Chaos, Solitons & Fractals 45 (2012) 805–809

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

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

Entropy estimation of the Hénon attractor

Chihiro Matsuoka a,⇑, Koichi Hiraide b

a Department of Physics, Graduate School of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama, Ehime 790-8577, Japanb Department of Mathematics, Graduate School of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama, Ehime 790-8577, Japan

a r t i c l e i n f o

Article history:Received 8 June 2011Accepted 18 February 2012Available online 31 March 2012

0960-0779/$ - see front matter � 2012 Elsevier Ltdhttp://dx.doi.org/10.1016/j.chaos.2012.02.013

⇑ Corresponding author.E-mail address: matsuoka.chihiro.mm@ehime-u.

a b s t r a c t

The topological entropy of the Hénon attractor is estimated using a function that describesthe stable and unstable manifolds of the Hénon map. This function provides an accurateestimate of the length of curves in the attractor. The estimation method presented herecan be applied to cases in which the invariant set is not hyperbolic. From the result ofthe length calculation, we have estimated the topological entropy h as h � 0.49703 forthe original parameters a = 1.4 and b = 0.3 adopted by Hénon.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The Hénon map [1] is a model that depicts fluid turbu-lence from the viewpoint of a dynamical system [2]; in thismap, the topological entropy of the attractor is the quan-tity that measures the complexity of fully developed tur-bulence [3,4]. Generally, when the topological entropy isknown, we can use it to calculate thermodynamical quan-tities such as the Gibbs measure or the pressure of asystem. Owing to the significance of entropy in thermody-namics of dissipative systems or the probability theory,several studies have calculated the topological entropy ofthe Hénon attractor by various methods. In the case of atwo-dimensional system, the topological entropy can bedefined as supremum of the growth ratio of lengths ofcurves in invariant sets (usually, the unstable manifold)in a system [5–7].

Mori and Fujisaka calculated the topological entropy hand the Housdorff dimension D of the Hénon map by themethod of map iteration and obtained the values ofh � 0.42 and D � 1.26. Biham and Wenzel introduced aHamiltonian associated with the Hénon map and esti-mated as h � 0.67 (note that their definition of topologicalentropy is slightly different from the one presented here)by counting the number of periodic points up to periodp = 28 [8]. Their numerical technique can also be applied

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ac.jp (C. Matsuoka).

to depict Julia sets in complex dynamical systems [9]. Gali-as and Zgliczynski [10] and Galias [11] calculated the topo-logical entropy by counting periodic points up to periodp = 30, in which they estimated the value h � 0.46486.Gelfert and Kantz adopted the algorithm of Biham andWenzel and use it to calculate the topological entropyand pressure for various values of dynamical parametersa and b [see (1)] by counting periodic points up to periodp = 31, by which they estimated as h � 0.44–0.46 for theHénon map [12]. Newhouse et al. performed numericalcalculations with rigorous error estimates and proved theresult of h P 0.46469 [13].

When the invariant set is hyperbolic, the topological en-tropy is usually given by counting the number of periodicpoints [14]; however, it is very difficult to obtain anaccurate value of topological entropy in cases where theinvariant set is not hyperbolic or the unstable manifold isnot a true attractor. Although a rigorous mathematicalproof does not exist, the invariant set of the Hénon mapis considered to be non-hyperbolic [15]; therefore, themethod of calculating periodic points [8,12] does not al-ways give an accurate entropy value. The method of count-ing parts of the stable and unstable manifolds created bythe map iteration presented by Newhouse et al. [13] givesan accurate value; however, complicated numerical calcu-lations are required for this method. Matsuoka and Hiraidefound a function that describes the stable and unstablemanifolds of the Hénon map and depicted these manifoldsusing this function [16]. This function provides the

806 C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809

accurate length of the unstable manifold; therefore, we cancalculate the topological entropy by using this functionregardless of whether or not the invariant set is hyperbolic,i.e., whether or not the unstable manifold is an trueattractor. In this paper, we report estimation of thetopological entropy of the Hénon map by adopting thisfunction. We mention that this estimation method isapplicable to various values of dynamical parameters aand b [see Definition (1)].

This paper is organized as follows. In Section 2, we pres-ent basic facts and notations adopted in this paper, andbriefly review the asymptotic expansion form to describethe stable and unstable manifolds of the Hénon map. InSection 3, we provide the definition of the topological en-tropy and perform the numerical estimate of that usingthe asymptotic expansion presented in the previous sec-tion. A figure of the stable and unstable manifolds of theHénon map represented by this asymptotic expansion isalso presented in this section. Section 4 is devoted toconclusion.

2. Asymptotic expansion of stable and unstablemanifolds of Hénon map

The Hénon map [1] is an invertible nonlinear transfor-mation defined by a polynomial map f : C2 ! C2

f ðx; yÞ ¼ ð1þ y� ax2; bxÞ; ð1Þ

where a and b are complex parameters; and we set a = 1.4and b = 0.3 when estimating the topological entropy.When a fixed point P = (xf,yf) is a saddle point, two eigen-values a1 and a2 at P are given by the solutions to thequadratic equation a2 + 2axfa � b = 0, where 0 < ja1j < 1and ja2j > 1. We define the stable and unstable manifoldsat P by

WsðPÞ ¼ fQ 2 C2jf nðQÞ ! P as n!1g

and

WuðPÞ ¼ fQ 2 C2jf nðQÞ ! P as n! �1g;

respectively. After shifting the fixed point P to the origin byx ? x + xf and y ? y + yf, we introduce a parameter t 2 C

that parameterizes the stable and unstable manifolds asf[x(t),y(t)] = [x(t + 1),y(t + 1)]. Then, the Hénon map (1)yields the difference equation

xðt þ 1Þ � kxðtÞ � bxðt � 1Þ ¼ �afxðtÞg2; ð2Þ

associated with y(t) = bx(t � 1), where k = �2axf.Using the Borel–Laplace transform, Matsuoka and Hira-

ide found solution x(t) to (2), whose asymptotic expansionis given as [16]

xðtÞ �P1N¼1

e�Nf1t bðNÞN;N�1ðN � 1Þ!ð�tÞN

; ð3Þ

where the + and � signs correspond to the stable andunstable manifolds, respectively, and f1 is given byf1 = �logja1j for the stable manifold and f1 = �logja2j + pifor the unstable manifold. The coefficient bðNÞN;N�1 is a con-stant given by a recurrence formula (Theorem 2 in [16])and is estimated as

bðNÞN;N�1

������ 6 KN

N ja1jN log N

N!

for the stable manifold and as

bðNÞN;N�1

������ 6 K 0NN ja2j�N log N

N!

for the unstable manifold, where KN > 0 and K 0N > 0 areconstants. These estimates guarantee the suppression ofdivergence of the factor eNf1tðN � 1Þ! for Re(f1t) > 0 in (3).Here, we mention that the asymptotic expansion (3) isnot a formal one; that is, this expansion is not the diver-gent series (for details, see Ref. [16]).

In order to depict the stable and unstable manifolds, weintroduce a large positive integer n such that

xðt � nÞ �P1N¼1

e�Nf1ðt�nÞ bðNÞN;N�1ðN � 1Þ!ð�tÞN

; ð4Þ

where the upper (lower) sign in �(±) corresponds to thestable (unstable) manifold. We denote the integer part ofjRe(t) � nj as M; then the integer M corresponds to themap iteration number (minus 1), i.e., f�(M+1)[x(t)] = x(t � n).When Re(t) > 0 and Re(t) � n < 0, the curve(x,y) = [x(t � n),bx(t � n � 1)] describes the stable mani-fold; when Re(t) > 0 and Re(t) � n > 0, the curve[x(t � n),bx(t � n � 1)] tends to the fixed point P owing tothe relation limM?1fM[x(t)] ? P. When Re(t) < 0 andRe(t) + n > 0, the curve (x,y) = [x(t + n),bx(t + n � 1)] de-scribes the unstable manifold; when Re(t) < 0 andRe(t) + n < 0, the curve [x(t + n),bx(t + n � 1)] tends to thefixed point P owing to the relation limM?1f�M[x(t)] ? P.In order to calculate the length of the unstable manifold,we adopt the expansion (4).

3. Topological entropy of the Hénon attractor

3.1. Definition and numerical estimate of topological entropy

Following Yomdin, and Newhouse et al. [5–7], we adoptthe topological entropy h as

h ¼ limM!1

1M

XM�1

k¼0

lnLkþ1

Lk; ð5Þ

where Lk is the length of the kth piece (the length obtainedby the (k + 1)th iteration of the map f) on the Hénon attrac-tor. Here, the length L of a curve is approximately given by

L ¼PK�1

m¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðtmþ1Þ � xðtmÞ½ �2 þ yðtmþ1Þ � yðtmÞ½ �2

q; ð6Þ

where the set [x(tm),y(tm)] denotes the coordinate onthe (x,y) plane that is parameterized by the parametertm, i.e., the discretized t. If the difference Mtm � tm+1 � tm

is sufficiently small, L in (6) can be expected to wellapproximate the true length of the curve Ltrue ¼R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðdx=dtÞ2 þ ðdy=dtÞ2

qdt. We adopt formula (6) for calcu-

lating the partial length Lk (k = 0,1,2, . . .) numerically. Here,we restrict ourselves to the curves in the unstablemanifold.

Table 1Length of curves in the unstable manifold. The first and third columns showthe iteration number M, and the second and fourth columns show thepartial length of the attractor, LM, at the iteration number M.

M LM M LM

0 7.1843 10�5 12 0.170081 1.3821 10�4 13 0.390142 2.6583 10�4 14 0.440083 5.1151 10�4 15 1.319124 9.8341 10�4 16 1.659985 1.8938 10�3 17 2.687246 3.6353 10�3 18 4.771247 7.0211 10�3 19 7.386648 1.3402 10�2 20 9.708609 2.6162 10�2 21 15.1093

10 4.8879 10�2 22 24.504511 9.9124 10�2

Fig. 1. Partial topological entropy h12,M versus map iteration number M,where the white circles show the calculation points of h12,M.

C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809 807

Since we cannot take M ?1 in (5) numerically, we de-fine the partial entropy hl,M for numerical calculations asfollows:

hl;M ¼1

M � l

XM�1

k¼l

lnLkþ1

Lk: ð7Þ

As is seen easily, h is given by h = liml?0 limM?1hl,M.We use the asymptotic expansion (4) to calculate the

length of the Hénon attractor. The algorithm for derivingthe length Lk in (7) is as follows. First, we calculate thecoefficient bðNÞN;N�1 in (4) by using the recurrence formulain Theorem 2 in Ref. [16]. In order to avoid a round-off er-ror, the calculation of bðNÞN;N�1 is performed with 300 digits.The upper limit of the summation in (4) is taken asN = 600. Then, we calculate x(t + n) (the unstable case) in(4) by using this bðNÞN;N�1, where f1 = �logja2j + pi� �0.6543 + pi for a = 1.4, b = 0.3, and a2 � �1.9237. Weset n = 20,000 here. For this n, we consider the region�20,000 6 Re(t) 6 �19,978, so the map iteration number(minus 1) M; i.e., the integer part of jRe(t) + nj, is in therange 0 6M 6 22.

As can be seen from the value of f1, the value of x(t + n)with respect to real t is not real for the unstable manifold;therefore, we must seek points such that the imaginarypart of (x,y) is close to zero on C2 in order to obtain real(x,y). To do this, we divide the complex t-plane into piecesand search the points at which the distance from the

real axis satisfies the condition sðtÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixiðtÞ2 þ yiðtÞ

2q

6 � ð�� 1Þ, where xi(t) = Im[x(t + n)] and yi(t) = Im[bx(t + n � 1)]. We select � = 10�3 in this study. The points thatsatisfy this condition correspond to tm in (6). Upon settingt = tr + i ti, we found that tm appears on a line with slope ti/tr � 4.8 [16]. The sets [x(tm),y(tm)] obtained by this algo-rithm accurately depict the unstable manifold of theHénon map (see Fig. 3).

For a small map iteration number M, we can find pointst = tm that satisfy s(t) 6 10�3 up to the neighborhood of thenext integer M + 1, when the starting point of the detectionis taken to be as M. For example, we can find such pointsapproximately up to M + 0.9 for M = 0 and 1 (tr = �20,000and �19,999, respectively), M + 0.8 for M = 2 and 3, andM + 0.7 for M = 4 and 5, starting from M. However, thesepoints gradually decrease in number owing to the round-off error, and finally, we few such points are found forM P 23. We find s(t) 6 10�3 points up to M + 0.1 forM = 22 (tr = �19,978); therefore, taking this value into ac-count, we select the interval M 6 tr 6M + 0.1 in tr (ti = 4.8tr)for all M. Dividing the interval 0.1 in one integer M intoequally spaced grid points with a distance ofMtr (Mti = 4.8Mtr), we calculate the length of curves for allM (0 6M 6 22). Here, we set Mtr = 1.0 10�4 for0 6M 6 7, Mtr = 5.0 10�5 for 8 6M 6 17, and Mtr = 1.0 10�5 for 18 6M 6 22; that is, we consider 1000, 5000, and10,000 grid points for 0 6M 6 7, 8 6M 6 17, and18 6M 6 22, respectively. The calculations for x(t + n) =f(M+1)[(x(t)] are performed with 32, 64, 128, and 300 digitsfor 0 6M 6 7, 8 6M 6 17, 18 6M 6 21, and M = 22,respectively (see Fig. 3).

By using the above algorithm, we find [x(tm),y(tm)] thatwould provide the unstable manifold and calculate the

length LM (0 6M 6 22) from formula (6). These lengthsfor a = 1.4 and b = 0.3 are listed in Table 1. The lengths LM

for 0 6M 6 11 are very small and the points [x(tm),y(tm)]for these M values stay in the neighborhood of the fixedpoint. Indeed, almost all parts of the Hénon attractor aredepicted by the points [x(tm),y(tm)] for 12 6M 6 22. Thepartial topological entropy hl,M in (7) for 0 6M 6 11 varieswidely and we cannot obtain an accurate value for it.Therefore, we do not adopt these LM (0 6M 6 11) valuesfor the topological entropy calculation. Fig. 1 shows thepartial topological entropy h12,M, for which l = 12 in (7).After varying at relatively small M values, the partial topo-logical entropy h12,M approaches a certain value. The h12,M

values for 19 6M 6 22 are estimated as h12,19 � 0.53874,h12,20 � 0.50556, h12,21 � 0.49853, and h12,22 � 0.49703.We regard this last h12,22 value as the asymptotic valueof the topological entropy h in the approximation up toM = 22, i.e.,

h � 0:49703: ð8Þ

Our result supports the estimate of h P 0.46469 by New-house et al. [13]. We remark that Fig. 1 mathematically

Fig. 3. Stable and unstable manifolds of Hénon map described byasymptotic expansion given by (4), where a = 1.4 and b = 0.3, and thefixed point is shifted to the origin.

808 C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809

converges with M ?1; however, the result up to M = 22 isbest possible in actual calculations at present.

The values obtained by Newhouse et al. and in the pres-ent study are larger than the value calculated by countingthe periodic points [12]. Katok [17] proved a theorem thatwhen the system is two-dimensional and the topologicalentropy h > 0, there exist hyperbolic subdynamical systemshaving entropy h� infinitely close to h. This h� is possible tocalculate by counting the number of periodic points. Thedifference between the result presented by Gelfert andKantz (h � 0.44–0.46) and ours (h � 0.49703) seems tosuggest that there exist numerous periodic points withlonger period in the Hénon attractor that cannot countwithin the period p = 31 considered in Ref. [12].

We show the topological entropy for various values of ain the neighborhood of a = 1.4 in Fig. 2, where the value ofb is fixed at b = 0.3. The accuracy of the calculation is iden-tical with that of Fig. 1. We find the devil’s staircase-likestructure for the topological entropy, which resembles inthe logistic maps in one-dimensional case. We see thatthe topological entropy in the neighborhood of a = 1.4 ismonotonically increase and there exist a few flat places;therefore, the dynamical system varies sensitively depend-ing on the value of a.

3.2. Stable and unstable manifolds

Fig. 3 shows the stable and unstable manifolds of theHénon map. We see that the unstable manifold intersectsthe stable manifold almost everywhere. These manifoldsare obtained from the asymptotic expansionx(t � n) = f�(M+1)[x(t)] in (4). The unstable manifold in thisfigure is depicted by all [x(tm),y(tm)] adopted for the calcu-lation of the length LM (0 6M 6 22): however, the attrac-tor remains unchanged visually, even if we take points[x(tm),y(tm)] only for 18 6M 6 22. For the stable manifold,f1 in (4) is given as f1 = �logja1j � 1.8582 fora = 1.4, b = 0.3, and a1 � 0.1560. The number of digits andthe upper limit in (4) for the calculation of bðNÞN;N�1 for thestable manifold are the same as those for the unstablemanifold. Since the value of x(t � n) with respect to real tis real for the stable manifold, we can depict the stable

Fig. 2. Partial topological entropy h12,M in the neighborhood of a = 1.4

manifold (x,y) = [x(t � n),bx(t � n � 1)] continuously withrespect to real t. We take the region 19,988 6 t 6 20,000for n = 20,000, i.e., the map iteration number (minus 1) M(the integer part of jt � nj), where 0 6M 6 11, for the sta-ble manifold. Here, 105 grid points are considered. Sincethe stable manifold is not an attractor, it is more difficultto take a large iteration number M, in contrast to the unsta-ble manifold. The stable manifold diverges in the firstquadrant as N ?1 in (4). We emphasize that M = 22, i.e.,the 23th iteration of f (223th order polynomial) is realizedfor calculating the topological entropy.

4. Conclusion

We have estimated the topological entropy of theHénon attractor by using a function that describes the sta-ble and unstable manifolds of the Hénon map. The topolog-ical entropy is estimated by calculating the length ofcurves in the attractor geometrically. Our estimation resultsupports the estimation made by Newhouse et al. [13].

; (a) 1.31 6 a 6 1.41 and (b) 1.3292 6 a 6 1.41, where b = 0.3.

C. Matsuoka, K. Hiraide / Chaos, Solitons & Fractals 45 (2012) 805–809 809

There also exists another function that can describe thestable and unstable manifolds of the Hénon map. Thisfunction, often called the Poincaré function, was an entirefunction and was first presented by Poincaré in 1890[18]. The existence and regularity (smoothness) of thePoincaré function have been investigated in detail by Cabréet al. [19–21]. This function is given in the form of a Taylorseries around a hyperbolic fixed point [21] and is applica-ble to depiction of the stable and unstable manifolds(although the detailed structure as found in Fig. 3 cannotbe obtained, especially in the stable manifold). Because itis expressed in the form of the Taylor series, the Poincaréfunction well describes the manifolds near the fixed point;however, as pointed out by Newhouse et al. [13], theapproximation by this function deteriorates fairly quicklywhen we consider points far from the fixed point. There-fore, we cannot calculate the topological entropy using thisfunction. It is important to mention that the Poincaré func-tion is different from the function presented in this paper.

The derivation of topological entropy is important, be-cause various physical quantities in thermodynamics ofdissipative systems are expected to be derived from topo-logical entropy. Our method is also applicable to systemsin which the attractor does not exist or the invariant setis not hyperbolic, as long as the fixed point is hyperbolic.We have calculated here up to 300 digits (M = 22). How-ever, the series (4) is an exact solution of Eq. (2) and math-ematically, we can calculate the topological entropy withanalytical accuracy by using this series. The function pre-sented here can capture the exponentially small or largeeffects and that enables us to investigate various analyticalstructures of stable and unstable manifolds in addition tocalculating topological entropy. We refer to the fact thatthe algorithm adopted in the present study cannot be ap-plied to the case in which the fixed point is elliptic, i.e.,the case of the eigenvalue jaj = 1. However, our algorithmis applicable to the entire range of parameters a and b ex-cept this case.

Acknowledgements

This work was partially supported by a Grant-in-Aid forScientific Research (C) from the Japan Society for the Pro-motion of Science.

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