Continuous iteration of dynamical maps R. Aldrovandi and L. P. Freitas Citation: Journal of Mathematical Physics 39, 5324 (1998); doi: 10.1063/1.532574 View online: http://dx.doi.org/10.1063/1.532574 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/39/10?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1774531301/x01/AIP-PT/JMP_ArticleDL_022614/aipToCAlerts_Large.png/5532386d4f314a53757a6b4144615953?x http://scitation.aip.org/search?value1=R.+Aldrovandi&option1=author http://scitation.aip.org/search?value1=L.+P.+Freitas&option1=author http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov http://dx.doi.org/10.1063/1.532574 http://scitation.aip.org/content/aip/journal/jmp/39/10?ver=pdfcov http://scitation.aip.org/content/aip?ver=pdfcov e the nt at the ed at the erse at the - if such atrix linear mpo- matrix real th the JOURNAL OF MATHEMATICAL PHYSICS VOLUME 39, NUMBER 10 OCTOBER 1998 This article is copyrig Continuous iteration of dynamical maps R. Aldrovandia) and L. P. Freitasb) Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 Sa˜o Paulo SP, Brazil ~Received 10 November 1997; accepted for publication 16 June 1998! A precise meaning is given to the notion of continuous iteration of a mapping. Usual discrete iterations are extended into a dynamical flow which is a homotopy of them all. The continuous iterate reveals that a dynamic map is formed by inde- pendent component modes evolving without interference with each other. An ap- plication to turbulent flow suggests that the velocity field assumes nonseparable values. © 1998 American Institute of Physics.@S0022-2488~98!03410-0# I. INTRODUCTION There are two main approaches to describe the evolution of a dynamical system.1 The first has its roots in classical mechanics—the solutions of the dynamical differential equations provid continuous motion of the representative point in phase space.2 The second takes a quite differe point of view: it models evolution by the successive iterations of a well-chosen map, so th state is known after each step, as if the ‘‘time’’ parameter of the system were only defin discrete values.3 It is possible to go from the first kind of description to the second through snapshots leading to a Poincare´ map. Our aim here is to present the first steps into a conv procedure, going from a discrete to a continuous descriptionwhile preserving the idea of iteration. This is possible if we are able to ‘‘interpolate’’ between the discrete values in such a way th notion of iteration keeps its meaning in the intervals. Iteration is a particular case of function composition: given a basic mapf (x)5 f ^1&(x), its first iterate isf ^2&(x)5@ f + f #(x)5 f @ f (x)#, its nth iterate isf ^n&(x)5 f @ f ^n21&(x)#5 f ^n21&@ f (x)#, etc. The question is whether or not, given the set of functionsf ^n&(x), an interpolationf ^t&(x) with real values oft can be found which represents the one-parameter continuous group~or semigroup! describing the dynamical flow of the system. In order to do it,f ^t& should satisfy the conditions f ^t&@ f ^t8&~x!#5 f ^t8&@ f ^t&~x!#5 f ^t1t8&~x!, ~I.1! f ^0&~x!5Id~x!5x. ~I.2! We shall find in what follows a map interpolationf ^t&(x) with these properties. It is a well known fact that Taylor seriesf (x), g(x) satisfying the conditionsf (0)50, g(0)50, and with nonvanishing coefficients ofx are invertible and constitute a group by composition.4 The neutral element is the identity functione(x)5Id(x)5x and two functionsf, g are inverse to each other it is true thatf @g(x)#5g@ f (x)#5Id(x)5x. In this case,g5 f ^21& and f 5g^21& ~we are taking the liberty of using the word ‘‘function’’ even for purely formal series and multivalued maps!. The clue to the question lies in the formalism of Bell polynomials, which attributes to every function f a matrixB@ f #, whose inverse represents the inverse function and such that the m product represents the composition operation. In other words, these matrices provide a representation of the group formed by the functions with the operation of composition. Co sition is thus represented by matrix product and, consequently, iterations are represented by powers. Furthermore, the representation is faithful, and the functionf is completely determined by B@ f #. Now, in the matrix group there does exist a clear interpolation of discrete powers by powers and the inverse way, going from matrices to functions, yields a map interpolation wi desired properties. a!Electronic mail: RA@AXP.IFT.UNESP.BR b!Electronic mail: LFREITAS@IFT.UNESP.BR 53240022-2488/98/39(10)/5324/13/$15.00 © 1998 American Institute of Physics hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 tion ts e se . of of the one rbu- . which ing: 5325J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig Section II is a short presentation of Bell polynomials, with only the minimum informa necessary for our present objective. It is shown how a matrixB@ f # can be found which represen each formal seriesf, and that the compositionf +g of two functions is represented by the~right-! product of the respective matrices:B@ f +g#5B@g#B@ f #. The identity matrix corresponds to th identity function,B@ Id#5I , the matrixB@ f ^n&# corresponding to thenth iterate, f ^n& is the nth powerBn@ f #, and the Lagrange inversef ^21& to a seriesf is represented by the respective inver matrix,B@ f ^21B21@ f #. The necessity of findingBt@ f # for noninteger ‘‘t’’ leads to the problem of defining functions of matrices, succinctly discussed in Sec. III. Given a matrixB, there exists a very convenient basis of projectors in terms of which any function ofB is defined in a simple way A method is given to obtain the members of this basis in a closed form, in terms of powersB. The procedure is applied to Bell matrices in Sec. IV to obtainBt@ f #5B@ f ^t&# for any value oft, from which the functionf ^t&(x) can be extracted and shown to satisfy conditions~I.1! and~I.2!. It turns out that, though it is quite natural to call ‘‘time’’ the continuous labelt, this ‘‘time’’ is related to a certain class of flows, among all those leading to a specific Poincare´ map. There is an extra bonus: The matrix decomposition in terms of projectors is reflected in a decomposition original map, and of its iterate, into a sum in terms of certain ‘‘elementary functions,’’ each with an independent and well-defined time evolution. An application, relating a simplified tu lence model to nonseparable solutions of the double harmonic oscillator, is given in Sec. V II. BELL MATRICES Given a formal series with vanishing constant term, g~x!5( j 51 ` gj j ! xj , ~II.1! its Bell polynomialsBnk@g# are certain polynomials5 in the Taylor coefficientsgi , defined by Bnk~g1 ,g2 ,...,gn2k11!5 1 k! H dn dtn @g~ t !#kJ t50 . ~II.2! Their properties are in general obtained from their appearance in the multinomial theorem, reads 1 k! S ( j 51 ` gj j ! t j D k 5 ( n5k ` tn n! Bnk~g1 ,g2 ,...,gn2k11!. ~II.3! Depending on the situation, one or another of the notations Bnk@g#5Bnk~g1 ,g2 ,...,gn2k11!5Bnk$gj%, ~II.4! is more convenient. The symbol$gj% represents the Taylor coefficient list ofg, with gj a typical member. Some properties coming immediately from the multinomial theorem are the follow Bn1@g#5gn , ~II.5! Bnn@g#5~g1!n, ~II.6! Bnk@cg~ t !#5ckBnk@g~ t !#, ~II.7! Bnk@g~ct!#5cnBnk@g~ t !#, ~II.8! wherec is a constant. Given two formal Taylor series, f ~u!5( j 51 ` f j j ! uj , g~ t !5( j 51 ` gj j ! t j , ~II.9! hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 ries - own rs are effect, n, 5326 J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig their composition F~ t !5@ f +g#~ t !5 f @g~ t !#5( j 51 ` tn n! Fn@ f ;g# ~II.10! will have the Taylor coefficientsFn given by the Faa` di Bruno formula, Fn@ f ;g#5 ( k51 n f kBnk$gj%. ~II.11! Other properties can be obtained from the double generating function eug~ t !215 ( n51 ` tn n! ( j 51 n ujBn j@g#. ~II.12! Series like~II.1! constitute a group under the composition operation (f +g)(x)5 f @g(x)#. The identity series ‘‘e’’ such thate(x)5Id(x)5x plays the role of the neutral element and each se g possesses an inverseg^21&, satisfyingg^21&+g5g+g^21&5e and given by the Lagrange inver sion formula. The simplest example of a Bell matrix and its inverse is given by the well-kn case of the Stirling numbers: matrices formed by the first and second kind of Stirling numbe inverse to each other, because they correspond to functions inverse to each other. In consider the series g~x!5 ln~11x!5( j 51 ` ~2 ! j 21 j xj , ~II.13! whose inverse is f ~u!5eu215( j 51 ` 1 j ! uj . ~II.14! A generating function for the Stirling numbers of the first kindsk ( j ) is 1 k! ~ ln~11x!!k5 ( n5k ` xn n! sn ~k!. ~II.15! It follows from ~II.3! that Bnk@ ln~11x!#5Bnk~0!,21!,2!,23!,...!5Bnk$~2 ! j 21~ j 21!! %5sn ~k!. ~II.16! For the Stirling numbers of the second kindSk ( j ), the generating function is 1 k! ~eu21!k5 ( n5k ` un n! Sn ~k!, ~II.17! from which Bnk@eu21#5Bnk~1,1,1,...,1!5Bnk$1%5Sn ~k!. ~II.18! The inverse property of Stirling numbers is(k5 j n sn (k) Sk ( j )5dn j , the same as ( k5 j n Bnk@ ln~11x!#Bk j@eu21#5dn j . ~II.19! The polynomialsBnk@g# are the entries of a~lower-!triangular matrixB@g#, with n as row index andk as the column index. From~II.5!, the function coefficients constitute the first colum hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 mpo- ent any ty’’ mming eries. t is ll study 5327J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig so that actuallyB@g# is an overcomplete representative ofg. From~II.6!, the eigenvalues ofB@g# are (g1) j . Triangular matrices form a group, of which the set of matrices (Bnk) constitutes a subgroup. Hereby comes the most fascinating property of Bell polynomials: The matricesB@g# 5(Bnk@g#), with the operation of matrix product, provide a representation of the series co sition group: B@g#B@ f #5B@ f +g#. ~II.20! It is in reality an antirepresentation because of the inverse order, but this does not repres problem. This property comes easily by using twice~II.3!, as 1 k! ~ f @g~ t !# !k5 1 k! S ( j 51 ` f j j ! gj D k 5 ( n5k ` g~ t !n n! Bnk@ f # 5 ( n5k ` ( j 5n ` t j j ! Bjn@g#Bnk@ f # 5( j 5k ` t j j ! ( n5k j Bjn@g#Bnk@ f #, from which Bjk@ f ~g~ t !!#5 ( n5k j Bjn@g#Bnk@ f #, ~II.21! which is just~II.20!. Associativity can be easily checked, and it is trivial to see that the ‘‘identi seriese(x)5x has the representativeBnk@e#5dnk , so thatB@e#5I . The seriesg(t) with g150 can be attributed to a matrix, but a singular one and, consequently, outside the group. Su up, infinite Bell matrices constitute a linear representation of the group of invertible formal s If we consider only the firstN rows and columns, what we have is an approximation, but i important to notice that the group properties hold at each orderN. The general aspect of a Be matrix can be illustrated by the caseN55: B@g#5S g1 0 0 0 0 g2 g1 2 0 0 0 g3 3g1g2 g1 3 0 0 g4 4g1g313g2 2 6g1 2g2 g1 4 0 g5 10g2g315g1g4 15g1g2 2110g1 2g3 10g1 3g2 g1 5 D . ~II.22! The result~II.19! is the best example of the general property B@ f #B@ f ^21I . ~II.23! It is evident that, given the seriesf, its inverse series can be obtained from B@ f ^21B21@ f # ~II.24! by simple matrix inversion. The inversion properties of Bell matrices have been used in the of cluster expansions for real gases.6 BecauseB@g^nBn@g#, Bell matrices convert function iteration into matrix power and provide a linearization of the process of iteration. hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 t e es , e, is ll of the n 5328 J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig Suppose now that we are able to obtain the matrixBt, with t an arbitrary real number. The continuous iteration ofg(x) will then be that functiong^t& such thatB@g^tBt@g#. By ~II.5!, its Taylor coefficients are fixed byg^t& n5B@g^t&#n15Bt@g#n1 . To arrive atBt, let us make a shor preliminary incursion into the subject of matrix functions. III. MATRIX FUNCTIONS Suppose a functionF(l) is given which can be expanded as a power seriesF(l) 5(k50 ` ck(l2l0)k inside the convergence circleul2l0u,r . Then the functionF(B), whose argument is now a givenN3N matrix B, is defined byF(B)5(k50 ` ck(B2l0)k and has a sens whenever the eigenvalues ofB lie within the convergence circle. Given the eigenvalu x1 ,x2 ,...,xN , the set of eigenprojectors$Zj@B#5uxj&^xj u% constitutes a basis in whichB is written B5( j 51 N xjZj@B#, ~III.1! and the functionF(B), defined as above, can also be written7 as the matrix F~B!5( j 51 N F~xj !Zj@B#. ~III.2! Thus, for example,eB5( j 51 N exjZj andBa5( j 51 N xj a Zj@B#. The basis$Zj@B#% depends onB, but is the same for every functionF. The Zj ’s, besides being projectors~that is, idempotents Zj 25Zj ), can be normalized so that tr(Zj )51 for eachj and are then orthonormal by the trac tr(ZiZj )5d i j . Other properties follow easily, for example, tr@F(B)#5( j 51 N F(xj )Zj and tr@BkZj #5(xj ) k. If B is a normal matrix diagonalized by a matrixU, UBU215Bdiagonal, then the entries ofZk are given by (Zk) rs5Urk 21Uks ~no summation, of course!. A set ofN powers ofB is enough to fix the projector basis. Using forF(B) in ~III.2! the power functions B05I , B1,B2,...,BN21, we haveI 5( j 51 N Zj ; B5( j 51 N xj Zj ; B25( j 51 N xj 2Zj ;...; Bk5( j 51 N xj kZj ;...;BN215( j 51 N xj N21Zj . For k>N, the Bk’s are no more independent. Th comes from the Cayley–Hamilton theorem,8 by which B satisfies its own secular equation D~x!5det@xI2B#5~x2x1!~x2x2!~x2x3!¯~x2xN!50. D(B)50 will give BN in terms of lower powers ofB, so that the higher powers ofB can be computed from the lower powers. Inversion of the above expressions for the powers ofB in terms of theZj ’s leads to a closed form for eachZj , Zj@B#5 ~B2x1!~B2x2!¯~B2xj 21!~B2xj 11!¯~B2xN21!~B2xN! ~xj2x1!~xj2x2!¯~xj2xj 21!~xj2xj 11!¯~xj2xN21!~xj2xN! . ~III.3! The functionF(B) is consequently given by F~B!5( j H) kÞ j B2xk xj2xk J F~xj !. ~III.4! Thus, in order to obtainF(B), it is necessary to find the eigenvalues ofB and the detailed form of its first (N21) powers. Though forN not too large theZj@B# ’s can be directly computed, we sha give closed expressions for them. These expressions involve some symmetric functions eigenvalues. Let us examine the spectrum$xk% of B in some more detail. The eigenvaluesxk will be called ‘‘letters’’ and indicated collectively by the ‘‘alphabet’’x5$x1 ,x2 ,x3 ,...,xN%. A monomial is a ‘‘word.’’ It will be convenient to consider both the alphabetx and its ‘‘reciprocal,’’ the alphabet x* 5$x* 1 ,x* 2 ,x* 3 ,...,x* N% where eachx* j521/xj . Notice that taking the reciprocal is a involution, x** 5x. A symmetric function in the variablesx1 ,x2 ,x3 ,...,xN is any polynomial hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 to ‘‘ ouwer 5329J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig P(x1 ,x2 ,x3 ,...,xN) which is invariant under all the permutations of thexk’s. Only one kind of them will be needed here, the ‘‘j th elementary symmetric functions,’’s j5sum of all words with j distinct letters: s0@x#51~by convention!, s1@x#5x11x21x31¯1xN , s2@x#5x1x21x1x31¯1x1xN1¯1x2x31x2x41¯1¯1xN21xN , ] sN@x#5x1x2x3¯xN21xN . The symmetric functions ofx andx* are related by ~2 ! jsN2 j@x#5sN@x#s j@x* #. ~III.5! Their generating function is ( j 50 N s j@x#t j5( j 51 N ~11xj t !5) j 51 N ~12t/x* j !5 1 sN@x* # ) j 51 N ~x* j2t !, so that ) j 51 N ~x* j2t !5sN@x* #( j 50 N s j@x#t j . We use the involution property and~III.5! to write the general expression ) j 51 N ~xj2t !5sN@x#( j 50 N s j@x* #t j5( j 50 N ~2 ! jsN2 j@x#t j . ~III.6! The j th eigenvalue is absent in the numerator of expression~III.3! for Zj . We shall need some results involving an alphabet with one missing letter. Lets j i @x# be the sum of allj-products of the alphabetx, but excludingxi . For example,sNi@x#5PkÞ i N xk . We put by conventions0i51 and find that ski@x#5 ( p50 k ~2 !pxi psk2p@x#5( j 50 k ~2xi ! k2 js j@x#. ~III.7! In the absence of thei th letter,~III.6! becomes ) j 51; j Þ i N ~xj2t !5sNi@x#( j 50 N s j i @x* #t j . ~III.8! The projectors~III.3! are then Zj@B#5 ) k51;kÞ j N xk2B xk2xj 5 (k50 N sk j@x* #Bk (k50 N sk j@x* #xj k , ~III.9! clearly written in the basis$I ,B,B2,...,BN21%. In our application to Bell matrices, it will be convenient to use instead the basis$B,B2,...,BN%. This is due to the fact that we shall prefer start from the matrixB@g#, corresponding tog, and not from the matrixI, corresponding to the identity function. The mappings of interest, like the logistic map for example, have a generalù’’ aspect and the identity map does not have the same end points: the identity map has Br9 hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 0. The he closed s of a t to t 5330 J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig degree 1, while the dynamical maps of interest can be characterized by their degree Cayley–Hamilton theorem implies( j 50 N s j@x* #Bj50, from which we obtain the identityB0 as I 52( j 51 N s j@x* #Bj . ~III.10! Replacing this identity in~III.9!, the projectors are recast into another form, Zi@B#5 I 1(k51 N ski@x* #Bk 11(k51 N ski@x* #xi k 5 (k51 N $ski@x* #2sk@x* #%Bk (k51 N $ski@x* #2sk@x* #%xi k . ~III.11! Using ~III.7! and ~III.5! we get Zi@B#5 (k51 N $( j 50 k21~xi ! j 2ks j@x* #%Bk ( j 50 N21~N2 j !~xi ! js j@x* # 5 (k51 N $( j 50 k21~xi ! j 2k~2 ! jsN2 j@x#%Bk ( j 50 N21~N2 j !~xi ! j~2 ! jsN2 j@x# , ~III.12! where we have also used( r 51 N $( j 50 r 21(xi) js j@x* #%5( j 50 N21(N2 j )(xi) js j@x* #. There is a good immediate check: ReplacingB by the eigenvalues we find, as expected, Zi@xk#5d ikI . ~III.13! The projectors are now clearly in the basis$B,B2,...,BN%. Actually, eachZi is now just that given in the basis$I ,B,B2,...,BN21% multiplied by (B/xi): instead of~III.3!, Zj@B#5 ~B2x1!~B2x2!¯~B2xj 21!B~B2xj 11!¯~B2xN21!~B2xN! ~xj2x1!~xj2x2!¯~xj2xj 21!xj~xj2xj 11!¯~xj2xN21!~xj2xN! . ~III.14! The Zi ’s and the powersBk can be seen as components of two formal column ‘‘vectors.’’ T linear conditionsBn5( j 51 N xj nZj are then represented by a matrixL5@xj n#, S B1 B2 • • • BN D 5S x1 x2 x3 • • xN x1 2 x2 2 x3 2 • • xN 2 • • • • • • • • • • • • • • • • • • x1 N x2 N x3 N • • xN N D S Z1 Z2 • • • ZN D , ~III.15! and what we have done has been to obtain its inverse: Zi@B#5 ( k51 N @L21# ikBk, ~III.16! @L21# ik5 ( j 50 k21~xi ! j 2k~2 ! jsN2 j@x# ( j 50 N21~N2 j !~xi ! j~2 ! jsN2 j@x# . ~III.17! It seems a difficult task to improve the above expressions, as it would mean knowing a analytical expression for the recurrent summation of the formS j 50 k21 ujs j@x* #. A closed expres- sion for s j@x* # would be necessary and, even for the simple alphabet consisting of power fixed lettera, which we shall find in the application to Bell matrices, this would be equivalen solving an as yet unsolved problem in Combinatorics. In effect, in terms of such an alphabe$aj% with N letters, the symmetric function is given bysk5S j 51 N(N11)/2 qj ,k,Naj , where qj ,k,N 5number of partitions ofj into k unequal summands, each one1qj ,k,Najuk, but have no known closed expression. They are calculated, one by one, just in this way.10 hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 lity is form their ed on gebras th men- of the f view, cacies e way, ction rices 5331J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig Bell matrices are not normal, that is, they do not commute with their transposes. Norma the condition for diagonalizability. This means that Bell matrices cannot be put into diagonal by a similarity transformation. As it happens, this will not be a difficulty because we know eigenvalues. That functions of matrices are completely determined by their spectra is justifi much more general grounds. Matrix algebras are very particular kinds of von Neumann al and it is a very strong result of the still more general theory of Banach algebras11 that functions on such spaces, as long as they can be defined, are fixed by the spectra. Another point wor tioning is that the infinite Bell matrices which constitute the true, complete representation group of invertible series will belong, as limits forN→` of N3N matrices, to a hyperfinite von Neumann algebra. Our considerations here are purely formal from the mathematical point o as we are only discussing formal series. We are not concerned with the topological intri involved in the convergence problems, though they surely deserve a detailed study. By th the infinite algebra generated by Bell matrices would provide a good guide in the study of fun algebras with the composition operation. IV. THE CONTINUOUS ITERATE We are now in a position to find, given a functiong, the matrixBt@g# and its corresponding function, the continuous iterateg^t&(x). Since many things—theB@g# spectrum, for example— will depend only on the first-order Taylor coefficient, we shall putg15a. By ~II.6!, the letters in the eigenvalue-alphabet ofB@g# will be simple powers ofa and the alphabeta5(a1,a2,...,aN) will have the reciprocala* 5(2a21,2a22,...,2a2N). The matrixL has entries (L ik)5(aik) and the projectors Zi@B#5 ( k51 N L21 ikBk@g# ~IV.1! now have the coefficients @L21# ik5 ( j 50 k21ai ~ j 2k!~2 ! jsN2 j@a# ( r 51 N ( j 50 r 21ai j ~2 ! jsN2 j@a# . We can verify easily that trZi@B#51 andBZi5aiZi . A consequence of the latter is f ~B!Zi5 f ~ai !Zi , ~IV.2! whose particular case BtZi5aitZi ~IV.3! will be helpful later on. To give an idea of their aspect, we show the projector mat Zi (N)@B@g## for N53: Z1 ~3!5S 1 0 0 g2 g1~12g1! 0 0 3g2 21g3~12g1! g1~12g1!2~11g1! 0 0 D , Z2 ~3!5S 0 0 0 2g2 g1~12g1! 1 0 23g2 2 g1 2~12g1!2 3g2 g1~12g1! 0 D , Z3 ~3!5S 0 0 0 0 0 0 3g2 22g1g3~12g1! g1 2~12g1!2~11g1! 23g2 g1~12g1! 1 D . hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 ey e ctions 5332 J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig Because they do not have the form~II.22!, they cannot be the Bell matrices of any function. Th inherit, however, a good property of the Bell matrices: for eachN, the projectorsZj (N) contain, in their higher rows, the projectorsZj (k) for all k,N. The upper-left 232 submatrices of theZk (3)’s above are justZk (2). We can take the first-column entries of the matrix~IV.1! as Taylor coefficients defining th functions Ri ~N!~x!5( r 51 ` xr r ! @Zi@B## r1 ~IV.4! 5 ( k51 N @L21 ikg^k&~x!#. ~IV.5! To each projectorZi corresponds such an ‘‘elementary function’’Ri (N)(x), a relationship reflect- ing in part that between the series and their Bell matrices. Taking the summation of allRi (N)’s in ~IV.4! and usingI 5( i 51 N Zi , we find ( i 51 N Ri ~N!~x!5( r 51 ` xr r ! F( i 51 N Zi@B#G r1 5( r 51 ` xr r ! d r15x. ~IV.6! Thus, just as the projectors give a decomposition of the identity matrix, the elementary fun provide a decomposition of the identity function, Id5( i 51 N Ri ~N!. ~IV.7! Applying to B@g# the general form~III.2! for the function of a matrix, we have Bt5( i 51 N aitZi@B#5 ( k51 N C~N! k~ t !Bk, ~IV.8! where C~N! k~ t !5( i 51 N aitL21 ik . ~IV.9! The coefficients of the continuum iterate function g^t&~x!5( r 51 ` gr ^t& xr r ! ~IV.10! will be gr ^t&5[Bt[g]] r15( i 51 N ait [Zi [B]] r15 ( k51 N C(N) k(t)B k[g] r15 ( k51 N C(N) k(t)gr ^k&. ~IV.11! Therefore, g^t&~x!5 ( k51 N C~N! k~ t !g^k&~x!5 ( k51 N F( i 51 N aitL21 ikGg^k&~x!. ~IV.12! Time dependence is factorized in the alternative form g^t&~x!5 ( k51 N aktRk ~N!~x!. ~IV.13! hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 f s 5333J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig One of the announced properties,~I.2!, follows immediately:g^0&(x)5Id(x). For t51, a sugges- tive decomposition of the function comes up: g~x!5 ( k51 N akRk ~N!~x!. ~IV.14! It remains to show that~IV.13! does satisfy property~I.1!. Notice first that, from the definition ~IV.4! of Ri (N) and the multinomial theorem~II.3!, Ri ~N!@g^t&~x!#5( r 51 ` @Zi@B## r1 @g^t&~x!# r r ! 5( r 51 ` ( j >r xj j ! Bjr @g^t&#@Zi@B## r1 5( j 51 ` xj j ! ( r 51 j Bjr t @g#@Zi@B## r1 5( j 51 ` xj j ! @BtZi # j 1 . ~IV.15! Therefore, g^t8&~g^t&~x!!5( i 51 N ait 8Ri ~N!@g^t&~x!# 5( r 51 ` xr r ! ( i 51 N ait 8@BtZi # r1 5( r 51 ` xr r ! FBt( i 51 N ait 8Zi G r1 5( r 51 ` xr r ! @BtBt8# r15( r 51 ` xr r ! g^t1t8& r15g^t1t8&~x!, just the result looked for. Notice also that, using~IV.3! and ~IV.4! in ~IV.15!, we obtain Rk ~N!@g^t&~x!#5aktRk ~N!~x! ~IV.16! and, consequently, Rk ~N!@g^t1t8&~x!#5ak~ t1t8!Rk ~N!~x!5akt8Rk ~N!@g^t&~x!#. ~IV.17! The function decomposition is preserved in time, as we can write g~x!5 ( k51 N Rk ~N!@g~x!#, g^t&~x!5 ( k51 N Rk ~N!@g^t&~x!#. ~IV.18! All the above expressions hold for each value ofN and give approximations to that order. O course, exact results would only really come out whenN→`. The exact expressions would, a long as they are defined, take forms like Rj~x!5 ( k>1 @L21 jkg^k&~x!#, ~IV.19! hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 irror he ral, en to r nal f ced to on the of a as e- 5334 J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig g~x!5 ( k>1 akRk~x!, ~IV.20! g^t&~x!5 ( k>1 aktRk~x!. ~IV.21! It is tempting to conjecture that the elementary functionsRi play, in function space and with the operation of composition, the role of projectors. In fact, this is not so. Their properties m only partially those of the projectorsZi . If we calculateRi@Rj (x)# using~IV.4! and the multino- mial theorem, we findRi@Rj (x)#5Ri(x j / j !). The composition gives rise to a change in t variable. Thus, the functionsRi fail even to satisfy the defining idempotent property: in gene Ri+RiÞRi . An explicit example ofRj (x), for which all these properties can be tried out, is giv below @see~V.6!#. We can introducee5 ln a and rewrite~IV.13! as x~ t !5g^t&~x0!5 ( k>1 eketRk~x0!, ~IV.22! a decomposition ofg^t&(x) into a sum of modes, each one evolving independently according xk~ t !5eketxk~0!5eketRk~x0!. ~IV.23! The imaginary ‘‘frequency’’ke plays the role of a ‘‘modular Lyapunov exponent’’ of thekth mode. We have thus a ‘‘multi-Hamiltonian’’ flowx(t) with one Hamiltonian for each projecto component: Equations~IV.16! and~IV.17! showRk (N) as a representation of the one-dimensio group engendered by thekth dynamical flow Rk@g^t&~x!#5eketRk@g^0&~x!#. ~IV.24! The functiong^t&(x) is actually a Lagrange interpolation in the convenient variable (g1) t 5at, which coincides with the discrete iterates at each integer value oft andkeeps the meaning o a continuum iterate in between. It is a homotopy of all the usual discrete iterations. As announ in Sec. III, we have used the basis$B,B2,B3,...,BN%, instead of$I ,B,B2,...,BN21%, because in the latter the matrix ‘‘I’’ would correspond to the identity function Id(x)5x, which does not have the same end points of the maps of interest. The variablet has been given the sense of a ‘‘time.’’ If it is really time,g^n&(x) will give the nth point of a Poincare´ map. Now, there are in principle~infinitely! many dynamical flows corresponding to a given Poincare´ map. The functiong^t&(x) as given above would correspond a class of them, that of the flows with equal intervals of time between successive points Poincare´ section. To consider other cases, we recall that any strictly monotonous function first-given ‘‘time’’ is another ‘‘time.’’ Finally, we cannot resist exhibiting a ‘‘thing of beauty.’’ By the very manner in which it h been obtained, the expression~IV.12! for the continuum iterate is equivalent to the highly mn monic determinantal equation U g^1&~x! a a2 ... ... aN g^2&~x! a2 a4 ... ... a2N g^3&~x! a3 a6 ... ... a3N ... ... ... ... ... ... g^N&~x! aN a2N ... ... aN2 g^t&~x! at a2t ... ... aNt U50. ~IV.25! Expansion along the first column or the last row and comparison with~IV.11! and ~IV.13! will give determinant expressions forC(N) k(t) andRk (N)(x). hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 y ic map n nd. he t e not 5335J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig V. AN APPLICATION TO TURBULENCE Let us describe an application of our basic result~IV.21!, which shows that a relationship ma exist between harmonic oscillators and turbulence. Consider the special case of the logist g~x!54x~12x!, ~V.1! which is helpful in modeling the velocity field in fully developed turbulence~the ‘‘poor man’s Navier–Stokes equation’’!.12 It takes the interval@0, 1# into itself. The coordinate transformatio x852x21, y852y21, takes it into a map@21,1#→@21,1#. We shall writev(x)52x21, in terms of which the logistic map takes the form g~x!5 1 2@12T2~2x21!#, ~V.2! whereT2(z)52z2215cos@2 arcosz# is the second-order Chebyshev polynomial of the first ki Using v ^21&(u)5 1 2(u11), we notice that T2~z!52v+g+v ^21&~z!. Iterating once this expression, we have@T2+T2#(z)52v+g+v ^21&@2v#+g+v ^21&(z). As v ^21&@2v(u)#512u, but g(12u)5g(u), we arrive atT2 ^2&(z)52v+g^2&+v ^21&(z). In the same way we obtain thenth iterate T2 ^n&~z!52v+g^n&+v ^21&~z!5122g^n&@ 1 2~11z!#. ~V.3! Now, these Chebyshev polynomials13 have a very interesting property, easily proved from t definition: they transform composition into index product: Tn@Tm~z!#5@Tn+Tm#~z!5Tnm~z!. Therefore,T2 ^n&(z)5T2n(z)5cos@2n arcosz#, and we obtain a closed expression for then-iterate of g: g^n&5v ^21&+@2T2n(z)#+v, or g^n&~x!5 1 2$12T2n~z!~2x21!%5 1 2$12cos@2n arcos~2x21!#%5sin2@2n21 arcos~2x21!#. ~V.4! We proceed to find a continuum version of this iterate. The result, g^a&~x!5 1 2$12cos[2a arcos~2x21!%, ~V.5! can be naively anticipated, but we shall prove it. In the present casea54, the eigenvalue alphabe is (41,42,43,...) andmatrix L21 jn in ~III.17! is such that(n>1 L21 kn4 np5dk p . We use the second expression in~V.4! in the form g^n&(x)52(1/2)(p>1@(2)p/(2p)! # 4np@arcos(2x21)#2p and calculate~IV.19!: Rk~x!5 ( n>1 L21 kng ^n&~x!52 1 2 ~2 !k ~2k!! @arcos~2x21!#2k ~V.6! @notice that~IV.6! is trivially satisfied#. Equation~IV.21! gives then g^a&~x!5 ( k>1 4kaRk~x!52 1 2 ( k>1 ~2 !k ~2k!! @2a arcos~2x21!#2k, which is the same as~V.5!. Let us now recall some facts about the double oscillator. Givenx5cosvt andy5cos(nvt), we have of coursey5Tn@x#, which solves the differential equation (12x2)y92xy82n2y50. This leads to the usual, ‘‘isolated,’’ finite Lissajous curves. When the two frequencies ar commensurate,y5cos(nvt), the differential equation becomes hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27 act rves is first um us hich . ld in oduct hich is proper- term- to ore, a degrees s, it f the mics. any h M. to the re 5336 J. Math. Phys., Vol. 39, No. 10, October 1998 R. Aldrovandi and L. P. Freitas This article is copyrig ~12x2!y92xy82n2y50, ~V.7! whose solutiony5cos@n arcosx# is not a polynomial in the general case. It is a well-known f that, whenn is not a rational number, each one of these transcendental, infinite Lissajous cu dense in the~x,y! square,14 and that the topology of the space of such trajectories is not even separable.15 Now, the related functions T2a~z!5cos@2a arcos~2x21!#5122g^a&@ 1 2~11z!# are just the continuum versions of the Chebyshev polynomials obtained from~V.5!. For each value of a, T2a(z) solves~V.7! with n52a, and constitutes one of the Lissajous curves. The continu parametera, which can be interpreted as ‘‘time’’ for the logistic map, is a label for Lissajo curves from the point of view of the double oscillator. It is just the set of its values w constitutes a nonseparable space: all trajectories witha irrational are ultimately indistinguishable However rough the ‘‘poor man’s’’ model may be, the result suggests that the velocity fie turbulent flow can assume a large amount of nonseparable values. VI. CONCLUDING REMARKS In the passage from functions to Bell matrices, composition is translated into matrix pr and iteration into matrix powers. Continuous powers of matrices have a sound meaning, w translated back into continuous iteration and makes of it also a sound concept, respecting ties ~I.1!–~I.2!. The usual discrete iterations are thus extended into a continuous flow. The by-term factorization of thex and t dependencies ofg^t&(x) reveals itself as a decomposition in independently evolving modes, one for each projector of the corresponding Bell matrix. Once it is established that iteration can have a continuous meaning and is, furtherm homotopy, new possibilities are made open to study, such as the use of the conservation of like Brouwer’s and Morse’s. It is our contention that, with the continuum version of iteration will be possible to get a better understanding of the detailed unfolding of bifurcations and o general relationship between the differential and the mapping approaches to chaotic dyna ACKNOWLEDGMENTS This work was supported by CNPg, Brasilia. The authors are grateful to Uriel Frisch for m a hint given during his inspiring lectures at I.F.T. They are also greatly indebted to Dr. Keit Briggs for a very useful mail interchange. Dr. Briggs suggested an interesting relationship Schröder equation, which is under study. 1J. L. McCauley,Classical Mechanics~Cambridge University Press, Cambridge, 1997!. 2See for instance V. I. Arnold,Les Méthodes Mathe´matiques de la Me´chanique Classique~MIR, Moscow, 1976!. 3See for instance S. N. Rasband,Chaotic Dynamics of Nonlinear Systems~Wiley, New York, 1989!. 4P. Henrici,Applied and Computational Complex Analysis~Wiley, New York, 1974!. 5L. Comtet,Advanced Combinatorics~Reidel, Dordrecht, 1974!, whose notations we have adopted. 6R. Aldrovandi and I. Monte Lima, J. Phys. A13, 3685~1980!; Astrophys. Space Sci.90, 179 ~1983!. 7See F. R. Gantmacher,The Theory of Matrices~Chelsea, New York, 1990!, Vol. I. The simple treatment sketched he holds only for matrices with all eigenvalues distinct. 8D. K. Faddeev and V. N. Faddeeva,Computational Methods of Linear Algebra~Freeman, San Francisco, 1963!. 9See for instance R. Aldrovandi and J. G. Pereira,An Introduction to Geometrical Physics~World Scientific, Singapore, 1995!. 10See Ref. 5, Chap. II.3. 11A. Kirillov, Élements de la The´orie des Repre´sentations~MIR, Moscow, 1974!; O. Bratelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I~Springer, New York, 1979!. 12U. Frisch,Turbulence~Cambridge University Press, Cambridge, 1996!. 13T. J. Rivlin, The Chebyshev Polynomials~Wiley, New York, 1974!. 14See, for example, R. Balescu,Equilibrium and Nonequilibrium Statistical Mechanics~Wiley, New York, 1975! ~appen- dix!. 15A. Connes,Noncommutative Geometry~Academic, London, 1994!. hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.174.147 On: Mon, 17 Mar 2014 13:14:27