov exponents is
independent
of
Ec.
Therefore,
we
derive that
Ec
< 0.008 the metric entropy
decreases with
Ec
as
h(Ec) =
~
+
~
'"
0.905 - 4.0Ec , whereas the
Kaplan-
Yorke dimension can
be
approximately written as
dKy{Ec)
=
4+(~
+~
+~
+A,4)J1AsI
'"
4.l83-2.01Ec.
For larger values
of
Ec,
linearity
is
not obeyed (except for ~ and ,14)' whereas a faster, monotonic
decrease
of
both the metric entropy and the Kaplan-Yorke dimensions are found.