Craig C. Martens and Gregory S. Ezra- A Simple Method for Determining the Number of Isolating...

8/3/2019 Craig C. Martens and Gregory S. Ezra- A Simple Method for Determining the Number of Isolating Integrals in Multidi

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Volume 10s. number 6 CHEhlICAL PHYSICS LETTERS 27 July 198

A SIMPLE METHOD F OR DETERMINING THE NUMBER OF ISOLATING INTEGRALS

IN MULTIDIMENSIONAL SYSTEMS: COMPUTATION OF THE POINTWISE DIMENSION

Craig C. MARTENS and Gregory S. EZRA

Received 7 Ilarch 1984; in final form 4 hlay 1984

The point~vise dimension i s calculated for trajectories of two- and three-dimensional Hamiltonians. enablin g the numberof isolating integrals to be determined directly. Results f or the H&on-Heiles system are in agreement wi th recent calcula-tions of the fractal and information dimensions by Stine and Noid.

1. Introduction

There is considerable current interest in the natureof the coexistence of regular (quasi-periodic) and irregu-lar (chaotic, stochastic) motions in the phase space ofgeneric classical Hamiltonian systems [ 1,2], especially thetransition from predominantly regular to global chaoticbehaviour found at high energies [ 1,2] _These ideas arewidely recognized to be of fundamental importance forchemistry [3], insofar as classical dynamics provides anaccurate description of intramolecular dynamics.

The classical mechanics of two-dimensional modelpotential surfaces has been intensively studied in recentyears [2,3] _ In particular, the Poincare surface of sec-tion has been found to be a useful diagnostic for regular-ity or irregularity of a trajectory [3] _By contrast. verylittle is known at present concerning the detailed classi-cal dynamics of multidimensional systems described byrealistic potential surfaces (for recent work, cf. refs.[4--71). Surface of section methods are difficult to ap-ply for (IV > 3)dimensional systems, so that emphasismust be placed on intrinsic indicators of dynamical be-haviour, such as the appearance of coordinate powerspectra, or rate of divergence of nearby trajectories (cf.refs. [ 1,3] )_

An important characteristic of any trajectory is the

number of isolating integrals associated with it (in addi-tion to the ener,v). This number determines the dimen-sion of the trajectory, where a generalized notion of di-mension such as the fractal dimension [8] must in gen-eral be used. The importance of the fractal dimension

0 00$2614/84/S 03.00 OElsevier Science Publishers B-V.(North-Holland Physics Publishing Division)

in this context has recently been emphasized byReinhardt [9]_ There are indications that for (N= 3)-dimensional coupled oscillator systems, for example,the chaotic re8ion of phase space is divided at inter-mediate energies into disjoint stochastic seas, eachof which is characterized by a particular number ofisolating integrals [lo] _ These disjoint regions thenmerge at higher energy into a single big sea. It isclearly important for understanding the structure ofchaos in multidimensional systems to have efficientmethods for direct computation of the dimension ofthe trajectory in phase space.

Stine and Noid [ 11,12] have recently described

methods for calculating both the fractal [ 1 l] and theso-called information dimension [ 121 of a trajectory[ 13]_ These are summarized in section 2. Both tech-niques rely on computationally intensive bin-countingand sorting operations. In this letter we apply a con-ceptually and computationally simple approach fordetermining the po in tw i se d im ens ion [ 13 1 of trajec-tories in two- and three-dimensional Hamiltonian sys-tems. This method has previously been used byGuckenheimer [ 141 to characterize attractors. The de-fiiition [ 131 of the pointwise dimension (cf. section 3)relies on the existence of both a metric and a probabil-ity measure (density of points) in phase space. For

Hamiltonian systems, the pointwise dimension is ex-pected to be equal to both the information and thefractal dimension [ 13,13]_ We have confirmed this re-sult for the particular two- and three-dimensional po-tentials treated below.

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Volume 108, nunlber 6 CHEMICAL PlIYSICS LETTERS 27 July 1981

9,. The fractal and information dimensions: methods versus In IV should give a straight line of slope 211 -kof Stine and Noid [12].

The idea for computing the fractal dimension of atrajectory [I I] or attractor [13,15] is very simple:

Phase space is divided into a large number of hyper-cubes (bins). The trajectory is then run for a time longenough to ensure that the number of bins visited bythe trajectory converges to a constant value (no noteis taken of the number of times a particular bin is visit-ed by the trajectory). Plotting the log of the numberor bins visited versus the log of the inverse size of thebin gives a straight line whose slope is the fractal di-mension of the trajectory. Thus, let fV be the numberof bins in each direction of phase space, 212 the dimen-sion of phase space. and N,,,k,Uc the number of uniquebins visited by the trajectory. We then have [ 1 I] :

In IV unique = (212 -- kj IIl N + COIlSt31lt, (1)

where 212 -k is the fractal dimension of the trajectorywith k the effective number of isolating integrals.

There are several drawbacks to this approach. Foriargc M (correspcnding to a large numberN2 of bins,ncccssary for accurate results) the trajectory must berun for a very long time to ensure that the number ofunique hypcrcubes visited has reached its asymptoticvalue. In orher words. the trajectory may tend to fillphase spw2 wry IIon-uniforlnly.wlii~i~ slows conver-!$ncc of A ,niclI,L The method also requires nluch com-putationally intensive bookkeeping to be done keeping

track of bin occupations. Finally_ all counting and sort-ing has IO bc done for a scrics of bin sized in order todetcrminc k.

The above methods have been applied by Stine andNoid to calculate the number of isolating integrals kfor quasi-periodic and chaotic trajectories of the 2D

H&on-Heiles system. Both approaches yield valuesk = 2 (quasi-periodic) and k = 1 (chaotic) within reasonable error limits. These results are expected on the ba-sis of surface-of-section studies [2,3] _

Nevertheless, both methods used by Stine and Noidrely on time-consuming bin-sorting operations. Thisfactor becomes increasingly important for (fV> 3)-d&mensional systems. We have therefore applied a muchsimpler approach to determine the so-called pointwisedimension of a trajectory- This method. due toGuckenheimer [ 141 and described in the next section.has minimal sorting and storage requirements yet yieldsvalues in agreement wit11 those of Stine and Noid forboth two- and three-dimensional systems.

We note also the recent calculation of the so-calledcorrelation exponent for attractors by Grassberger andProcaccia [ 161. For the systems treated here the corre-lation exponent is identical with the pointwise dimen-sion, but is computed using a different and (we found)more time-consuming algorithm.

3. Computation of the pointwise dimension

To ~llcviatc the first problem mentioned above,Srine and Noid developed a method for calculating theiIiforniaIion dimension of a trajectory. In this proce-Jurc. b i n s t l la t a r c v i s i t e d f r e q u e n t l y b y t h e t r a j e ct o r yarc ivzighlcd more st rorigly than those visited less fre-quently. It is therefore possible to attain convergencewith a sliortcr trajectory icmgtll than for the fractal di-mension. In addition to a metric in phase space, a prob-ability mcasurc (number of trajectory points in :I givenbiII/Iotal number of points 011 trajectory) is required.

ifgi is Ilic probabili(y of accessing hypcrcubc i, ZIplot111tlic infoIniaIion entropy

Suppose we have a probability measure ~1 density

of points on a trajectory) in phase space. The point-wise dimension d,, is then the exponent with which thetotal probability contained in a region (sphere) arounda given point decreases as the radius of the sphere de-creases [ I;]_ hlorc precisely, if S, (x0) is a sphere of aradiuss centered at point x0 on the trajectory, then

(3

If d,, (x0) is independent ofxo for almost all x0, thenI,, = d, (x-0) is called the poitltwise dimension.

A practical procedure for computingd,(xo) is as

rollows [ 141:(i) Choose an initial point x0 = (qo, po) in phase

space.s = (2) (ii) Run a trajectory to obtain a series of points

S(tj = (q(t), p(f)) with x(r = 0) =x0_ The trajectory


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Volume 108, number 6 CHEMICAL PHYSICS LETTERS 27 July 1984

must be run for a time long enough to accumulate anappreciable density of points in the vicinity ofxo.Thisturns out to be quite feasible for two- and three-dimen-sional systems examined below. but is of course a ma-jor limitation of the method as applied to higher di-

mension problems, since it necessitates accurate inte-gration over long times of trajectories having possiblestochastic character.

(iii) At each point on the trajectory, calculate D(t)= 11x(t) -x011, the distance of the point x(t) from thereference point x0_ For a perturbed oscillator Hamii-tonian of t he foml

H(q_p) = If0 + V(q)_

wliere

(4

a natural metric in phase space is:

II

For all the systems treated below Wi = 1 (i= I. 3and 3). so that the metric (6) reduces to the Cartesiandistance in phase space. We have also verified that theprecise form of the metric adopted is not crucial forcomputation of the pointwise dimension. For exam-ple, taking 0.5 wf instead of of in the metric (6) hasno effect on the calculated value ofdt,.

(iv) As the trajectory is run. the pointsx(f) are bin-ned according to the value of the distanceD(r). At theend of the run, this discrete distance distribution is in-tegrated to give a binned distribution of the total num-ber of points on the trajectory within a given distanceof the reference point x0_ Note that no large-scale sort-ing operations are involved, regardless of the length ofthe trajectory or the dimension of phase space.

(v) The log of the number of points within a givendistance is then plotted against the log of the distance.The slope of the straight line portion of the graph atsmall distances (see section 4) is d,(xo).

(vi) Assuming that $,(x0) is the same for all pointsx0 on the trajectory, the pointwise dimension d, canbe defined as the average of d,,(x,) over a set of refer-ence points {x0). We have used sets of 10 referencepoints (chosen by taking points at every 100th time-step at the beginning of a trajectory run; 100 timestepsis approximately 0.5 characteristic periods) to obtain

the straight line plots shown in section 4. Keepingtrack of the distance along the trajectory from eachreference point requires very little extra computa-tional effort.

The assumption that dp(xO) is constant over rhe

whole trajectory allows us to improve statistics by aver-aging over a set of {x0}_ Examination of log-log plotsfor individual values of x0 indicates that this assump-tion is valid for the trajectories shown in section 4,which have well defined regular or chaotic character.It would however be of great interest to study the vari-ation of d,(xo) with phase space point xo for a trajec-tory corresponding to a patchy torus, as suggestedby Reinhardt [9] _ Such a trajectory would presumablyhave local regions corresponding to different point-wise dimensions.

4. Results

4.1. T~coifirltctuiorral H&m--Hcilcs potential

We have applied the method described in the previ-ous section to calculate the pointwise dimension oftrajectories for the standard degenerate two-dimen-sional H&on-Heiles Hamiltonian:

H =+


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Volume 108. number 6 CHEhIICAL PIiYSICS LITERS 7 July 1984

a

-li.s

cg

05

I

i

1

rirnc src ps. In this plot the distanc e D cxtcnds all theway to its maximum relevant v n l u e D,,,;,, , i.e. the sizeof Lhc whole triijcctory i n p h a s e s p a c e , 3 s c m b e seeniron1 111~ aturation of the number of points at largeD. A well defined straight line portion of the plot isapparent at small values of D u p to ~0.1 Dll l i ly, andrbr slope of this section defines the pointwise dimen-sion (I,,. Regions of the plot for larger D up to D,,,,,contain information on the global phase space structureof the torus, but are not analysed any further here. AHplots shown from IIOW o11 extend only up to D 50.1D 111,, \ ~

Log-log plots for each of the three trajectories of

40

a

-49O-0

b-

1:;::. I. Contiguration spxe projections of tlm tluee twodi-mcnsional IICncm-licilcs trajectoriss cxaminetl: (:~)Trmxsin~:quasi-periodic. (II) Iibrating quasipcriodic, (c) cliaotic.

fig. 1 are shown in fig. 3. The plot of fig. 3a has least-squares slope 2.00, so that to the nearest integer valuethe pointwise dimension of the precessing quasi-period-ic trajectory is 2. This corresponds to X-=4-2 =2, sothat there is one isolating integral in addition to theenergy, as expected for a quasi-periodic trajectory onan invariant torus. The straight line portion of thecurve for the librating trajectory does not extend as

far as for the precessing case. Since the librating trajec-tory corresponds to a long thin torus, this is a resultof large local curvature of the trajectory manifold inphase space. The plot of fig. 3b corresponds to 2.5 X1O5 points, and has least-squares slope 2.03. The librat-

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Volunw IO& nurnbcr 6 CI1EMICAL PHYSICS LETTERS 27 July 1964

JO 10 8.0 30

LN(D)

I:@. 2. 1101 l log (number of trajcrlory points within 3 dis-tmcc D of rckrcncc point) versus log fl for 111s r;~jcctory offig. la, nvcragcd over ;I set of 10 rcfcrence p o i n 1 s . D extendsa11 l lC w3y to D,, , ,, . t l i c masimmn cstcnt of the trujcstoryin phase spncc. Note the well defined str:lighl line portion ntsmall D. and the mluration at large D. he poillL!visc dimcn-sion is the slope of the plot at small D.

ing trajectory therefore also has 2 isolating integrals.The plot for the chaotic trajectory is shown in fig. 3c(5 X IO5 points) and has least-squares slope 2.98. Thechaotic trajectory therefore has k = 1, and no isolatingintegrals in addition to the energy.

4.2. nlrec-di~nensional model porerlfiul

To test the applicability of the method for three-dimensional systems, we follow ref. [ 1 ] and introducean additional degree of freedom:

H=H++ (p; + C&Js), 0)

where we take w3 = 1.0. Although the third degree offreedom is uncoupled, it serves to increase the dimen-sionality of phase space.

Trajectories were run for the three-dimensional sys-tem of eq. (8) with initial conditions for modes 1 and2 identical with those of the trajectories of fig. 1; formode 3. we set q3 = 0.0, p3 = 0.3. All trajectories were

run for 1 O X IO6 time steps.Log-log plots for the three trajectories run areshown in fig. 4. Least-squares values of d, are 3.06.2.99, and 3.94, corresponding to integer k values 3,3,and 2, respectively. These values are just those expect-

LX(D)

Fig. 3. Log (number of points) versus IosD for the tr3jectoriesof fig. I. The msGrnutn value of D shown is _


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Volume 108, number 6 CHEhlICAL PHYSICS LET2-ERS 27 July 1984

00 isLX(D)

ed for the addition of an uncoupled degree of freedomto the two-dimensional system treated above.

These results establish the practicability of the meth.od for three-dimensional Hamiltonian systems, andopen the way for application to more realistic molecu-

lar potentials.

5. Conclusion

We have applied a conceptually and computational-ly simple method to calculate the pointwise dimensionof trajectories of two- and three-dimensional Hamil-tonian systems. The method provides the number ofisolating integrals for the trajectory directly. Our calcu-lations for the H&on-Heiles potential confirm expec-

tations based on surface of section studies, and are inagreement with the recent results of Stine and Noid[I 1 13]_ We have also established the applicability ofthe method for N=3 systems_

Work in progress is concerned with computation ofthe pointwise dimension for trajectories of realisticthree-dimensional surfaces.

Acknowledgement

Acknowledgement is made to the donors of thepetroleum Research Fund, administered by the

American Chemical Society, for support of this research.

References

A.J. Licbtcnbcrg and M.A. Lieberman. Repulxr andstochtistic motion (Springer. Berlin. 1963).51. Tabor, Advan. Chom. Phys. 46 (198 1) 73;P. Brumer, Advan. Chcm. Phys. 47 (198 I) 20 1.I>.\\. Noid, h1.L. Koszykowski and R.A. hlrtrcus. Ann.Rev. Phys. Chem. 32 (198 1) 267;S.A. Rice. Advnn. Chcm. Phys. 47 (198 1) 117.D. Carter nnd P. Brumer. J. Chcm. Phys. 77 (1982) 4206.SC. lsrantos and J.N. Rlurrell, Chcm. Phys. 55 (1981)205 ;S.C. l.arantos. Cbem. Phys. 71 (1982) 157; Chem. Phys.

Letters 92 (1982) 379: J. Phys. Chcm. 87 (1983) 5061.K.N. Swamy and W.L. Hasc, Chcm. Pbys. titters 92(1982) 371.J. Santanwrh and G.S. Ezra. unpublished.l3.B. Wandelbrot, Fractnls: form, chance and dimension(Freeman, San Francisco. 1977); The fractsl geometryof nature (Freeman, S;in Irancisco, 1982).


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Volume 108, number 6 CHEMICAL PHYSICS LETTERS 27 July 1984

[9] W.P. Reinhardt, in: The mathematical analysis of physicalsystems, ed. R. hlickens (Van Nostrand, Princeton), tobe published.

[IO] G. Contoupoulos. L. Gaigani and A. Ciorgilli, Phys. Rev.A18 (1978) 1183.

[ 111 J.R. Stine and D.W. Noid, J. Phys. Chem. 87 (1983)3038.

1121 .R. Stine and D.W. Noid, Chem. Phys. Letters 100(1983) 282.

[ 131 J.D. Farmer, E. Ott and J.A. Yoke, Physica 7D (1983)153.

[14] J. Guckenbeimer and G. Buznya, Phys. Rev. Letters 51(1983) 346;J. Guckenheimer, in: Contemporary mathematics, Vol.28 (American Mathematical Society, 1984).

[IS] H.S. Greenside, A. Wolf, J. Swift and T. Pignataro,Phys.Rev. A25 (1982) 3453.

[16 J P. Grassberger and I. Procaccia, Phys. Rev. Letters 50(1983) 346; Plrysica 9D (1983) 189.

[17] C.\V_ Gear, SIAM. J. Num. Anal. 3B (1964) 69.[IS] P. Brumer, J. Comput. Phys. 14 (1974) 391.

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