CHINESE JOURNAL OF PHYSICS
VOL. 37, NO. 6
DECEMBER 1999
Stochastic Responses of the Stable Period-p Orbits in
One-Dimensional Noisy Map Systems
T&-Nan Wang, Rue-Ron Hsu, Han-Tzong Su, Wei-Fu Lin, Jyh-Long Chern,
and Chia-Chu Chen
Nonlinear Science GTOU~, Department of Physics, National Cheng Kung University,
Tainan, Taiwan 701, R.O.C.
(Received June 27, 1999)
We derive the correlation functions for the stochastic response of period-p orbits in
a generic one-dimensional map system under the influence of weak external, additive
white noise. Two approaches, a recurrence scheme and an integral method, are provided. Besides the supercycle, we recognise that the fluctuations in the elements of the
period-p orbits are colored noise, which have a non-vanishing time correlation. The results also indicate that the stochastic responses will be stationary after a large number
of period-p iterations. These analytical results are confirmed by numerical simulations.
PACS. 02.50.F~ - Stochastic analysis.
PACS. 05.4O.+j - Fluctuation phenomena, random processes, and Brownian motion.
I. Introduction
The role of dynamic noise in a deterministic nonlinear system has been studied
extensively [l-9]. These results indicate that, in a noisy dynamical system, not only do
the higher periods become obliterated by the increasing noise level but the bifurcation
points themselves become blurred. In particular, at the points of bifurcation, the Lyapunov
exponents no longer vanish as they do in the deterministic limit. Meanwhile, through both
numerical and analytical studies, some universal scaling behaviors have been discovered at
the onset of chaos even in noisy dynamical systems [2-51. Mayer-Kress and Haken carried
out a numerical investigation of the distribution function of the stochastic response in a
logistic model with uniform white noise using an integral recursion formula [6]. However the
analytical form of the correlation function, or the distribution function, of the stochastic
response of the period-p orbits in a generic noisy map system was not presented.
Later on, in a review paper [7], Crutchfield, Farmer and Huberman studied the
stochastic response of a noisy one-dimensional map, x,t1 = TV + p,, where f(x) =
Z( 1 - Z) and p, is the external noise. They proposed that the stochastic response should be
regarded as the fluctuations of the control parameter T. This means that the noisy map can
be rewritten as xntl = (T + qn)f(xn), where q,, is the response parametric noise. However,
when we numerically study the bifurcation diagrams of the noisy logistic map and the noisy
Ikeda map, see Fig. 1, we notice that the mean values of 2, in the elements of the noisy
stable orbits coincide with those of the stable orbits without noise. This fact implies that
535
@ 1999 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
~T~~HA~TICRESP~NSES
536
OF
VOL.
THESTABLEPERIOD-p...
37
,, ,/--- ---- -. ---:y0.6
---_.__ _
I
0.6 -
0.2
I
2
I
2.2
2.4
2.6
2 6
,
3
3.2
3 4
0
0.2
04
0 6
0.6
1
1 2
(b)
(4
FIG. I. The coincidence between the mean value for each element in a noisy stable orbit
and those stable orbits without noise, are demonstrated in (a) a noisy logistic map,
z,+i = rz,(I - xn) + 71,, and (b) a noisy Ikeda map, z,+i = 1 - $1 - cos(;z,)) + qn.
The amplitude of the white noise q,, is 0.001. The cross (x) denotes the average for 10,000
ensembles of each element in a period-p orbit after 1,000 times of the period-p iterations.
The solid line denotes the noise-free orbit elements.
the dispersion should be regarded as the fluctuation in the elements of the noise-free periodp orbits {2$)11 5 m 5 p} of the one-dimensional map z,+~ = fr(zn). In this map,
the element of the period-p orbits Z$I and the control parameter T obey the periodicity
condition, 22) = f,“(Zg)), and the stability condition, /A@)[ E 1 n~=,(~j,(p~)l < 1.
Based on this observation, we will derive a general formula for the correlition functions
of the elements in the period-p orbits by using the recurrence scheme and the integral
method. The results indicate that, beside the supercycles, the stochastic responses are
colored in nature which means that the correlations between the kp-th step and the (Ic +Qth step are non-vanishing. This also shows that the correlation functions, or the distribution
functions, of the stochastic response are stationary after kp iterations when k >> 1. T h e
plan of this paper is as follows. We derive the correlation functions by using the recurrence
scheme in section 2, and by using the integral method in section 3. Numerical verifications
in two interesting systems, a logistic map and an Ikeda map, are demonstrated in section
4. Finally, some concluding remarks are provided.
II. Recurrence scheme
Here, we consider a generic one-dimensional nonlinear map in which the external
additive noise is involved dynamically,
X n+1
=
fr(Xn)
+
%I,
(1)
where 7, is white noise with zero mean, (~~7,) = D6,, and (77,) = 0. For examples, D is
$ for uniform white noise with a noise level c and D = u2 for Gaussian white noise with
a standard deviation cr.
TIAN-NAN WANG, RUE-RON HSU,
VOL. 37
537
To derive the stochastic response in each element of a fluctuating period-p orbit, we
let the initial state 2, be located in the neighbourhood of z?), i.e. x, = s?)+ trn, J,, > 1.
After kp iterations, the stochastic response in the m-th element of period-p can be regarded
as a fluctuation in the m-th element of the noise-free period-p orbits, that is
xkP+m
-(‘)
= xkp+,
t
(kpfm
=
22’ t [kp+m.
(2)
Here, tkp+m is the stochastic response in the m-th element of period-p after k times of
period-p iteration.
From now on, the noise level and stochastic response will be assumed to be small
in comparison with the noise-free period-p orbits suck that the perturbation scheme is
applicable. Substituting Eq. (2) and z(k+r)p+m = 5:) + t(k+r~~+~ into the p times iteration
of Eq. (l), i.e.
x(k+l)p+m =
leads to a
fr(* * *fdxkp+m) +
vkp+m) + Tkp+m+l...) + ~kp+m+p-l,
(3)
recurrence relation between [(k+r)p+m and <kp+m,
h+l)p+m =
M(P)(kp+,
+
(4)
fit;‘) .
Here, we have denoted
p-1
n Mf$
&f(P) -
(5)
i=O
and
k,m
forp= 1,
qk+m,
&p-l) =
-
~;i:(n/r,‘p ,-%kp+m+j-1)
t
qkp+rn+p_l,
for p >
2,
(6)
where
(7)
and
(8)
By iterating the recurrence relation, Eq. (4), one obtains
t kp+m
=
(M(p))k&n + &M(p))i-l@‘;‘,
i=l
For simplicity, but without loss of generality, we set the initial state to be
that is <,,, = 0. Therefore, the stochastic response, after kp iterations, becomes
538
STOCHASTIC RESPONSES OF THE STABLE PERIOD-p . .
VOL. 37
i=l
As a result, the mean fluctuation of the stochastic response is zero when (nk) = 0. Here,
(77k) 3 lmk, h CL;’ 77:) denotes the ensemble average. The correlation function of the
stochastic response between the kp-th step and the (k + e)p-th step can be written as
= 7, ~,(M’p’) i’- l(M’p’) i-l(LI~~+L;)_i,,,H~_~,~).
i’= l i=l
(t(k+f)p+mtkp+m)
(11)
Using the input white noise properties, we find, for the case p > 2, that
CH,‘,prt:_il,~~~<,~)
p-l
p-l
j’= l j=l
(12)
p-l p - l
=
c c M,(~,‘)J&‘) ~j,j~6i,i~-~D
I,m
+ bi,il-tD
j'zl j=l
p-1
1 +C(~~~1))2
j=l
Therefore, the correlation function (for p 2 2) between the kp-th step and the (k + j)p-th
step is
(Mqy
(hk+~)p+mtkp+m)
=
1 - ( kwqZk)
1 - (M(p))2
P-l
1 + c(Mj;1’)2
j=l
(13)
It should be noted that the term C~~:(M~~1))2 in Eq. (13) will be zero for the most trivial
casep- 1.
III. Integral approach
As has been shown in previous section, by using the recurrence scheme, one can
derive the correlation function of the stochastic response of a dynamical map contaminated
by uniform or Gaussian white noise. Due to the excellent form of Gaussian white noise
2
P(%)
= A
I/!%2
exp
-2L
{
26
I
’
(14)
one also can derive its stochastic response by using an integral approach.
For simplicity, we assume that the initial state is located at 5, = ~2). This means
that the probability distribution of z is P(zm) = S(z, - 32)) at the beginning. After one
iteration, z,+~ = fV(zm) + qrn, the distribution function of z,+~ disturbed by the Gaussian
noise has become
539
TIAN-NAN WANG, RUE-RON HSU, ...
VOL. 37
=
&gexp
1
(X m+1 - fT@~‘>)”
(15)
*
&2
-
1
After the next iteration, x,+~ = fr(x,+r) + ~,,,+r, one can write down the distribution
function of z~,+~,
= / dXm / dXm+lP(Xm+1
P(Xm+z)
- fr(Xm))~(Xm+2
(16)
- fr(Xcm+l))P(X,).
Now, we introduce the kernel K(z,+~, x,) which is defined as
(17)
hnK(xm+~, xm>P(xm>.
P(X m+t) =
The kernel can be read from Eq. (16) and (17), it is
K(X
m+l,
xm) =
1 j=l
E dxcm+j j=O‘Ii K(xm+j+mn+j),
(18)
where
KC” m+j+l
T
xm+3.)
Cx
-
fr(xm+j))2
= &exp - m+J+l 20-2
1
>
(19)
.
Then, the distribution function of x++~, after kp-iterations beginning with CC, = z?), is
p(Xkp+m)
=
.dXm+J &--#p
(x
-
m+l+l
fr(xm+j))2
2a2
6(x,
_
m
&P))
*
(20)
To obtain the distribution function by integration, we change the variable x,+j to
2 m+j = xrn+j
-(‘)
+ [m+j* As mentioned in section 2, [m+j is the corresponding stochastic response and is assumed to be small enough that the linear perturbation scheme is applicable.
Therefore, we have dxm+j = d[m+j and
(21)
and the distribution function can be expressed as
P(Xkp+m) = J ‘b ’ {
j=O
Using the formula
dt,+j & exp { - (‘m+j+1 -2:“j’.“)2}}
6(tm).
(22)
STOCHASTIC RESPONSES OF THE STABLE PERIOD-p.. .
540
(z - W2
2 4
1
=
1
VOL. 37
(23)
J27r(a,2 + a20i
and integrating out d&,, . . . dEckp_l)+m step by step, we f?naUy have
J&
P(xkp+m) =
exp { -(z?$;f”l”)
(24
>
where
(25)
This result indicates that the stochastic response in each element of a period-p orbit for
the input Gaussian noise is still a Gaussian-like distribution with mean ~2) and deviation
gkp+m. This is consistent with the e = 0 case in Eq. (13).
It is expected that the stochastic response will inherit time correlations from the
deterministic dynamical system. Here, we define the time correlation between the kp-th
step and the (Ic + 1)p-th step of stochastic responses as
((x(k+f)p+m
- (x(k+f)p+m))(2kp+m - (xkp+m)))
(26)
= (t(k+f)p+mtkp+m)
=
J
From
dZ(k+f)p+m
Eq. (18) and
J
dxkp+&k+f)p+m~kp+m~(~(k+f)p+m,
xkp+“@@(xkp+m).
Eq. (23), one find
K(x(k+f)p+m,
xkpfm)
=
_ (h+++m
-;o~‘p))ftkp+-‘z
(27)
fP
where
(28)
Therefore, the correlation between the (k $ e)p-th step and the kp-th step is
(‘t(k+f)p+mtkp+m)
=
(M(P))fdp+m.
(29)
TIAN-NAN WANG, RUE-RON HSU, ...
VOL. 37
541
These results are exactly the same as Eq. (13) which is obtained from the recurrence scheme.
IV. Numerical verification
To check the correctness of our analytical derivation, we compare the analytical results
to the numerical results for two interesting systems: a logistic map and an Ikeda map. First,
let us consider the noisy logistic map which is contaminated by uniform white noise with
amplitude u = 0.001. Fig. 2(a) shows that the ratio between the standard deviation of
the stochastic response and the input noise, dK-$Z, in each m-th element of the
period-l, 2, 4 orbits in the noisy logistic map, z,+r = rz,(l - 2,) t qn, for 1.0 2 T 5
3.54. The normalized correlation function between the (k t 1)p-th step and the kp-th step,
JT, in each m-th element of the period-2 orbit in the logistic map for ! = I, 2,
are presented in Fig. 2(b) and 2(c), respectively. Similar comparisons for the case of the
noisy Ikeda map. zntl = 1 - ~(1 - COS(~Z,)) + qn, are demonstrated in Fig. 3(a)-(c).
When Gaussian white noise is added into the dynamical system, it is worth examining
the distribution function P(z kptm) directly. Fig. 4(a)-(c) show the numerical and analytical
distribution functions P(z++,,, ) of the m-th element of the period-2 orbit in the logistic
model with control parameters T = 3.1, l+ & and 3.3, respectively.
As one can see from the graphs, all the numerical results agree with the analytical
derivation very well, so long as the amplitude of the input noise is small enough, and the
dynamical system is not too close to the bifurcation point.
V. Concluding remarks
From the derivation of the general formula for the correlation functions for the elements in the period-p orbits, by using the recurrence scheme and the integral method, we
found that the recurrence approach is good for any white noise with zero mean, but the integral approach is only feasible for Gaussian white noise. However, the integral method gives
us the distribution function, with this, one can easily write down the higher moments of the
correlation function directly. Besides the supercycles ( M (P) = 0), the stochastic responses
in the elements of a period-p orbit are colored, having the correlation time r:P) = - 1.
This time correlation can be regarded as an inheritance from the deterministic prd”,!ZiZs
of the dynamical system.
The general formula Eq. (13) also indicates that the amplitude of the stochastic
response is always larger than the level of the input noise. The response of the supercycle
is the most attractive one in that the response of each element is independent of Ic, the
period-p iteration times, that is
P-1
(bptmbp+m) =
1
t
{
~(iq$l))2 D*
j=l
1
(30)
The minimum response occurs at the m = 1 elements of the supercycles in which M&-l) =
0, and the response is identical to the original input white noise (see Fig. 2(a) and Fig. 3(a)).
VOL. 37
STOCHASTIC RESPONSES OF THE STABLE PERIOD-p .
542
zQ r
k=lQ
,=I0
k:lQ
p=Z
p=4
1
I
k=lO,p=2.!=2
3
3.05
3.1
3.15
w
3.2
3.25
3.3
3.35
3.4
3
3.05
3.1
3.15
3.2
3.25
33
3.35 3 4
(4
FIG. 2. JFr via T for k = 10, f? = O,l, 2 are shown in (a), (b) and (c), respectively
(see text). Here, the dynamical system is the noisy logistic map with a noise amplitude
c = 0.001. The diamond (0) denotes the numerical simulation for 10,000 ensembles and
the solid lines indicate the analytical results.
VOL. 37
TIAN-NAN WANG, RUE-RON HSU,
543
L=,O
i=lO
L=Io
p=l
p=*
p-4
P-4
FIG. 3. Similar to Fig. 2 (a)-(c), but for the Ikeda map x,+1 = - ~(1 - cos(Tjzn)) + vn, i n s t e a d
of the logistic map.
544
STOCHASTIC RESPONSES OF THE STABLE PERIOD-p..
VOL. 37
15
PhP+‘n)
10
(b)
(4
20
15
P(w-t~ )
10
5
0
0
0.2
0.1
0.0
0.6
x
(4
FIG. 4. The distribution function P(x~~+~) for the m-th element of period-2 in a logistic map with
control parameters: (a) r = 3.1, (b) 1+ @and (c) T = 3.3. Here, k = 10 and the standard
deviation of the input Gaussian white noise is g = 0.1. The diamond (0) denotes the
numerical simulation and the solid lines indicate the analytical results.
Moreover, from the stability condition IAI(P)I < 1, one can deduct that
P-l
t
1I
c(M;;1))2
j=l
D,
(31)
for k >> 1. This means that after a large number of period-p iterations the correlation
functions on each element becomes stationary. Therefore, if one defines the long-time
average after large k (transient) as
the correlation function almost matches that computed by the ensemble average. Fig. 5
shows that Eq. (30) is a good approximation for the correlation function evaluated by the
long-time average of the stochastic response of the logistic map after the transient steps
TIAlY-NAN WANG, RUE-RON HSU,
VOL. 37
20
545
r-
p=l
p=2
FIG. 5. The diamond (0) denotes the numerical results for a correlation function evaluated by
a long-time average for different r in a logistic map. Here, we take 50,000 data for the
long-time average after 1,000 transient steps. The solid lines indicate the analytical results
of Eq. (30) for X: = 1,000, e = 0.
k = 1,000, if the neighborhood of the bifurcation point is excluded. This approximation has
been reported in our previous papers [lo-111. Indeed, those numerical results also indicate
that noise leads to a shift of the bifurcation points. This feature could account for the
failure of our schemes near the bifurcation points. It may imply that one should regard the
response of noise as parametric noise in the neighborhood of the bifurcation points, such
as Crutchfield has done before, or deal with the response by some other method [12].
As an application, we have used the recurrence scheme to study the stochastic response of colored noise and the noise cascading in a unidirectionally coupled element system
[ll]. Those results offer us an opportunity to develop suitable methods to control noise in
the system, see reference [ll].
The recurrence method can be extended to study the stochastic response of a periodic
orbit in a continuous time system, as long as one can write down the Poincare map for the
dynamical system. In fact, Svensmark and Samuelsen [12] obtained a similar result for a
period-l case in a noisy dynamical system, see Eq. (11) in reference [12]. They also found
546
STOCHASTIC RESPONSES OF THE STABLE PERIOD-p..
VOL. 37
another type of auto-correlation function in a noisy dynamical system under the influence of
a period-2 resonant perturbation. On the other hand, using the Floquet theory, Wiesenfeld
[13] obtained the noise response of for special model of continuous time system as an integral
of the noise source and the fundamental matrix, and gave a generic correlation function for
the period-l orbit. Both papers focus on the influence of noise in the vicinity of perioddoubling, but pay little attention to the correlation function for each element of the period-p
orbits.
Note in addition: After the completion of this manuscript, Prof. Weiss informed us
of his related works published about a decade ago [14]. In his paper, he calculated the
frrst and the second moments of the probability distribution for an N-dimensional noisy
map, up to the second order approximation. The ! = 0 case in our results is the first
order approximation of a one-dimensional noisy map of his result in slightly different form.
However, he did not state the response of each element in the stable orbits cxphcitly.
Moreover, he did not compute the time correlation function between the Icp-th step and the
(Ic +!)p-th step for the case 1 # 0, which reveals the inheritance of the dynamical properties
from the deterministic structure.
Acknowledgements
We thank Professor J. B. Weiss for informing us of his excellent work. This work was
partially supported by the National Science Council, Taiwan, R.O.C. under the contract
numbers. NSC 89-2112-M006-007.
References
[ 1 ] E. N. Lorenz, Ann. NY Acad. Sci. 35’7, 282 (1980).
[ 2 ] J. P. Crutchfield and Huberman, Phys. Lett A’77, 407 (1980).
[ 3 ] J. P. Crutchfield, M. N auenberg, and J. Rudnick Phys. Rev. Lett. 46, 933 (1981).
[ 4 ] B. Shraiman, C. E. Wayne, and P. Martin, Phys. Rev. Lett. 46, 935 (1981).
[ 5 ] M. J. Feigenbaum and B. Hasslacher, Phys. Rev. Lett. 49, 605 (1982).
[ 6 ] G. Mayer-Kress and H. Haken, J. Stat. Phys. 26, 149 (1981).
[ 7 ] J. P. Crutchfield, J. D. Farmer, and B. A. Huberman, Phys. Rep. 92, 45 (1982).
[ 8 ] H. G. Schuster, Deterministic Chaos: an introduction, 3rd ed., (Weinheim; New York; Basel;
Cambridge; Tokyo; VCH, 1995).
[ 9 ] T. Kapitaniak, Chaos in systems with noise, 2nd ed., (World Scientific, Singapore, 1990).
[lo] R.-R. Hsu et al., Phys. Rev. Lett. 78, 2936 (1997).
[ll] R-R. Hsu et al., Chin. J. Phys. 37, 292 (1999).
[12] H. Svensmark and M. R. Samuelsen, Phys. Rev. A36, 2413 (1987).
[13] K. Wiesenfeld, J. Stat. Phys. 38, 1071 (1985).
[14] J. B. Weiss, Phys. Rev. A35, 879 (1987).