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Author's personal copy

On the numerical simulation of propagation of micro-level inherentuncertainty for chaotic dynamic systems

Shijun Liao ⇑Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, ChinaState Key Lab of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, ChinaSchool of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

a r t i c l e i n f o

Article history:Received 8 October 2011Accepted 17 November 2012

a b s t r a c t

In this paper, an extremely accurate numerical algorithm, namely the ‘‘clean numericalsimulation’’ (CNS), is proposed to accurately simulate the propagation of micro-level inher-ent physical uncertainty of chaotic dynamic systems. The chaotic Hamiltonian Hénon–Heiles system for motion of stars orbiting in a plane about the galactic center is used asan example to show its basic ideas and validity. Based on Taylor expansion at ratherhigh-order and MP (multiple precision) data in very high accuracy, the CNS approachcan provide reliable trajectories of the chaotic system in a finite interval t 2 [0,Tc], togetherwith an explicit estimation of the critical time Tc. Besides, the residual and round-off errorsare verified and estimated carefully by means of different time-step Dt, different precisionof data, and different order M of Taylor expansion. In this way, the numerical noises of theCNS can be reduced to a required level, i.e. the CNS is a rigorous algorithm. It is illustratedthat, for the considered problem, the truncation and round-off errors of the CNS can bereduced even to the level of 10�1244 and 10�1000, respectively, so that the micro-level inher-ent physical uncertainty of the initial condition (in the level of 10�60) of the Hénon–Heilessystem can be investigated accurately. It is found that, due to the sensitive dependence oninitial condition (SDIC) of chaos, the micro-level inherent physical uncertainty of the posi-tion and velocity of a star transfers into the macroscopic randomness of motion. Thus,chaos might be a bridge from the micro-level inherent physical uncertainty to the macro-scopic randomness in nature. This might provide us a new explanation to the SDIC of chaosfrom the physical viewpoint.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Using high performance digit computers, a lot of com-plicated problems in science, finance and engineering havebeen solved with satisfied accuracy. However, there existsome problems which are still rather difficult to solve evenby means of the most advanced computers. One of them isthe propagation of micro-level inherent physical uncer-tainty of chaotic dynamical systems.

It is well-known that all numerical simulations are not‘‘clean’’: there exist more or less numerical noises such astruncation and round-off errors, which greatly depend onnumerical algorithms. In most cases, such kind of numeri-cal noises are much larger than the micro-level inherentphysical uncertainty of dynamic systems under consider-ation, so that the micro-level inherent uncertainty is com-pletely lost in the numerical noise. This becomes moreserious for chaotic dynamic systems, which have the sensi-tive dependence on initial conditions (SDIC), i.e. very tinychange of initial condition leads to great difference ofnumerical simulations of chaotic systems so that long-term prediction is impossible. Thus, very fine numericalalgorithms need be developed to accurately simulate the

0960-0779/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.chaos.2012.11.009

⇑ Address: State Key Lab of Ocean Engineering, Shanghai Jiao TongUniversity, Shanghai 200240, China.

E-mail address: [email protected]

Chaos, Solitons & Fractals 47 (2013) 1–12

Contents lists available at SciVerse ScienceDirect

Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier .com/locate /chaos


propagation of micro-level inherent physical uncertaintyof chaotic dynamic systems. This is the motivation of thisarticle.

In this article, a kind of numerical algorithm in a ratherhigh accuracy, called the ‘‘clean numerical simulation’’(CNS), is proposed to accurately simulate propagation ofmicro-level inherent physical uncertainty of chaotic dy-namic systems. Here, the word ‘‘clean’’ means that thetruncation and round-off errors can be controlled to anarbitrary level that is much less than the micro-level inher-ent physical uncertainty of the initial condition so that thenumerical noises can be neglected in a given finite intervalof time for the propagation of uncertainty. A chaotic Ham-iltonian system proposed by Hénon and Heiles [10] is usedto show its validity. The basic ideas of the so-called cleannumerical simulation (CNS) are given in Section 2, fol-lowed by the investigation of the micro-level uncertaintyof the system in Section 3 and its propagation in Section 4from statistical viewpoint. Conclusions and discussions aregiven in Section 5.

2. The numerical algorithm of the CNS

2.1. Basic ideas

Hénon and Heiles [10] proposed a Hamiltonian systemof equations

€xðtÞ ¼ �xðtÞ � 2xðtÞyðtÞ; ð1Þ€yðtÞ ¼ �yðtÞ � x2ðtÞ þ y2ðtÞ; ð2Þ

to approximate the motion of stars orbiting in a planeabout the galactic center, where the dot denotes the differ-entiation with respect to the time t. Its solution is chaoticfor some initial conditions, such as

xð0Þ ¼ 1425

; yð0Þ ¼ 0; _xð0Þ ¼ 0; _yð0Þ ¼ 0; ð3Þ

as mentioned by Sprott [23]. Without loss of generality, letus use this chaotic system to describe the basic ideas of theCNS and to illustrate its validity.

It is well-known [5,8,11,13,14,22,23,29] that chaotic dy-namic systems have the sensitive dependence on initialconditions (SDIC), i.e. a tiny change of initial conditionsleads to great difference of numerical simulations at largetime, so that long-term prediction of chaos is impossible.It is well-known that all numerical simulations containthe unavoidable truncation and round-off errors at eachtime-step. Generally speaking, most of traditional numeri-cal simulations of chaos are mixed with these numericalnoises and thus are not ‘‘clean’’. Because these numericalnoises of traditional numerical approaches are generallymuch larger than the micro-level inherent physical uncer-tainty of initial condition, the propagation of such kind ofphysical uncertainty of chaotic dynamic systems has neverbeen studied accurately, to the best of the author’sknowledge.

For numerical simulations of chaotic dynamic system,we must take rigorous account of numerical errors androunding, because ‘‘what is observed on the computerscreen would be completely unrelated to what was meant

to be simulated’’, as pointed out by Galatolo et al. [7]. Themethods of shadowing may gain accurate numerical simu-lations closed to true trajectories of hyperbolic dynamicsystems, but fail to have long shadowing trajectories forthose with a fluctuating number of positive finite-timeLyapunov exponents, as pointed out by Dawson et al. [3].Besides, it is found that numerical simulations of chaoticsystems given by low-order Runge–Kutta methods orTaylor expansion approaches have sensitive dependencenot only on initial conditions but also on numerical algo-rithms, so that different numerical schemes might lead tocompletely different long-term predictions, as pointedout by Lorenz [15,16] and Teixeira et al. [20,24].

In order to gain reliable chaotic solutions in a long inter-val of time, Liao [12] developed a numerical techniquewith extremely high accuracy, called here the ‘‘cleannumerical simulation’’ (CNS). Using the computer algebrasystem Mathematica with the 400th-order Taylor expan-sion for continuous functions and data in accuracy of800-digit precision, Liao [12] gained, for the first time,the reliable numerical results of chaotic solution of Lorenzequation in a long interval 0 6 t 6 1175 LTU (Lorenz timeunit). The basic ideas of the CNS are simple and straightfor-ward. Since the order of Taylor expansion is very high, thecorresponding truncation error is rather small. Besides,since all data are expressed in the accuracy of large-num-ber digit precision, the small enough round-off error isguaranteed. Thus, as long as the order of Taylor expansionis high enough and the digit-number of data is long en-ough, both of the truncation and round-off errors can bemuch smaller than the micro-level inherent physicaluncertainty so that the propagation of micro-level uncer-tainty of the initial condition can be simulated accuratelyin a long enough interval of time. Here, the ‘‘clean’’ numer-ical simulation means that the truncation and round-off er-rors can be controlled to an arbitrary level so that thenumerical noises can be neglected in a given finite intervalof time, as shown later. Currently, Liao’s ‘‘clean’’ chaoticsolution [12] of Lorenz equation is confirmed by Wanget al. [27] to be a reliable trajectory of Lorenz equation inthe interval 0 6 t 6 1175 LTU, who used parallel computa-tion with the multiple precision (MP) library: they gainedreliable chaotic solution of Lorenz equation up to 2500LTU by means of the 1000th-order Taylor expansion anddata in the accuracy of 2100-digit precision. Note that,similar to the so-called shadowing trajectories given bythe shadowing approach [21], such kind of ‘‘clean’’ numer-ical simulations given by the CNS are close to true trajecto-ries of chaotic systems.

The CNS is based on Taylor expansion at a rather high-order. Let (xn,yn) and ð _xn; _ynÞ denote the position and veloc-ity at the time tn = nDt, where Dt is a constant time-step.Assume that x(t), y(t) are M + 1 times differentiable onthe open interval (t, t + Dt) and continuous on the closedinterval [t, t + Dt]. According to Taylor theorem, we have

xðt þ DtÞ ¼ xðtÞ þXM

n¼1

anðtÞðDtÞn þ RxMðtÞ; ð4Þ

yðt þ DtÞ ¼ yðtÞ þXþ1n¼1

bnðtÞðDtÞn þ RyMðtÞ; ð5Þ

2 S. Liao / Chaos, Solitons & Fractals 47 (2013) 1–12


where

anðtÞ ¼1n!

dnxðtÞdtn ¼ xðnÞðtÞ

n!; bnðtÞ ¼

1n!

dnyðtÞdtn ¼ yðnÞðtÞ

n!ð6Þ

and

RxMðtÞ ¼ aMþ1ðn1ÞðDtÞMþ1

; t 6 n1 6 t þ Dt; ð7ÞRy

MðtÞ ¼ bMþ1ðn2ÞðDtÞMþ1; t 6 n2 6 t þ Dt; ð8Þ

are remainders of x(t) and y(t), respectively. Assuming that

jaMþ1ðtÞj < l; jbMþ1ðtÞj < l; t > 0; ð9Þ

it holds obviously

RxMðtÞ

�� < lðDtÞMþ1; Ry

MðtÞ�� < lðDtÞMþ1

: ð10Þ

Thus, we have the following theoremTheorem of truncation error If x(t), y(t) are M + 1 times

differentiable on the open interval (t, t + Dt) andcontinuous on the closed interval [t, t + Dt], and besides ifjx(M+1)(t)j/(M + 1)! < l and jy(M+1)(t)j/(M + 1)! < l for t > 0,where l > 0 is a constant, then the Taylor expansion

xðt þ DtÞ � xðtÞ þXM

n¼1

anðDtÞn; ð11Þ

yðt þ DtÞ � yðtÞ þXþ1n¼1

bnðDtÞn; ð12Þ

have the truncation errors less than l(Dt)M+1.The round-off error is determined by the accuracy of

data. To avoid large round-off error, all data are expressedin high accuracy of long-digit precision. For example, onecan use data in accuracy of 2M-digit precision, where Mis the order of Taylor expansions (11) and (12). Thus, forlarge enough M, the round-off error are rather small. Forexample, in case of M = 70, all data are expressed in accu-racy of 140-digit precision so that the correspondinground-off error is in the level of 10�140. Such kind of highprecision data can be gained easily by means of computeralgebra system like Mathematica and Maple, or the multi-ple precision (MP) library for FORTRAN and C. Obviously,the larger the value of M, the smaller the truncation andthe round-off errors. In this meaning, we can control thetruncation and round-off errors to a required level.

The coefficients an and bn can be calculated in a recur-sive way. Assume that a0 ¼ xn; b0 ¼ yn; a1 ¼ _xn; b1 ¼ _yn areknown. Substituting the Taylor expansions (11) and (12)into the original governing Eqs. (1) and (2) of the Hénonand Heiles system [10] and equaling the like power ofDt = t � tn, we have the recursion formula

anþ2 ¼ �an þ 2

Pnk¼0akbn�k

ðnþ 1Þðnþ 2Þ ; ð13Þ

bnþ2 ¼ �bn þ

Pnk¼0ðakan�k � bkbn�kÞðnþ 1Þðnþ 2Þ ð14Þ

for n P 0. Then, we have the Mth-order Taylorapproximation

xnþ1 �XM

k¼0

akðDtÞk; ynþ1 �XM

k¼0

bkðDtÞk ð15Þ

and

_xnþ1 �XM�1

k¼0

ðkþ 1Þakþ1ðDtÞk; ð16Þ

_ynþ1 �XM�1

k¼0

ðkþ 1Þbkþ1ðDtÞk ð17Þ

at the time tn+1 = (n + 1)Dt. Besides, all data are expressedhere in the accuracy of 2M-digit precision (we use the com-puter algebra system Mathematica). In this way, one gainsrather accurate numerical simulations of x(t) and y(t) stepby step in a finite interval of time, with extremely smalltruncation and round-off errors at each time-step, as veri-fied below.

For short time, both of the truncation and round-off er-rors are so small that the numerical results are often closeto the true trajectory. This is the reason why most ofnumerical results of chaotic systems given by different ap-proaches match well in a short time from the beginning. Itis widely believed by the scientific community that suchkind of numerical results of chaos in a short time is reli-able. However, due to the sensitivity on initial conditionsof chaotic dynamic system, the truncation and round-offerrors are amplified quickly so that the numerical resultsdepart greatly from the true trajectory after a critical timeTc. Here, Tc denotes such a maximum time that numericalresults gained by means of different numerical approaches(for example, with different M and Dt of the CNS) are closeto the true trajectory of chaotic solution in the interval0 6 t 6 Tc. In other words, the numerical results are‘‘clean’’, i.e. without observable influence by the round-off and truncation errors, and thus is reliable in the finiteinterval t 2 [0,Tc]. Here, the so-called critical predictabletime Tc is similar to the so-called shadowing time for theshadowing approach [3,21]. Mathematically, let u1(t) andu2(t) denote two time-series given by different numericalapproaches. The so-called ‘‘critical time’’ Tc is determinedby the criteria of decoupling

1� u1

u2

�� > d; _u1 _u2 < ��; at t ¼ Tc; ð18Þ

where � > 0 and d > 0 are two small constants (� = 1 andd = 5% are used in this article). In this paper, the criticaltime Tc is determined by the CNS approach, i.e. the valuesof M, Dt and the accuracy of data. Obviously, the larger M,the smaller Dt and the higher accuracy of data, the longertime interval [0,Tc] in which the numerical results matchwell with the true trajectory. For given reasonable Dt andhigh accuracy of data, the larger the value of M, the largerTc. So, Tc for given M is determined by comparing the cor-responding CNS result with that obtained by means of alarger value of M with the same initial condition, the sameDt and the same accuracy of data.

The key step of the CNS is to provide a good estimationof the critical time Tc, which is an important characteristiclength-scale of time for the CNS. Without loss of generality,we use in this article the Mth-order Taylor expansions (11)and (12) with Dt = 1/10 and the data in accuracy of (2M)-digit precision. Comparing different CNS results given bydifferent M, we gain the different values of Tc for differentM by means of the criteria (18). Then, by means of

S. Liao / Chaos, Solitons & Fractals 47 (2013) 1–12 3


regression analysis, it is found that Tc can be approximatelyexpressed by

Tc � 32ð1þMÞ: ð19Þ

For details of how to gain the above estimation of Tc, pleaserefer to Liao [12]. Seriously speaking, given two time seriesu1(t) and u2(t), different small values of � andd might give alittle different value of Tc. However, it is found that theestimation expression of Tc is not sensitive to the valuesof � and d, mainly because chaotic systems are sensitiveto numerical noises. Thus, (19) provides us a good estima-tion of the critical time Tc. For the sake of guarantee, it isbetter to choose a little larger value of M than that esti-mated by (19) in practice. For example, in order to gainreliable chaotic solution of the Hénon and Heiles system[10] in the interval 0 6 t 6 2000, say, Tc = 2000, we usethe 70th-order1 Taylor expansion (with Dt = 1/10) and thedata in accuracy of 140-digit precision. It should be men-tioned here that (19) is consistent with the conclusion aboutmethods of shadowing [21]: the shadowing time havepower law dependencies on the level of numerical noise.

Thus, given an arbitrary value of Tc, we can always cal-culate such a corresponding order M of Taylor expansionsthat the corresponding CNS result is reliable in the intervalt 2 [0,Tc], as verified below. In other words, given the criti-cal time Tc, the choice of the time-step Dt and the order Mof Taylor expansion for reliable trajectories in t 2 [0,Tc] isunder control. In this meanings, the CNS approach can beregarded as a ‘‘rigorous’’ one.

2.2. Validity of numerical simulations

As mentioned before, the larger the order M of Taylorexpansion and the more accurate the data, the better thecorresponding CNS results of chaotic system (1) and (2).The CNS results at t = 500, 1000, 1500 and 2000 given byM = 70 in case of the initial condition (3) are listed inTable 1.

Is it a reliable trajectory of the chaotic system (1)–(3) inthe interval 0 6 t 6 2000? To verify its validity, we re-peated computations by means of Dt = 1/10 and M = 100,150, 200, 300, 500, respectively, and found that all of themgive exactly the same trajectory in the interval0 6 t 6 2000, as listed in Table 1. Besides, even using asmaller time-setp Dt = 1/20 and Dt = 1/100 of the chaoticsystem (1)–(3), we always gain the exactly same trajectoryin the finite interval 0 6 t 6 2000 by means of M = 100,150, 200, 300 and 500, respectively. All of these indicatethat the CNS approach indeed provide us a reliable trajec-tory of the chaotic system (1) and (2) under the initial con-dition (3) in the finite interval 0 6 t 6 2000.

To verify the CNS results, let us further consider the le-vel of truncation and round-off errors. In case of M = 70 andD t = 1/10, the round-off error is in the level of 10�140. Thecorresponding truncation error of the CNS approach can beroughly estimated in the following way. According to our

CNS results, the maximum values of ja70j and jb70j are6.1 � 10�34 and 6.7 � 10�34, respectively. Since two diver-gent series decouple quickly due to the sensitive depen-dence on numerical noises, the Taylor series should beconvergent in the interval t 2 [0,Tc], i.e.

ja71jDtja70j

< 1;jb71jDtjb70j

< 1:

Thus, we have the estimation

ja71j < ja70j=Dt < 6:1� 10�33; jb71j < jb70j=Dt

< 6:7� 10�33:

Although there exist some uncertainty in the above deduc-tion, we have many reasons to assume that2

ja71j < 10�29; jb71j < 10�29; ð20Þ

i.e. l = 10�29. Then, according to (10), the truncation errorsshould be less than 10�100, which is rather small. Similarly,the truncation errors in case of Dt = 1/10 and M = 100, 150,200, 300 and 500 are less than 10�145, 10�219, 10�294,10�444 and 10�744, respectively, as shown in Table 2.

Similarly, in case of M = 70 and Dt = 1/20, the maximumCNS results of ja70j and jb70j are 6.1 � 10�34 and6.9 � 10�34, respectively, so that the two inequalities in(20) still hold, say, we have the same constantl = 10�29

for (9) to be valid, although the smaller time step Dt isused. It is found that, in case of M = 70 with much smallertime-step Dt = 1/100, the corresponding maximum CNSvalues of ja70j and jb70j are 6.2 � 10�34 and 7.1 � 10�34,which are very close to those found in case of M = 70 withDt = 1/10 and Dt = 1/20, so that we still have the same con-stant l = 10�29 for (9) to be valid! In fact, according to our

Table 1Reliable numerical results of Hénon and Heiles’ chaotic system (1)–(3)given by M = 70 and Dt = 1/10 with data in accuracy of 140-digit precision.

t x(t) y(t)

500 0.19861766 �0.238424311000 �0.04915404 �0.319716481100 �0.48949729 �0.040521611200 �0.04886847 0.777974911300 0.03097135 0.324012541500 0.03489977 0.434081692000 0.44371428 �0.30558921

Table 2Estimated level of the truncation and round-off errors of the CNS results ofthe chaotic system (1)–(3) in case of Dt = 1/10.

M Constant l for (9) Truncation error Round-off error

70 10�29 10�100 10�140

100 10�44 10�145 10�200

150 10�68 10�219 10�300

200 10�93 10�294 10�400

300 10�143 10�444 10�600

500 10�243 10�744 10�1000

1 The estimation formula (19) gives M � 62 for Tc = 2000. Consideringthat (19) is an estimation formula for the chaotic Hamiltonian Hénon-Heiles system, we choose M = 70 so as to ensure that the CNS results areindeed reliable trajectories in the interval 0 6 t 6 2000.

2 Here, we multiply the values at the right-hand side of the aboveexpressions by 104 and replace the number 6.1 and 6.9 by 1.0 for the sake ofsimplicity.



numerical simulations based on the CNS approach, it isfound that, for the same M but different time-stepDt 6 1/10, the two inequalities in (9) indeed hold withthe same constant l, as shown in Tables 2–4. All of theseverify the validation of (9) and therefore the correction ofour estimation for the truncation errors.

Note that, the larger the order M of Taylor expansionand the smaller the time-step Dt, the smaller the trunca-tion and round-off errors, as shown in Tables 2–4. Espe-cially, in case of M = 500 and Dt = 1/100, thecorresponding truncation error is in the level of 10�1244

and the round-off error is in the level of 10�1000, respec-tively, which are much smaller than those given byM = 70 and Dt = 1/10, so that we have many reasons to be-lieve that the numerical result given by M = 500 andDt = 1/100 is much closer to the true trajectory of chaoticsystem (1) and (2) under the given initial condition (3).However, it should be emphasized that all of our CNS re-sults given by M P 70 and Dt 6 1/10 are the same as thoselisted in Table 1. In other words, the CNS provides us thechaotic results that are independent of not only the orderM of Taylor expansion but also the time-step Dt and thedata precision. This guarantees that our CNS results givenby means of 70th-order Taylor expansion and data in accu-racy of 140-digit precision are indeed a true, reliable tra-jectory of the chaotic dynamic system (1) and (2) withthe initial condition (3), at least in the interval t 2 [0,2000].

According to Tables 2–4, the truncation and round-offerror of the CNS approach can be decreased to the levelof 10�1244 and 10�1000 (by means of Dt = 1/100 andM = 500), respectively. Thus, theoretically speaking, thetruncation and round-off error of the CNS approach canbe reduced to a required level. Besides, the CNS results gi-ven by Dt = 1/10 and M = 70 agree well (in the accuracy of8-digit precision) with all of the CNS results by the largerM P 70 and/or the smaller time-step Dt 6 1/10. All ofthese indicate that the CNS results give the reliable

trajectories of the chaotic system, and the CNS is a rigorousapproach.

In addition, to show the sensitive dependence on initialcondition, let us consider a different initial condition

xð0Þ ¼ 1425

; yð0Þ ¼ 10�60; _xð0Þ ¼ 0; _yð0Þ ¼ 0 ð21Þ

with a rather tiny difference of y(0), i.e. y(0) = 10�60, fromthe previous initial condition (3). The corresponding CNSresults given by Dt = 1/10, M = 70 and data in accuracy of140-digit precision are listed in Table 5. To verify that itis a reliable trajectory of the chaotic system 1, 2 and 21in the interval 0 6 t 6 2000, we repeat the CNS approachby means of Dt = 1/10, 1/20 and M = 100, 150, 200, 300,500, respectively, and always obtain the exactly same re-sults in the interval t 2 [0,2000] as those listed in Table 5.Thus, the CNS approach indeed provides the true trajectoryof the chaotic dynamic system (1), (2) and (21) in the re-stricted interval 0 6 t 6 2000. Note that, the initial condi-tion (21) with y(0) = 10�60 has a very tiny difference from(3) with y (0) = 0. According to Tables 1 and 5, the two reli-able (or shadowing) trajectories corresponding respec-tively to the different initial conditions (3) and (21),match well each other in the interval 0 6 t 6 1100. Evenat t = 1200, they still match in accuracy of 5-digit precision.However, due to the sensitive dependance on initial condi-tion, the two reliable (or shadowing) trajectories com-pletely depart from each other thereafter, although theirinitial conditions have only a tiny difference in the mi-cro-level 10�60.

All of these indicate that the CNS results given byDt = 1/10, M = 70 and data in the accuracy of 140-digit pre-cision are indeed reliable in the interval t 2 [0,2000]. Inother words, the CNS results given by M = 70 and Dt = 1/10can be regarded as a kind of ‘‘shadowing trajectory’’ of thechaotic system, as mentioned by Dawson et al. [3], but in arestricted interval 0 6 t 6 2000.

It should be emphasized that the difference 10�60 is in-deed rather small, which is however much larger than thetruncation error in the level of 10�100 and the round-off er-ror in the level of 10�140 of the CNS approach. Due to thisreason, the CNS provides us a tool to accurately investigatethe propagation of the micro-level inherent physicaluncertainty of chaotic Hénon–Heiles system, which is atthe level of 10�60 that is much larger than the numericalnoises of the CNS, as shown below.



70 10�29 10�122 10�140

100 10�44 10�176 10�200

150 10�68 10�265 10�300

200 10�93 10�355 10�400

300 10�143 10�535 10�600

500 10�243 10�895 10�1000



70 10�29 10�170 10�140

100 10�44 10�245 10�200

150 10�68 10�369 10�300

200 10�93 10�494 10�400

300 10�143 10�744 10�600

500 10�243 10�1244 10�1000

Table 5Reliable numerical results of Hénon and Heiles’ chaotic system (1) and (2)under a different initial condition (21) given by M = 70 and Dt = 1/10 withdata in accuracy of 140-digit precision.

t x(t) y(t)

500 0.19861766 �0.238424311000 �0.04915404 �0.319716481100 �0.48949729 �0.040521611200 �0.04886067 0.777978961300 0.42344110 �0.241514411500 �0.17190612 �0.213495142000 0.03364286 0.17136302



3. The micro-level physical uncertainty

Many, although not all, mathematical models have clearphysical background. A good model for physical problemsoften remains the fundamental properties and providesus a way to investigate and predict some of related physi-cal phenomenon. For example, the law of Newtonian grav-itation can describe and predict the motion of the moon ora satellite accurately. Besides, many CFD (computationalfluid dynamics) software based on mathematical modelscan predict the flows about a ship and an airplane in anacceptable accuracy. So, many of mathematical models re-veal physical truths of the related phenomenon.

Eqs. (1) and (2) provide us a model for the motion of astar orbiting in a plane about the galactic center, whichhas very clear physical background. In general, a goodmathematical model should remain the key physical char-acteristics of the corresponding natural phenomena. Sincethe Hénon–Heiles system has been widely accepted by sci-entific community, we have many reasons to believe that(1) and (2) as a mathematical model process the funda-mental physical characteristics of the motion of a starorbiting in a plane about the galactic center.

The kinetic status of a star is determined by its positionand velocity. In the frame of Newtonian gravity law, it isbelieved that the kinetic status of a star is inherently exactand the uncertainty of position and velocity come from theimperfect measure equipments which provide limitedknowledge. However, according to de Broglie [4], this tra-ditional idea is wrong: the position of a star contains inher-ent uncertainty. Besides, the quantum fluctuation mightinfluence the existence of the so-called ‘‘objective random-ness’’, which is independent of any experimental accuracyof the observations or limited knowledge of initial condi-tions, as suggested by Consoli et al. [2]. Furthermore, ‘‘allthe sources of complexity examined so far are actuallychannels for the amplification of naturally occurring ran-domness in the physical world’’, as suggested by Allegriniet al. [1].

It is a common belief of the scientific community thatthe microscopic phenomenon are essentially uncertainand random. To show this point, let us consider some typ-ical length scales of microscopic phenomenon widely usedin modern physics. For example, Bohr radius

r ¼ �h2

mee2 � 5:2917720859ð36Þ � 10�11 ðmÞ

is the approximate size of a hydrogen atom, where ⁄ is a re-duced Planck’s constant, me is the electron mass, and e isthe elementary charge, respectively. Besides, Comptonwavelength Lc = ⁄/(mc) is a quantum mechanical propertyof a particle, i.e. the wavelength of a photon whose energyis the same as the rest-mass energy of the particle, wherem is the rest-mass of the particle and c is the speed of light.It is the length scale at which quantum field theory be-comes important. The value for the Compton wavelengthof the electron is

Lc � 2:4263102175ð33Þ � 10�12 ðmÞ:

In addition, the Planck length

lP ¼ffiffiffiffiffiffi�hGc3

r� 1:616252ð81Þ � 10�35 ðmÞ ð22Þ

is the length scale at which quantum mechanics, gravityand relativity [6] all interact very strongly, where c is thespeed of light in a vacuum, G is the gravitational constant,and ⁄ is the reduced Planck constant. Especially, accordingto the string theory [19], the Planck length is the order ofmagnitude of the oscillating strings that form elementaryparticles, and shorter length do not make physical senses. Be-sides, in some forms of quantum gravity, it becomes impos-sible to determine the difference between two locations lessthan one Planck length apart. Therefore, in the accuracy ofthe Planck length level, the position of a star is inherentlyuncertain, so is its velocity. Note that this kind of micro-scopic physical uncertainty is inherent and has nothing todo with the Heisenberg uncertainty principle [9] and theability of human being.

On the other hand, according to de Broglie [4], any abody has the so-called wave-particle duality, and thelength of the so-called de Broglie wave is given by

k ¼ hmv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v

c

� �2r

; ð23Þ

where m is the rest mass, v denotes the velocity of thebody, c is the speed of light, h is the Planck’s constant,respectively. Note that, the de Broglie’s wave of a bodyhas non-zero amplitude, meaning that the position isuncertain: it could be almost anywhere along the wavepacket. Thus, according to the de Broglie’s wave-particleduality, the position of a star is inherent uncertain, too.

Therefore, it is reasonable for us to assume that the mi-cro-level inherent fluctuation of position of a star shorterthan the Planck length lp is essentially uncertain and/orrandom.

To gain the dimensionless Planck length lp, we use thedimeter of Milky Way Galaxy as the characteristic length,say, dM � 105 (light year) �9 � 1020 (m). Obviously,lp/dM � 1.8 � 10�56 is a rather small dimensionless number.As mentioned above, two (dimensionless) positions shorterthan 10�56 do not make physical senses. Thus, it is reason-able to assume the existence of the inherent uncertainty ofthe dimensionless position and velocity of a star in the nor-mal distribution with zero mean and the micro-level stan-dard deviation 10�60. Strictly speaking, such kind of micro-level inherent physical uncertainty should be added to theobserved values (x0,y0,u0,v0) of the initial conditions, espe-cially for chaotic dynamic systems whose solutions arerather sensitive to initial conditions.

Therefore, strictly speaking, the initial condition shouldbe expressed as follows

xð0Þ ¼ x0 þ ~x0; yð0Þ ¼ y0 þ ~y0; _xð0Þ ¼ u0 þ ~u0; _yð0Þ¼ v0 þ ~v0;

where x0, y0, u0, v0 are observed values of the initial posi-tion and velocity of a star orbiting in a plane about thegalactic center, and ~x0; ~y0; ~u0; ~v0 are the corresponding mi-cro-level inherent uncertain ones, respectively. Assumethat (x0,y0,u0,v0) is exactly given and the inherent uncer-tain term ð~x0; ~y0; ~u0; ~v0Þ is in the normal distribution with



zero mean and a micro-level deviation r = 10�60. The rea-sons for the above assumptions are described above.

Compared to the scale of the initial data x0 = 14/25, thedeviation 10�60 is indeed rather small. However, by meansof the CNS approach with the 70th-order Taylor expansionand the MP data in accuracy of 140-digit precision, thepropagation of such kind of micro-level uncertainty canbe accurately studied, because the corresponding trunca-tion error (in the level of 10�100) and round-off error (inthe level of 10�140) of the CNS approach is much smallerthan the micro-level uncertainty (in the level of 10�60),as verified in Section 2.

4. Statistic property of chaos

Without loss of generality, let us consider the case ofthe observed values

x0 ¼1425

; y0 ¼ 0; u0 ¼ 0; v0 ¼ 0

of the initial conditions, corresponding to a chaotic motion[22]. The so-called observed values can be regarded as themean of measured data. Let

hxðtÞi ¼ 1N

XN

i¼1

xiðtÞ; rxðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N � 1

XN

i¼1

½xiðtÞ � hxðtÞi�2vuut

denote the sample mean and unbiased estimate of stan-dard deviation of x(t), respectively, where N = 104 is thenumber of total samples, xi(t) is the ith sample given bythe CNS using Dt = 1/10, M = 70 with the MP data in accu-racy of 140-digit precision, and a tiny random termð~x0; ~y0; ~u0; ~v0Þ with the micro-level deviation r = 10�60 inthe initial condition.

According to Section 2, all of these 104 trajectories givenby the CNS approach are reliable in the intervalt 2 [0,2000]. The standard deviations rx(t) and ry(t) of

x(t), y(t) are as shown in Figs. 1 and 2, respectively. Notethat there exists an interval 0 6 t 6 Td with Td � 1000, inwhich rx(t) and ry(t) are in the level of 10�14 so that onecan accurately predict the position (x,y) of a star, even ifthe corresponding motion is chaotic and the initial condi-tion contains uncertainty. Similarly, the velocity of the starcan be also precisely predicted in 0 6 t 6 Td, as shown inFig. 3 for the standard deviation ru(t) of _xðtÞ. Thus, when0 6 t 6 Td, the behavior of the chaotic system looks like‘‘deterministic’’ and ‘‘predictable’’, even from the statisticviewpoint. When t > Td, the standard deviations of the po-sition and velocity begin to increase rapidly, and thus thesystem becomes random obviously: the position (x,y) andvelocity ð _x; _yÞ of the star are strongly dependent upon their

time

Standarddeviation

σ x

0 500 1000 1500 200010-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Fig. 1. The standard deviation rx of x in case of x0 = 14/25, y0 = 0, u0 = 0,v0 = 0 and the uncertain term ð~x0; ~y0; ~u0; ~v0Þ in the normal distributionwith zero mean and a micro-level deviation r = 10�60.

time

Standarddeviation

σ y

0 500 1000 1500 200010-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Fig. 2. The standard deviation ry of y in case of x0 = 14/25, y0 = 0, u 0 = 0,v0 = 0 and the uncertain term ð~x0; ~y0; ~u0; ~v0Þ in the normal distributionwith zero mean and a micro-level deviation r = 10�60.

time

Standarddeviation

σ u

0 500 1000 1500 200010-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Fig. 3. The standard deviation ru of _x in case of x0 = 14/25, y0 = 0, u0 = 0,v0 = 0 and the uncertain term ð~x0; ~y0; ~u0; ~v0Þ in the normal distributionwith zero mean and a micro-level deviation r = 10�60.



micro-level inherent physical uncertainty ð~x0; ~y0; ~u0; ~v0Þ ofthe initial condition. In other words, due to the SDIC ofchaos, the unobservable micro-level inherent uncertaintyof the position and velocity of a star transfers into the mac-roscopic randomness of the motion. This suggests thatchaos might be a bridge from the micro-level uncertaintyto macroscopic randomness! Therefore, the micro-levelinherent uncertainty of the position and velocity mightbe an origin of the macroscopic randomness of motion ofstars in our universe. Possibly, this might provide us anew, physical explanation and understanding for the SDICof chaos. For this reason, each ‘‘big bang’’ [18] will createa completely different universe!

Besides, it is found that the standard deviations of theposition and velocity become almost stationary whent > Ts, where Ts � 1300, as shown in Figs. 1–3. Thus, whenTd < t < Ts, the system is in the transition process from the‘‘deterministic’’ behavior to the stationary randomness. Itis interesting that the stationary standard deviations ofx(t) and y(t) are about 1/3, and their stationary means hxiand hyi are close to zero. It means that, due to SDIC of chaosand the micro-level inherent uncertainty of position andvelocity, a star orbiting in a plane about the galactic centercould be almost everywhere in the galaxy at a given timet > Ts.

Write the fluctuations x0(t) = x � hxi and y0(t) = y � hyi.The stationary cumulative distribution functions (CDF) ofx0, y0 are almost independent of time, as shown in Figs. 4and 5. Besides, the stationary CDF of the fluctuation x0 israther close to the normal distribution with zero meanand the standard deviation of x0, as shown in Fig. 4. But,the stationary CDF of the fluctuation y0 is obviously differ-ent from the normal distribution, as shown in Fig. 5.

Similarly, we investigate the influence of the observedvalues (x0,y0,u0,v0) and the standard deviation r of theuncertain terms ð~x0; ~y0; ~u0; ~v0Þ in the initial condition bymeans of the CNS approach. It is found that Td decreases

exponentially with respect to r. Besides, the stationarymeans and standard deviations of x; y; _x; _y, and the CDFsof x0 and y0, are independent of the observed values(x0,y0,u0,v0). Thus, when t > Ts, all observed informationof the initial condition are lost completely. In other words,when t > Ts, the asymmetry of time breaks down so thatthe time has a one-way direction, i.e. the arrow of time.So, statistically speaking, the Hénon–Heiles system hastwo completely different dynamic behaviors before andafter Td: it looks like ‘‘deterministic’’ and ‘‘predictable’’without time’s arrow when t 6 Td, but thereafter rapidlybecomes obviously random with a arrow of time.

Consoli et al. [2] suggested that the objective random-ness ‘‘might introduce a weak, residual form of noise whichis intrinsic to natural phenomena and could be importantfor the emergence of complexity at higher physical levels’’.Our extremely accurate numerical simulations based onthe CNS approach support their viewpoint: the micro-leveluncertainty and the macroscopic randomness might have arather close relationship.

5. Conclusions and discussions

In this paper, an extremely accurate numerical algo-rithm, namely the ‘‘clean numerical simulation’’ (CNS), isproposed to accurately simulate the propagation of mi-cro-level inherent physical uncertainty of chaotic dynamicsystems. The chaotic Hénon–Heiles system describing themotion of a star orbiting in a plane about the galactic cen-ter is used as an example to show the validity of the CNSapproach.

In the frame of the CNS approach, the truncation error isestimated by (9), the round-off error is determined by thedigit-length of data, and the critical time Tc is explicitlydetermined by (19) (for the chaotic Hénon–Heiles system).So, given an arbitrary value of Tc, we can always find outthe required order M of Taylor expansion and the data in

x’

CDF

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1Dashed line: normal distribution

Solid line: t = 1500

Symbols: t = 2000

Fig. 4. The CDF of x0 , compared to the normal distribution (dashed line)with zero mean and the standard deviation of x0 at t = 2000. Solid line:CDF of x0 at t = 1500; symbols: CDF of x0 at t = 2000.

y’

CDF

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1Dashed line: normal distribution

Solid line: t = 1500

Symbols: t = 2000

Fig. 5. The CDF of y0 , compared to the normal distribution (dashed line)with zero mean and the standard deviation of y0 at t = 2000. Solid line:CDF of y0 at t = 1500; symbols: CDF of y0 at t = 2000.



accuracy of 2M-digit precision so as to gain a reliable tra-jectory of the chaotic Hénon-Heiles system in the finiteinterval t 2 [0,Tc] by means of Dt = 1/10. In addition, theCNS results in the interval t 2 [0,Tc] are verified very care-fully by means of Taylor expansion at higher-order andMP data in more accuracy, as shown in Section 2. As shownin Section 2, all of the CNS results (for the same initial con-dition) given by Dt = 1/10, M = 70 and data in accuracy of140-digit precision are exactly the same as those given byM = 100, 150, 200, 300, 500 and Dt = 1/20, 1/100, respec-tively, so that they are indeed reliable, true trajectories ofthe chaotic Hénon–Heiles system. Besides, as the order Mof Taylor expansion increases and the time-step Dt de-creases, the truncation and round-off errors decreasemonotonously. For example, as illustrated in Tables 2–4,the truncation error is in the level of 10�100 in case ofDt = 1/10 and M = 70, and decreases to the level 10�1244

in case of Dt = 1/100 and M = 500. In addition, the round-off error is simply in the level of 10�2M, where M denotesthe order of Taylor expansion. So, theoretically speaking,one can control the truncation and round-off error to arequired level. In these meanings, the CNS approach is arigorous one.

The Hénon–Heiles system of (1) and (2) as a mathemat-ical model has clear physical background: it has beenwidely accepted and used by the scientific community todescribe the motion of a star orbiting in a plane aboutthe galactic center. The status of a star is dependent uponits position and velocity. However, according to de Broglie[4], the position of a star contains micro-level inherentphysical uncertainty, as discussed in Section 3. So, strictlyspeaking, the Hénon–Heiles system of (1) and (2) is notdeterministic in essence. Due to the SDIC of chaos, suchkind of micro-level physical uncertainty transfers intomacroscopic randomness of motion, as illustrated in Sec-tion 4 by means of the CNS approach. Therefore, the mi-cro-level inherent physical uncertainty and macroscopicrandomness might have a close relationship: chaos mightbe a bridge from the micro-level inherent physical uncer-tainty to macroscopic randomness! This conclusion agreeswith the viewpoint of Consoli et al. [2] who suggested thatthe objective randomness ‘‘might introduce a weak, resid-ual form of noise which is intrinsic to natural phenomenaand could be important for the emergence of complexityat higher physical levels’’.

The CNS approach provides us an extremely precisenumerical approach for chaotic dynamic systems in a givenfinite interval t 2 [0,Tc]. According to (19), Tc ? +1 asM ? +1. In other words, if the initial condition were ex-act, then long-term prediction of chaos would be possiblein theory3: given an arbitrary value of Tc, we could gainthe reliable chaotic trajectory of the Hénon–Heiles systemin the finite interval 0 6 t 6 Tc by means of the Mth-orderTaylor expansion with data in accuracy of 2M-dight preci-sion, as long as M > Tc/32 and Dt 6 1/10. Qualitatively, theconclusion has general meanings and holds for other chaoticmodels such as Lorenz equation. Besides, it is consistent

with Tucker’s elegant proof [25,26] that there indeed existsan attractor of Lorenz equation. Thus, theoretically speaking,there is no place for the randomness in a truly deterministicsystem. However, most models related to physical problemscontain more or less physical uncertainty, and thus, strictlyspeaking, are not deterministic. For such kind of physicalmodels with inherent uncertainty, the accurate long-termprediction of trajectories of chaotic system has no physicalmeanings, because their long-term trajectories are inherentlyrandom that comes from the micro-level inherent physicaluncertainty, as illustrated in this article.

Traditionally, it is believed that the long-term predic-tion of chaos is impossible, mainly due to the impossibilityof the perfect measurement of initial conditions with anarbitrary degree of accuracy. This is the traditional expla-nation to the SDIC of chaos. Here, we provide a new expla-nation for the SDIC of chaos from the physical viewpoint:initial conditions of some chaotic systems with clear phys-ical meanings might contain micro-level inherent physicaluncertainty, which might propagate into macroscopic ran-domness. Different from the traditional explanation of theSDIC, which focuses on the measurement, the new explana-tion emphasizes the inherent micro-level uncertainty andits propagation with chaos. Besides, it should be empha-sized that such micro-level inherent physical uncertaintyof chaos was completely inundated with the numericalnoises of the traditional numerical methods based onlow-order algorithms, and thus has never been studied indetails. This shows the validity and potential of the CNSto precisely simulate complex physical phenomena withthe SDIC, such as weather prediction and turbulence.

Finally, for the easier understanding of the CNS, let usconsider the map

f ðxÞ ¼modð2x;1Þ ð24Þ

with the initial value x0 = p/4. It is well-known that thismap has the sensitivity dependence on initial condition,i.e. SDIC. The results of the nth mapping, i.e. xn = f(xn�1)with x0 = p/4, are expressed by both of the decimal andbinary systems in Table 6. In binary system, the mappingxn corresponds to such a kind of left shift: shifting x0 leftto the position of its 2nd digit ‘‘1’’ gives x1, and to the posi-tion of its 3rd digit ‘‘1’’ gives x2, and so on, as shown in Ta-ble 6. In general, xn (in binary system) corresponds to theleft shift of x0 (in binary system) to its position of the(n + 1) th digit ‘‘1’’. Since p/4 is exactly known in binarysystem, its position of the nth digit ‘‘1’’ is deterministic, de-noted by P2(n). So, in binary system, xn is exactly the leftshift of x0 to its P2(n + 1)th digit ‘‘1’’. So, mathematically

Table 6Mapping of f(x) = mod (2x,1) with x0 = p/4, expressed in decimal and binarysystems.

n xn in decimal system xn in binary system

0 0.785398163397448. . . 110010010000111111011010l. . .

1 0.570796326794896 . . . 10010010000111111011010 . . .

2 0.141592653589793 . . . 10010000111111011010 . . .

3 0.283185307179586 . . . 10000111111011010 . . .

..

. ... ..

.

3 Unfortunately, the required CPU time increases exponentially as Mincreases, so that it is practically impossible to give reliable, true trajectoriesof chaos in a very large interval.



speaking, this mapping is deterministic and xn is exactlyknown. However, in practice, one had to take x0 = p/4 ina finite accuracy, which leads to uncertainty. Assume thatx0 is in accuracy of N0 binary digits. Then, x1, x2, x3 hasN0 � 1, N0 � 4, N0 � 7 significance binary digits, respec-tively, as shown in Table 6. In general, xn is in the accuracyof N0 � P2(n + 1) binary digit precision. Obviously, whenP2(n + 1) > N0, the mapping xn losses its accuracy at all.However, even at one million times of mapping, i.e.n = 106, we can gain the accurate enough result x1000000 inthe accuracy of one million of binary digit precision, aslong as we take the initial value x0 = p/4 in the accuracyof 106 + P2(106 + 1) binary digit precision! This simpleexample illustrates that we do can gain reliable resultsfor dynamic systems with SDIC in a finite times of mappingor a finite interval, as long as initial conditions are accurateenough. This also explains why the CNS is based on ratheraccurate data, using the computer algebra system Math-ematica or the multiple precision library.

However, a chaotic dynamic system has no such kind ofelegant property of mod (2x,1) mentioned above, since itsexact solution is unknown in general. Thus, the above ap-proach based on the left shift has no general meanings. As-sume that one knows the SDIC of the mapping f(x) = mod(2x,1), but has no ideas about its left-shift property inthe corresponding binary system. How to gain reliable se-quence xn = f(xn�1) by means of x0 = p/4? A general,straight-forward way is to compare two sequences givenby x0 = p/4 in different accuracy of N-digit precision (indecimal system), where N = 15, 20, 25, 30 and 1000,respectively, as shown in Table 7. For example, by compar-ing the two sequences of xn given by x0 in accuracy of 15and 20-digit precision, one is sure due to the SDIC thatthe sequence xn given by x0 in accuracy of 15-digit preci-sion is reliable at n 6 15 in accuracy of 8 significance digits.Similarly, using x0 in accuracy of 20 and 25-digit preci-sions, one gains reliable xn at n 6 30 and n 6 40 with 8 sig-nificance digits, respectively. Note that the sequence xn

given by x0 in accuracy of 25-digit precision agrees wellwith that by x0 in accuracy of 30-digit precision for a finitenumber of mappings xn, where 0 6 n 6 40. Thus, one hasmany reasons to believe that the finite sequence x0, x1,. . ., x40 given by means of x0 in accuracy of 25-digit preci-sion is reliable. This is indeed true, because it completelyagrees with the ‘‘exact’’ sequence given by x0 in accuracyof 1000-digit precision, as shown in Table 7. The key pointis that, to gain reliable sequence x0,x1, . . ., x40 with the finite

number of mappings, we need use x0 in accuracy of only25-digit precision: it is unnecessary to use x0 in higheraccuracy. Similarly, using x0 in accuracy of 40-digit preci-sion, one can gain reliable sequence

x0; x1; x2; � � � ; x100

in accuracy of 8 significance digits. Furthermore, using x0

in accuracy of 60-digit precision, one can gain reliablesequence

x0; x1; x2; . . . ; x166

in accuracy of 8-digit precision as well. And so on. Thus, wecan gain reliable xn of finite but many enough mappings byusing accurate enough x0. In other words, the reliability andprecision of the finite sequence

x0; x1; x2; . . . ; xn

given by the mapping f(x) = mod (2x,1) with SDIC is undercontrol. It is true that, using x0 in 60-digit precision, x100000

is incorrect and thus has no meaning. However, the keypoint is that the corresponding sequence

x0; x1; x2; . . . ; x166

of a finite number of mappings is reliable in accuracy of 8-digit precision, which might be enough for one’s purpose.In essence, we seek for a kind of relative reliability and pre-dictability of chaotic dynamic systems, although very long-term accurate prediction of any a chaotic dynamic systemis absolutely impossible in theory. It is true that, using x0 inaccuracy of any a given precision, there absolutely existssuch a large enough n that xn totally losses its accuracy.However, we can guarantee the reliability and predictabil-ity of a given finite sequence (such as x0,x1,x2, . . . ,x166) byusing x0 in a reasonable accuracy (such as 60-digit preci-sion). It should be emphasized that such kind of compari-son approach is valid for any chaotic dynamic systems.So, it has general meanings and thus is practical. Note thatthe same comparison approach is used in the CNS de-scribed in Section 2 and Section 3. This example clearly ex-plains why the CNS based on such kind of comparison isindeed reasonable and valid.

It is important to provide a practical numerical ap-proach to gain reliable chaotic solutions of dynamic sys-tems in a long enough interval of time. Using CNS with400-order Taylor expansion, data in accuracy of 800-digitprecisions and Dt = 10�2, Liao [12] gained, for the firsttime, a reliable chaotic solution of Lorenz equation in a

Table 7xn given by the mapping f(x) = mod (2x,1) with the initial value x0 = p/4 in different accuracy of N decimal digit precision.

n N = 15 N = 20 N = 25 N = 30 N = 1000

5 0.1327412287 0.1327412287 0.1327412287 0.1327412287 0.132741228710 0.2477193189 0.2477193189 0.2477193189 0.2477193189 0.247719318915 0.9270182076 0.9270182075 0.9270182075 0.9270182075 0.927018207520 0.664582643 0.6645826427 0.6645826427 0.6645826427 0.664582642725 0.2666446 0.2666445682 0.2666445682 0.2666445682 0.266644568230 0.532626 0.5326261849 0.5326261849 0.5326261849 0.532626184935 0.0440 0.044037917 0.0440379171 0.0440379171 0.044037917140 0.409 0.40921335 0.4092133503 0.4092133503 0.4092133503



rather long time interval 0 6 t 6 1000, whose correction isconfirmed by Wang et al. [27]. As mentioned by Wang [28],in order to gain reliable chaotic solution of Lorenz equationin the interval 0 6 t 6 1000 by means of the 4th-orderRunge–Kutta method, one had to use multiple precisiondata in 10000-digit precision and a rather small time-stepDt = 10�170, which however needs about 3.1 � 10160 yearsby today’s high-performance computer! Therefore, thelow-order Taylor expansion approaches are not practicalto gain reliable chaotic solution of Lorenz equation in sucha long time interval. There exist some ‘‘rigorous’’ simula-tions [26] assuring that the real orbits of chaotic systemare ‘‘enclosed’’ in a computed region of space, such as[x(t) � d,x(t) + d], where d should be a small constant: re-sults with large d is useless in practice, even though it isobtained by ‘‘rigorous’’ methods. Due to SDIC, it is obviousthat one had to use rather small d to gain such a rigorouschaotic solution of Lorenz equation in 0 6 t 6 1000 bymeans of the enclosing approach: possibly d might be inthe level of 10�480, since the corresponding initial condi-tion must be accurate in 480 digit precision, as pointedout by Liao [12]. However, to the best of author’s knowl-edge, it is still an open question whether or not the ‘‘rigor-ous’’ method based on enclosing [26] can give such areliable, accurate enough chaotic solution of Lorenz equa-tion in the interval 0 6 t 6 1000 by means of a reasonableCPU time. Besides, to the best of author’s knowledge, it isalso an open question whether or not the enclosing ap-proach is practical for physical problems like those consid-ered in this article: note that the CNS is successfully usedto gain 10000 samples of reliable chaotic solutions givenby different initial conditions with 10�60-level uncertainty.So, compared to other approaches, the CNS is not only reli-able but also practical.

Indeed, the propagation of round-off and truncation er-rors of a chaotic dynamic system is rather complicated andthus is unknown in general cases. As pointed out by Parkerand Chua [17], a ‘‘practical’’ way of judging the accuracy ofnumerical results of a non-linear dynamic system is to useat least two (or more) ‘‘different’’ routines to integrate the‘‘same’’ system. This is mainly because, due to the SDIC ofchaotic dynamics systems, departure of two chaotic simu-lations indicates the appearance of large enough trunca-tion and round-off errors. In practice, the comparisonapproach provides us a time interval 0 6 t 6 Tc, in whichthe same results should be reliable, mainly due to SDICof chaos. Certainly, such kind of critical time Tc must becarefully checked by as many different approaches aspossible, as shown in Section 2.2. In fact, such kind ofcomparison approach is widely accepted by scientificcommunity [17,24,27,28]. And the CNS is based on suchkind of strategy. Using a metaphor, it is like measuringthe height of a man: the better the equipment, the moreaccurate the result, although we can not provide an ‘‘exact’’value of the height. Although it is difficult to measure theheight of a man in accuracy of 10�10 meter, it is rather easyto ensure that whether a man is higher than 1.85 m or not,as long as all measures given by all equipments give us thesame answer to this question: such kind of precision is rel-atively rough but enough in many cases of everyday life.Similarly, the CNS seeks for reliable, accurate enough

simulations of chaotic dynamic systems in a finite time-interval.

In summary, the CNS provides us a practical way to gainreliable, accurate enough solutions of chaotic dynamic sys-tems with a high enough precision in a finite but long en-ough time interval.

Acknowledgement

The author would like to express his sincere thanks tothe two reviewers for their comments, which greatlyheighten the quality of this article. Thanks to Prof. Y.L.Bei (Chinese Academy of Sciences), Prof. M.G. Xia (ChineseAcademy of Sciences), Prof. H.R. Ma (Shanghai Jiao TongUniversity) for their valuable discussions.

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