PHYSICAL REVIEW E 73, 037201 共2006兲
Chaos edges of z-logistic maps: Connection between the relaxation and sensitivity
entropic indices
Ugur Tirnakli1 and Constantino Tsallis2,3
1
Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
3
Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil
共Received 25 February 2005; revised manuscript received 7 November 2005; published 15 March 2006兲
2
Chaos thresholds of the z-logistic maps xt+1 = 1 − a 兩 xt兩z 共z ⬎ 1 ; t = 0 , 1 , 2 , . . . 兲 are numerically analyzed at
accumulation points of cycles 2, 3, and 5 共three different cycles 5兲. We verify that the nonextensive
q-generalization of a Pesin-like identity is preserved through averaging over the entire phase space. More
av
precisely, we computationally verify limt→⬁具Sqav 典共t兲 / t = limt→⬁具lnqav ␰典共t兲 / t ⬅ ␭qav , where the entropy Sq
sen
sen
sen
⬅ 共1 − 兺i pqi 兲 / 共q − 1兲 共S1 = −兺i piln pi兲, the sensitivity to the initial conditions ␰ ⬅ lim⌬x共0兲→0⌬x共t兲 / ⌬x共0兲, and
av
av
⬍ 1, and the coefficient ␭qav ⬎ 0 depend on both z
lnqx ⬅ 共x1−q − 1兲 / 共1 − q兲 共ln1x = ln x兲. The entropic index qsen
sen
and the cycle. We also study the relaxation that occurs if we start with an ensemble of initial conditions
homogeneously occupying the entire phase space. The associated Lebesgue measure asymptotically decreases
as 1 / t1/共qrel−1兲 共qrel ⬎ 1兲. These results 共i兲 illustrate the connection 共conjectured by one of us兲 between sensiav ␣n
兲 , where the positive numbers 共An , ␣n兲
tivity and relaxation entropic indices, namely, qrel − 1 ⯝ An共1 − qsen
av
av
共cycle n兲 = Bnqsen
共cycle 2兲 + ⑀n.
depend on the cycle; 共ii兲 exhibit an unexpected scaling, namely, qsen
DOI: 10.1103/PhysRevE.73.037201
PACS number共s兲: 05.45.Ac, 05.20.⫺y
Boltzmann-Gibbs 共BG兲 entropy and the corresponding
statistical mechanics generically require strong chaos for
their applicability and 共well-known兲 usefulness. This type of
requirement was first used in 1872 by Boltzmann himself
关1兴. Indeed, his “molecular chaos hypothesis” allowed him to
arrive at the celebrated distribution of energies at thermal
equilibrium now known as the Boltzmann weight. Today, we
know that this requirement essentially amounts, for classical
nonlinear dynamical systems, to having at least one positive
Lyapunov exponent. In the case of many-body Hamiltonian
systems, such a condition is satisfied when the interactions
are short ranged. Such systems typically exhibit three basic
exponential functions 关2兴, namely, 共i兲 the sensitivity to the
initial conditions diverges exponentially with time, 共ii兲 physical quantities exponentially relax with time to their value at
the stationary state 共thermal equilibrium兲, and 共iii兲 at thermal
equilibrium, the probability of a given microstate exponentially decays with the energy of the microstate. These three
exponentials of different, though related, nature can be summarized in the following differential equation:
dy/dx = a1y
关y共0兲 = 1兴,
共1兲
whose solution is y = ea1x 共the subindex 1 will become transparent soon兲. Let us make explicit the point. The first physical interpretation concerns the sensitivity to the initial conditions of, say, a one-dimensional case and is defined as
␰共t兲 ⬅ lim ⌬x共t兲/⌬x共0兲,
⌬x共0兲→0
共2兲
where ⌬x共t兲 is the distance, in phase space, between two
copies at time t. If the system has a positive Lyapunov exponent ␭1, then ␰ diverges as ␰ = e␭1t. In other words, in this
case we have 共x , y , a1兲 ⬅ 共t , ␰ , ␭1兲. The second physical interpretation concerns the relaxation of some 共characteristic兲
physical quantity O共t兲 to its value O共⬁兲 at thermal equilib1539-3755/2006/73共3兲/037201共4兲/$23.00
O共t兲−O共⬁兲
rium. With the definition ⍀ ⬅ O共0兲−O共⬁兲 , we typically have
⍀ = e−t/␶, where ␶ is the relaxation time. In other words, in
this case we have 共x , y , a1兲 → 共t , ⍀ , −1 / ␶兲. This relaxation
occurs precisely because of the sensitivity to initial conditions, which guarantees strong chaos. Krylov was apparently
the first to realize 关3兴, over half a century ago, this deep
connection. The third physical interpretation is given by pi
−␤E j
= e−␤Ei / Z 共with Z ⬅ 兺W
兲, where Ei is the eigenvalue of
j=1e
the ith quantum state of the Hamiltonian 共with its associated
boundary conditions兲 and pi is the probability of occurrence
of the i-th state when the system is in equilibrium with a
thermostat whose temperature is T ⬅ 1 / k␤ 共Gibbs’ canonical
ensemble兲. In other words, in this case we have 共x , y , a1兲
→ 共Ei , Zpi , −␤兲.
A substantially different situation occurs when the maximal Lyapunov exponent vanishes. In this case the typical
differential equation becomes
dy/dx = aqy q
关y共0兲 = 1; q 苸 R兴,
共3兲
whose solution is y = eaqqx, the q-exponential function being
defined as follows: exq ⬅ 关1 + 共1 − q兲x兴1/共1−q兲 if the quantity between brackets is nonnegative, and zero otherwise 共ex1 = ex兲.
The sensitivity to the initial conditions is given in this case
␭ t
by 关4,5兴 ␰ = eq qsen 共sen stands for sensitivity; this expression
sen
stands in fact for the upper bound of ␰兲. In other words, we
have 共x , y , q , aq兲 ⬅ 共t , ␰ , qsen , ␭qsen兲. The relaxation is typi−t/␶
cally expected 关6兴 to be characterized by ⍀ = eq qrel 共rel
rel
stands for relaxation兲. In other words, in this case we have
共x , y , q , aq兲 ⬅ 共t , ⍀ , qrel , −1 / ␶qrel兲. For the long-standing
metastable states 关7兴 that precede thermal equilibrium for
long-range interacting Hamiltonians, it is expected that 关8兴
−␤
E
−␤q E j
stat 兲 共stat stands for
pi = eq qstat i / Zqstat 共with Zqstat ⬅ 兺W
j=1eq
stat
stat
stationary兲. In other words, in this case we have
037201-1
©2006 The American Physical Society
PHYSICAL REVIEW E 73, 037201 共2006兲
BRIEF REPORTS
共x , y , q , aq兲 ⬅ 共Ei , Zqstat pi , qstat , −␤qstat兲. For systems that are
at, or close to, the edge of chaos we typically have qsen 艋 1,
qrel 艌 1, and qstat 艌 1. For the BG case, where there are one
or more positive Lyapunov exponents, we recover the confluence qsen = qrel = qstat = 1.
The entire q-triplet should be either measurable or calculable for Hamiltonian 共or even more complex兲 systems. And
indeed it has recently been measured in the solar wind 关9兴.
However, the generic relation among these three q indices is
still unknown. For dissipative systems such as, say, the
z-logistic map, no qstat exists. Therefore, the problem reduces
to only two q indices, namely, qsen and qrel. Their generic
relation also is unknown. In the present paper, we provide
numerical evidence of such a connection.
Before entering into the details of the present calculation,
let us briefly review the connection with the entropy Sq, the
basis of a current generalization of BG statistical mechanics
referred to as nonextensive statistical mechanics 关10兴. This
entropy is defined as follows:
W
Sq ⬅
1 − 兺 pqi
i=1
q−1
W
= 兺 pilnq共1/pi兲
共4兲
i=1
where the q-logarithm function, inverse of the q-exponential,
1−q
is defined as lnqx ⬅ x 1−q−1 共ln1x = ln x兲 and S1 = SBG
W
⬅ −兺i=1
piln pi.
If we partition the phase space of a one-dimensional map
共at its edge of chaos兲 into W small intervals, randomly place
N initial conditions into one of those windows, and then run
the dynamics for each of those N points, we get, as time t
W
Ni共t兲
evolves, an occupancy characterized by 兵Ni共t兲其 关兺i=1
= N兴. With pi共t兲 ⬅ Ni共t兲 / N we can calculate Sq共t兲 for any
value of q. From this, we can calculate the entropy production per unit time 关11兴, defined as follows:
Kq ⬅ lim lim lim Sq共t兲/t.
t→⬁ W→⬁ N→⬁
共5兲
It has been proved 关12兴 that only Kqsen is finite 共Kq = 0 for q
⬎ qsen and Kq diverges for q ⬍ qsen兲. Furthermore, if we consider the upper bound of Kqsen with regard to the choice of
the little window within which we put the N initial conditions, we obtain the Pesin-like identity Kqsen = ␭qsen. Several
aspects of this problem have already been verified for various one-dimensional unimodal maps 关13–16兴, as well as for
a two-dimensional conservative map 关17兴. The influence of
averaging was recently studied 关18兴. It was verified that,
while the q-generalized Pesin-like identity is preserved, the
av
共av stands for average兲. The
value of qsen is changed into qsen
main goal of the present paper is to exhibit that a simple
av
and qrel by making use of the
relation exists between qsen
z-logistic map family defined as
xt+1 = 1 − a兩xt兩z
共6兲
where 共z ⬎ 1 ; 0 ⬍ a 艋 2 ; 兩xt 兩 艋 1 ; t = 0 , 1 , 2 , . . . 兲.
In our simulations, we first check whether the qrel values
corresponding to cycles 3 and 5 共denoted 5a, 5b, and 5c兲 are
different from those of cycle 2 obtained in 关6兴. To accom-
FIG. 1. Occupied volume as a function of discrete time. After a
transient period, which is the same for all Nbox values, the powerlaw behavior emerges. For each case, the evolution of a set of
10Nbox copies of the system is followed.
plish this task, we analyze the rate of convergence to the
critical attractor when an ensemble of initial conditions is
uniformly distributed over the entire phase space 共the phase
space is partitioned into Nbox cells of equal size兲 and we
found that, for all cycles that we studied, the volume W共t兲
occupied by the ensemble exhibits a power-law decay with
the same exponent value for given z. As an example, the case
of cycle 3 for z = 2 is given in Fig. 1. The same kind of
behavior is obtained for other z values and cycles, which
exhibits that qrel does not depend on the cycle 共see also Table
I兲. It is worth mentioning at this point that very recent extensive calculations 关19兴 showed that the qrel values should indeed be slightly larger than the values reported in 关6兴. But,
since this is just a systematic shift for all z values, this would
have no relevant effect for any scaling function based on qrel
values. Then we concentrate on the ensemble averages of the
sensitivity function ␰共t兲 by considering two very close points
关throughout this work we take ⌬x共0兲 = 10−12兴 and calculating
its value from Eq. 共2兲. This procedure is repeated many times
with different values of x randomly chosen in the entire
phase space. Finally an average is taken over all lnq␰共t兲 values. For cycles 3, 5a, 5b, and 5c, we obtain the behavior of
具lnq␰典共t兲 as a function of t, for various values of z. We then
av
deduce qsen
by identifying the linear time dependence as ilav
av
and ␭qav values
lustrated in Fig. 2. We verify that the qsen
sen
depend on both z and the cycle, whereas qrel only depends on
z. Finally, to investigate the entropy production for the cycles
3, 5a, 5b, and 5c, we use the same procedure as in 关15兴 for
cycle 2 of the z-logistic maps. It is numerically verified that,
as seen for a typical case in Fig. 3, for each value of z, and
for each cycle, the finite entropy production per unit time
occurs only for a special value of q which precisely coincides with the one obtained from the sensitivity function. In
av
av
addition to this, we obtained Kqav = ␭qav , which clearly
sen
sen
broadens the region of validity of the usual Pesin-like identity 关20兴.
037201-2
PHYSICAL REVIEW E 73, 037201 共2006兲
BRIEF REPORTS
TABLE I. z-logistic map family for cycles 2, 3, and 5a.
z
1.75
1.75
1.75
2
2
2
2.5
2.5
2.5
3
3
3
5
5
5
2
3
5a
2
3
5a
2
3
5a
2
3
5a
2
3
5a
av
av
ac
av
qsen
qrel
␭ qav
K qav
1.355060. . .
1.747303. . .
1.607497. . .
1.401155. . .
1.779818. . .
1.631019. . .
1.470550. . .
1.828863. . .
1.669543. . .
1.521878. . .
1.862996. . .
1.699440. . .
1.645533. . .
1.931072. . .
1.773088. . .
0.37± 0.01
0.92± 0.01
0.96± 0.01
0.36± 0.01
0.88± 0.01
0.93± 0.01
0.34± 0.01
0.82± 0.01
0.88± 0.01
0.32± 0.01
0.78± 0.01
0.84± 0.01
0.28± 0.01
0.68± 0.01
0.73± 0.01
2.25± 0.02
2.25± 0.02
2.25± 0.02
2.41± 0.02
2.41± 0.02
2.41± 0.02
2.70± 0.02
2.70± 0.02
2.70± 0.02
2.94± 0.02
2.94± 0.02
2.94± 0.02
3.53± 0.03
3.53± 0.03
3.53± 0.03
0.26± 0.01
0.48± 0.01
0.42± 0.01
0.27± 0.01
0.49± 0.01
0.42± 0.01
0.28± 0.01
0.48± 0.01
0.38± 0.01
0.29± 0.02
0.44± 0.01
0.34± 0.01
0.30± 0.02
0.36± 0.01
0.27± 0.01
0.26± 0.02
0.47± 0.02
0.40± 0.02
0.27± 0.02
0.48± 0.02
0.40± 0.02
0.28± 0.02
0.47± 0.01
0.37± 0.01
0.29± 0.03
0.44± 0.01
0.35± 0.01
0.30± 0.03
0.37± 0.01
0.25± 0.02
Cycle
Finally, for all cycles that we studied, we numerically
av
and
verified that a simple scaling relation exists between qsen
qrel 共see Fig. 4兲, namely,
qrel共cycle n兲 − 1 ⯝ An关1 −
av
qsen
共cycle
n兲兴
␣n
共7兲
sen
sen
We also notice 共see Fig. 5兲 an unexpected scaling behavior, namely,
av
av
qsen
共cycle n兲 = Bnqsen
共cycle 2兲 + ⑀n ,
共8兲
where n = 2 , 3 , 5a , 5b , 5c, the values of 共An , ␣n兲 being given
in the caption of Fig. 4; both numbers depend on the cycle.
For example, ␣n ⯝ 5.1, n = 2, and quickly approaches zero
when the cycle increases; An also decreases when the cycle
increases. This kind of relation between these two q indices
is seen here in a model system. It is clearly consistent with
the confluence expected for BG systems. This is to say, when
av
there is a positive Lyapunov exponent, we obtain qrel = qsen
= 1.
with 共⑀n , Bn兲 given in the caption of Fig. 5.
Summarizing, we have discussed a paradigmatic family of
one-dimensional dissipative maps, and have shown that its
共averaged兲 sensitivity to the initial conditions and its relaxation in phase space follow a simple path. This path is consistent with current nonextensive statistical mechanical concepts, and considerably extends the validity of Pesin-like
identities. The sensitivity to the initial conditions is characav
⬍ 1, which monotonically approaches unity
terized by qsen
with increasing cycle size 共at least for the specific cycles that
we have studied here兲, and decreases with z. It is further
FIG. 2. Time dependence of 具lnq␰典.
FIG. 3. Time dependence of 具Sq典.
037201-3
PHYSICAL REVIEW E 73, 037201 共2006兲
BRIEF REPORTS
av
FIG. 4.
Straight lines: qrel共cycle 2兲 − 1 = 13.5关1 − qsen
av
5.1
0.54
共cycle 2兲兴 ,
qrel共cycle 3兲 − 1 = 4.6关1 − qsen共cycle 3兲兴 ,
and
av
qrel共cycle 5a兲 − 1 = 4.1关1 − qsen
共cycle 5a兲兴0.39. Inset: All three cycles
av
5
are
shown
together:
关qrel共cycle 5b兲 − 1 = 4.2关1 − qsen
av
共cycle 5b兲兴0.30, and qrel共cycle 5c兲 − 1 = 4.7关1 − qsen
共cycle 5c兲兴0.23兴.
Notice that the Boltzmann-Gibbs limiting case appears as a special
point attained for all cycles studied here.
av
characterized by ␭qav , which exhibits a maximum as a funcsen
tion both of the cycle size and of z. The relaxation is characterized by qrel ⬎ 1, which monotonically increases with z,
and does not depend on the cycle. This numerical study has
enabled us to exhibit two interesting relations, namely, Eqs.
共7兲 and 共8兲. These results are expected to illuminate, among
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共2004兲.
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av
av
FIG. 5. Straight lines: qsen
共cycle 3兲 = 2.5qsen
共cycle 2兲 − 0.03,
av
av
av
5a兲 = 2.5qsen共cycle 2兲 + 0.03,
qsen共cycle 5b兲 = 1.75qsen
av
av
共cycle 2兲 + 0.34, and qsen
共cycle 5c兲 = 0.8qsen
共cycle 2兲 + 0.71.
av
qsen
共cycle
other things, the case of long-range-interacting Hamiltonian
systems, the situation of which is even more complex since a
third entropic index qstat is expected 共which would characterize the energy distribution at metastable states兲. Analytic
approaches to the present scalings are certainly most welcome.
This work is partially supported by TUBITAK 共Turkish
agency兲 under Research Project No. 104T148. It also is partially supported by Pronex, CNPq, and Faperj 共Brazilian
agencies兲, and SI International and AFRL 共U.S.A agencies兲.
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Physica A 261, 534 共1998兲. Full bibliography can be found in
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