10−4 10−3 10−2 10−1 100

D

6.0

6.5

7.0

7.5

8.0

8.5

9.0

<x

2 >

(a)

10−6 10−5 10−4 10−3

D

0.54

0.55

0.56

0.57

0.58

0.59

< s

in x

>

Dc of [1]

(b)

10−7 10−6 10−5

D−Dc

10−5

10−4

10−3

10−2

∆G

slope=2.1

(c)

FIG. 1. (a) hx2i for the linear map vs D with a � 0:7 and b �0:75 (diamonds) and analytical fit hx2i � x2 �

11�a2 D (full line).

(b) hsinxi for the logistic map with a � 3:8008 and a polynomialfit for D � 1:4� 10�5 (full line). (c) Scaling G� �D�Dc�

�

with Dc 7� 10�6 (circles) and a least-squares fit (full line).All numerical averages are performed by 108 iterations.

PRL 94, 219402 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending3 JUNE 2005

Comment on ‘‘Shadowability of Statistical Averages inChaotic Systems’’

Lai et al. [1] investigate systems with a stable periodicorbit and a coexisting chaotic saddle. They claim that, byadding white Gaussian noise (GN), averages change withan algebraic scaling law above a certain noise thresholdand argue that this leads to a breakdown of shadowabilityof averages. Here, we show that (i) shadowability is notwell defined and, even if it were, no breakdown occurs. Weclarify misconceptions on (ii) thresholds for GN. We fur-ther point out (iii) misleading ideas on the effect of noiseon averages. Finally, we show that (iv) the lack of a properthreshold can lead to any (meaningless) scaling exponent.

(i) Shadowing deals with macroscopic error propagationof noise bounded by a small value (e.g., computer accuracyof 10�16), comparing a noisy and a ‘‘true’’ trajectory [2].Since the authors use GN, one cannot properly speak ofshadowing. However, even for bounded noise, their claimsare still doubtful. In a recent work, the error of an averagewas found to be amplified by a factor of up to 1012,although the trajectory never exits the attractor [3]. Thisrenders a reliable computation truly unfeasible. In [1],though, the effect on the averages is only of the same orderas the variation of the noise level D. Furthermore, theaverages in Figs. 1a and 2a in [1] do change even lessbelow D � 10�2:5. Therefore, shadowability is not com-promised at all.

(ii) The authors of [1] mention a threshold for GN abovewhich the periodic orbit and the chaotic saddle wouldbecome connected. Yet, such a threshold does not exist,since the mean first exit time h�i is given by Kramers’ lawh�i � exp�UD �, where D is the noise variance and U iseither the potential [4] or, for nonequilibrium and chaoticsystems, the quasipotential difference [5]. Thus, h�i varieswith D, yielding a different average for every noise level.

(iii) Generally, averages depend on the noise for allnoise levels, implying that no threshold can exist, evenwith only one metastable state. For the linear map xn�1 �axn � b� n with a fixed point x �

b1�a and white GN

one gets hx2i � x2 �1

1�a2D. This is pictured in Fig. 1(a),

fitting perfectly the data. Although the average appears tobe constant for low noise, it depends, in fact, on the noisefor all D. The same applies also to nonlinear systems [cf. fitin Fig. 1(b)] and bounded noise.

(iv) Because the value of the threshold is arbitrary, anyscaling can be achieved, just by tuning Dc, as fittings of theform of Eq. (1) of [1] are very sensitive to the value of Dc.To verify this, we show in Fig. 1(b) the logistic map xn�1 �axn�1� xn� �Dn with hn; mi � nm as in [1]. Theputative threshold of Dc � 10�5 is marked by an arrow.The graph is evidently not constant there. With Dc 7�10�6 (first arrow), a scaling G� �D�Dc�

� results,

0031-9007=05=94(21)=219402(1)$23.00 21940

where G is as in [1]. This yields � 2:1 [Fig. 1(c)], inclear contrast to � 1 reported in [1]. Therefore, noreliable (i.e., any) exponent can be obtained, since Dc isnot well defined from the outset.

As to the reply, the authors claim to have a theoretical

justification for the existence of a threshold through Dc ��������������������������= ln��1

p, with � the quasipotential [see (ii)] and �

the probability resolution. Contrary to Ref. [2] of the reply,where � drops out since only proportionalities under pa-rameter variation are considered, Dc in [1] depends forfixed parameters on � and hence � is not uniquely defined.Applying this to (iv) with Dc � 10�5 of [1] and � 7:8� 10�10 (not shown) yields � � 0:0004. But a finerresolution � � 10�7 gives Dc � 7� 10�6, with � 2:1[cf. (iv)], whereas � � 10�2 results in Dc � 1:3� 10�5

and � 0:21 (not shown). Again, this invalidates anyproper scaling, as the finite observation time (i.e., resolu-tion) is arbitrary and does change the claimed universalityof the scaling exponent to an arbitrary value. Furthermore,we are neither claiming nor implying anywhere that theLyapunov exponent is positive for weak noise.

Suso KrautInstituto de FısicaUniversidade de Sao PauloCaixa Postal 66318, 05315-970 Sao Paulo, Brazil

Received 12 March 2004; published 31 May 2005DOI: 10.1103/PhysRevLett.94.219402PACS numbers: 05.45.-a

2-1

[1] Y. C. Lai et al., Phys. Rev. Lett. 89, 184101 (2002).[2] C. Grebogi et al., Phys. Rev. Lett. 65, 1527 (1990).[3] T. D. Sauer, Phys. Rev. E 65, 036220 (2002).[4] H. A. Kramers, Physica (Amsterdam) 7, 284 (1940).[5] R. Graham and T. Tel, Phys. Rev. Lett. 52, 9 (1984);

R. Graham, A. Hamm, and T. Tel, ibid. 66, 3089 (1991).

2005 The American Physical Society