- Email: [email protected]
Predicting orbits of the Lorenz equation from symbolic dynamics
PHYSlCA ELSEVIER
Physica D 109 (1997) 191-198
Predicting orbits of the Lorenz equation from symbolic dynamics Wei-Mou Zheng 1 Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China
Abstract
To a good approximation the family of maps proposed by Sparrow (1982) for the Lorenz system characterizes the dynamical behavior of the system very well. Such maps are numerically constructed. Symbolic dynamics of the maps is discussed. The procedures to find admissible sequences at given kneading sequences are proposed. By means of the symbolic dynamics allowed orbits are predicted for the Lorenz equation at a typical combination of parameters, and numerically located. PACS:
05.45.+b
1. Introduction
The Lorenz system Jc = - - c r x + cry,
~ = - - x y + r x -- y ,
~ = x y -- b x ,
(1)
where parameters r, cr and b are, respectively, the Rayleigh number, Prandtl number and geometric ratio, has been attracting great attention since the 1960s after the model of fluid dynamics was prop0sedto show that dynamical systems with a few degrees of freedom can possess stochasticity [ 1]. The numerical simulation of the model suggests the existence of contracting foliations due to a strong dissipation in the system. By identifying each contracting foliation with a single point the dimension of the problem can be reduced by 1 at the price of the loss of unique histories for the trajectories. In this simplification one may view the intersection of a trajectory on the attractor with a transversal surface as a one-dimensional object, and then construct a one-dimensional map. An alternative onedimensional map used by Lorenz is extracted by recording the successive local maximum values of the z-coordinate, and plotting them one against the next. In the monograph by Sparrow [2] on the Lorenz equations a family of maps was proposed as 'an obvious choice' for modeling the behavior of the system. We shall call this family of maps the Lorenz-Sparrow (LS) map. In fact, the map may easily be obtained from the return map of the upwards Poincar~ section of a trajectory in the plane z = r - 1, which contains the stationary points. It is our purpose to develop symbolic dynamics theory for the LS map, and then apply it to study the Lorenz system. l Corresponding author. 0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved SO 167-2789(97)00169-3
PII
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Symbolic dynamics of one-dimensional maps is well-understood [3,4]. After numerically constructing the LS map in Section 2; we describe the main ingredients of symbolic dynamics such as kneading sequences, ordering rules and admissibility conditions. When kneading sequences are known for a map, we may predict all the allowed orbits based on the admissibility conditions. However, except for simple maps with single kneading sequence such as the unimodal map and anti-symmetric cubic map, it has never been addressed in the literature how to construct admissible sequences for given kneading sequences. In Section 3 based on the continuity argument we give several propositions for determination of admissible sequences for a given map. Section 4 is devoted to the application of symbolic dynamics for the LS map to the Lorenz system. Finally, in Section 5 we make a few concluding remarks.
2. The LS map and its symbolic dynamics In the study of the Lorenz model the best known path in parameter space is the r-line of the varying Rayleigh number for fixed ~r = 10 and b = 8/3. Let us consider r = 125 as a specific example. The upward Poincar6 section of the attractor in the plane z = r - 1 is shown in Fig. 1. When viewing the attractor as a one-dimensional object every point can be identified with its x-coordinate. By making the transformation
=
x - 36.786 x+36.786
for'x > 0, forx < 0 ,
(2)
we interchange the left subinterval with the right subinterval. The return map for successive ~ is shown in Fig. 2 which belongs to the family of maps proposed by Sparrow for the Lorenz model. We now discuss symbolic dynamics of such maps. For the LS map on the interval shown in Fig. 2, which we denote by f , we divide the interval into four subintervals N, R, L and M at the two critical points C and D, and the middle discontinuous (breaking) point B. The mapping function is anti-symmetric, and is monotonic in each subinterval. The subsequent iterations of an initial point then generate a symbolic sequence that is obtained by registering which subinterval the successive points of an orbit fall in. Assigning the natural order of initial points to their symbolic sequences we may define the order of sequences. We then have the first ordering rule for sequences (3)
N
where, as shown in Fig. 2, letters N. R, L and M are the codes for the four subintervals from left to right, and C, B and D are dividing points. In subintervals R and L the mapping function increases, whereas in subintervals N and M it decreases. From the monotonicity of the mapping function branches we have the second ordering rule for sequences EN... ON...>
< EC... OC...
< ER... > OR...
< EB... > OB...
< EL... > OL...
< ED... > OD...
< EM.... > OM....
(4)
where E (or O) represents a finite string of N. M, R and L containing an even (or odd) total number of letters N and M. We call such strings E and O even and odd. respectively. Due to the symmetry the map is completely determined by two sequences: sequence K of the critical value f ( C ) and sequence H of the rightmost point f ( B _ ) . These may be called the kneading sequences of the map. From the symmetry the sequence/4 of the leftmosl point f ( B + ) is just the mirror image of H. i.e. sequence/1 can be obtained from H by interchanging letters R, N and C with L, M and D, respectively. Similarly, the sequence/( of the other critical value f ( D ) is the mirror image of K. The kneading sequences of the map determine all the
W.-M. Zheng/Physica D 109 (1997) 19l-.198
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30 r
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Fig. 2. The iterated map of ~ extracted from the Poincar6 map of Fig. 1. Compared with Fig. 1, the intervals left and right Io Ihc origin zu'e interchanged here. admissible sequences which c o r r e s p o n d to real orbits o f the map. In terms o f K and H the admissibility conditions for an a l l o w e d s e q u e n c e I o f the map f can be derived to be
K < Ir <_ H,
fI < Nln < N K ,
(5a)
1:1 < It <_ K ,
Mf~ < Mlm <_ H,
(5b)
where lr stands for any element in the set R of all subsequenccs of I that h)llow a letter R, i.e. 1, ~ R = {YI1 = X R Y } .
(6)
The m e a n i n g o f It, lm and In is analogous. O f course, the k n e a d i n g sequences t h e m s e l v e s must also satisfy the admissibility conditions (5). We say that two such sequences which satisfy conditions (5) and then form a k n e a d i n g
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pair are compatible to each other. From conditions (5a) we have K = min{Kr, K n } , i,e, K is shift minimal with respect to R and N. A sequence which has such a property shall, for brevity, be called shift minimal later on.
When we examine the admissibility of a given sequence ! by means of conditions (5) an alternative way is to check only conditions (5a) for both I and its mirror image [ , i.e. we need to look at subsequences following R and N only. We may now define the LS map more precisely by a compatible kneading pair K and H associated with the admissibility conditions (5). While the LS map family with two parameters described by K and H is rather general~ the r-line of the Lorenz system only explores a special path of the parameter plane. Sometimes the Lorenz system may correspond to certain reduced cases. When r (< 30.1) is small enough the branches M and N will disappear. Points C and D reduce to the end points of the interval. Sequence K is then the minimal in R. Furthermore, K and H become dependent on each other, and satisfy H = L/~. When r is large enough, the subinterval N U R always maps to L U M, and vice versa. Sequences K and H are then related by K = L / t . In these two reduced cases conditions (5) are still the same. In this paper we shall consider only the typical case of r = 125.
3. Propositions for admissible sequences The main goal of symbolic dynamics is to determine all the possible motions of the system under study. That is, for a given kneading pair K and H of the LS map we determine all admissible sequences. In practice all the allowed short periodic sequences up to a certain period are very important. In principle, one can generate all possible sequences, and then check their admissibility. However, we may derive some propositions which will make the job much easier. All these propositions are based on the argument of continuity. Due to the symmetry we may consider only periodic sequences whose non-repeating fundamental strings end with R and N. (i) If 11 = U0Ul " • • U n / x " " " a n d 1 2 ~ - U o U 1 • • • U n V ' ' " a r e admissible for a given kneading pair where u i , / X , v {N, R, L, M} and/X ¢ v, then 11 = u o u l . • . u n r is also admissible where ~ 6 {C, B, D} is between/X and v. Of course, C must be followed by K and D by/~, and B should be interpreted as either R H or L/~. This proposition is rather obvious. For example, let u i - 1 = R . Both K <_%u i u i + l . . . U n / x . . . < H and K <_ u i u i + l . . , U n P . . . < H are satisfied. Thus, K < u i u i + l . . . U n T < H is also valid since u i u i + 1 " " U n ~ is between u i u i + 1 . . . U n / x a n d btibli+l
" • • UnP.
It is natural that, under the condition of this proposition, if U C ~ u o u l . - . u n C is admissible, so are U R K and U N K . In fact, the admissibility of U C always implies that of U t K , where t stands for R and N. Assume K = SOSl • • • Sn • "" a n d u i - 1 : R . From the admissibility of U C we have c~ =---s o s l • • • S n - i + l < u i " ' " U n t ~ Ott. If o t ¢ ~r, then U t K satisfies the condition due to K. If ot = ~r, then from the admissibility of the U C string ot must be odd. So, we have to prove S n _ i + 2 S n _ i + 3 • . • > K , w h i c h i s , of course, satisfied. Similarly, we can prove that U t K satisfies the condition due to K for u i - 1 = N . T h e condition due to H can be checked in a similar way to verify that the condition is also satisfied. (ii) If I1 = U O U l . • • u n B = U B and I2 = U C are admissible, so is ( U R ) ~ . As we know, U B a n d U C imply U R H a n d U R K , respectively. Since for 0 < k < n + 1 the kth shift of ( U R ) ~ is always between the kth shifts of U R H and U R K , the admissibility of ( U R ) ~ is then guaranteed. Similarly, one can prove that if U C a n d U N I S I a r e admissible, so is ( U N ) ~ . (iii) If UOUl • • • u n C =- U C a n d U / X . . . a r e admissible, and, in addition, U / x - . . > ( U t ) c~ > U t K , where t {R, N}, then ( U t ) ~ is also admissible. We regard this ordering of the three admissible sequences as 'right'. The proof of this proposition is rather trivial, and is thus omitted. When U / x . . . is just some shorter u o u l . . . u k r , where k < n and r 6 {C, B, D}, the wrong
W.-M. Zheng/Physica D 109 (1997) 191-198
195
ordering will imply a non-admissible ( U t ) °c. For example, for r = C from UOUl .. • u k C > U C string UOUl . . . Uk must be odd, The ordering would lead to uk+l " • " u n t ( U t ) °~ < K , which forbids ( U t ) ec. Similarly, if U C and U v . . . are admissible, and U w K > ( U w ) ~ > U v . . . , where w 6 {R, N}, then ( u o u l . . . u ~ w ) ~ is also admissible. Here, when U v . . . is generated from U with some Un-i replaced by v {C, B, D}, the wrong ordering also means that ( U w ) °° is non-admissible. Furthermore, one can prove: (iv) If U l z . . . , U D and U v . . . are admissible, U t and U w are, respectively, the greater and smaller of U M and U L , then ( U t U w ) °c is admissible when U / z . . - > ( U t U w ) ec > UtB2, and ( U w U t ) ~ is admissible when Uwf~ > (Uwfrt) ~ > Uv....
In this case, we have also the similar statement that the wrong ordering implies the non-admissibility of the periodic sequence under examination. We have discussed admissible sequences associated with sequences ending with C or D. The next proposition is related to a sequence ending with B. (v) If I1 = u o u l " " u n B = U B and /2 = U R . . . are admissible, and I~ = u o u l ' . . u k v , where k < n and 6 {C, B, D}, then ( U R ) ~z is admissible when it is between 11 and 12, otherwise it is non-admissible. Similarly, if I1 = U B and I2 = U L . . . are admissible, ( U L ( J R ) ~z is admissible when it is in between, and non-admissible otherwise.
4. Application to the Lorenz equation We now determine allowed periodic orbits up to period 6 of the Lorenz system at parameters r = 125, cr = 10 and b = 8/3. It is not difficult to find numerically the kneading sequences, which are K = RLRLL
NLRLR...,
H = MRRLR
MLRLR....
(7)
To avoid counting the same period more than once, we regard the shift minimal of all its shifts as its representative, and consider only such sequences. Thus, the non-repeating fundamental string of a periodic sequence in this form ends with either R or N. Due to the symmetry, if a sequence is admissible, so is its mirror image. We may keep only one of the two. Furthermore, we may zonsider only sequences starting with L or R. A sequence starting with N will violate the conditions due to K. The general form of fundamental strings starting with M is M U l q M U z t 2 . . . M U n t n -- W , where ti ~ { R , N } a n d U i is a blank or a string of L and M. String Ua cannot start with M, otherwise its mirror image N N . . • would contradict with K. If the first letter of U1 is L, then the shift minimal of 1~, i.e. the mirror image of W, starts with R. If U1 is a blank, tl cannot be N, otherwise W = ( M N ) °z would contradict with H. In this case, tl = R implies that the shift minimal of !~ starts with L or R. We shall use the propositions given in the Section 3 to generate allowed shift minimal periodic sequences. The greatest sequence starting with L is L K , and the smallest shift minimal sequence is K. The interval J1 = [K, L K ] bounded by these two sequences is the image of the interval J0 = [ N D , R D ] which includes C at the center. Due to the discontinuity of the mapping function at B, after iterations the single-piece interval J0 or J1 may map to multiple pieces, i.e. every time a subinterval contains B its image usually becomes two pieces. Denoting the shift operator by a , then the end points of subintervals must be either a i c , o r c r i B , o r okD, where i, j and k are non-negative integers including zero. When a period n sequence exists, a subinterval in the nth image of J0 will overlap with J0, and contain a region inside which sequences take the non-repeating string of the period as their first n letters. We put L / ( at the top and K at the bottom. Comparing the two sequences, we see that there exists B, which is interpreted as either L/£ = L N L L R L N . • • or R H = R M R R L R M . . . , in between. Comparing further L/£ and L / 4 we find LLIYI = L L N L L R L . . . , LB, LRH = LRMRRLR... and L C between L/~ and L/~. Near L C , which is regarded as either L R K = L R R L R L L . . . or L N K = L N R L R L L . , ., both ( L R ) ~ and ( L N ) ~ are rightly ordered, so they are admissible. Up to this step, the result is shown in the first column of Table 1.
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Table 1 Generating admissible periodic sequences up to period 6 for K = R L R L L N L . LK : LLR~R
LLRLRRMR
LLIZl = LLNLLRLN LB L R H = LRMRRLRM
111111111111 LLCRLRLL LLNI~N
LRLRLR LC = LCRLRLLN LNLNLN
LLNLLCRL IllI~HIIIltHH
• • and H = M R R L R M L .
••
LLNLLRLN LI=I = L N L L R L N B RH = RMRRLRM RMLNRM RD = R D L R L R R M RLLRRL K = RLRLLNLR
There exists L L C between L / ( and L B . ( L L R ) °° results in a wrong ordering, but ( L L N ) c~ does not. So, °° is admissible, but ( L L R ) °° is not. Comparing L K with L L C , we see that there exist no shift minimal sequences up to period 6 in between. (In fact, we may find admissible sequences ( L L R L N ) °°, ( L L N L N ) °~ and (LLNLR) °° there, and all are not shift minimal.) Although L L N L L C exists between ( L L N ) °~ and L B , sequence (LLNLLR) °~ is wrongly ordered, hence forbidden. The above steps are shown in the second column of Table 1. We may continue the proceedures to find more allowed sequences until any nearby two sequences coincide in the first six letters. When both of them do not come from any periodic sequence we should check whether the periodic sequence made of the leading six letters is between the two. The period may be shorter than 6, e.g. when the leading string is R M L N R M , the periodic sequences to be checked are ( R M L N ) °° a n d ( R M L N R M ) °~. If such a periodic sequence is between the two non-periodic sequence, the sequence is allowed, otherwise it is forbidden. We keep only those periodic sequences which are shift minimal. After we examine all periodic sequences associated with allowed sequences ending with C, B and D, as well as those not associated with any sequences ending with C, B and D, we can find all the allowed periodic sequences. They are listed in the first column of Table 2, where only non-repeating strings are given, and for a sequence and its mirror image only one of the two is kept. All these orbits are numerically located [7], and the corresponding orbit points are also listed in the next two columns of the same table. From the table we see that some orbits, e.g. (RMLNLN) ~ a n d ( R M L N ) c~, are very close, and would be easily missed from a numerical search by using homogenously distributed initial points if the precision is not high enough. The knowledge of symbolic dynamics is very useful in locating orbits. We roughly determine the critical point C at x = 19.9029 from 2000 x-values on the attractor by sorting pairs ( x n , X n + l ) according to Xn. (The point C is the minimal point of Xn+l as a function of Xn.) We then assign letters L, M, N and R to intervals (--40.0, --19.9029), (--19.9029, 0.0), (0.0, 19.9029) and (19.9029, 40.0), repectively. From pairs (x, y) of several points on the attrctor sorted according to x, we pick up some points which are roughly equally spaced in x, and then generate sequences for each point. From the ordering rule we know where to look for a certain sequence. We may scan attractor points at a low precision, and generate sequences at a higher precision when the x-coordinate of a point falls in the expected subinterval. If a sequence is found coincidant with the target one in the first few, say 7, letters, we can use the point as an initial trial point for the Newton method to locate the target orbit. In this way we can determine orbits very efficiently. (LLN)
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Table 2 Periodic orbits up to period 6 at r = 125, b = 8/3 and a = 10 Sequence
x
y
LLN LR LN LNLR LNLRLN LNLRLR LNLRMR LNLLN LNLLR RMRLN RMRLR RMLNLR RMLNLN RMLN RLLR RLLN RLLNLN RLLNLR RLRMLN RLRMLR RLRLLR RLRLLN
-23.7940 -25.8026 -32.9691 --34.1507 -34.1665 -34.5169 -35.6238 -35.6728 -36.3245 35.5826 34.9728 32.8501 32.7998 32.7873 28.1808 27.4689 27.3435 27.3254 27.2031 27.1660 26.5101 26.4647
-41.8178 -48.7146 -76.6436 -81.7424 -81.8116 -83.3486 -88.2773 -88.4978 -91.4554 88.0914 85.3643 76.1318 75.9185 75.8651 57.0741 54.2912 53.8014 53.7205 53.2534 53.0988 50.5501 50.3789
:
Furthermore, from the admissibility conditions we may construct allowed chaotic sequences. For example, it can be verified that any sequences consisting of only the segment L R and L N L R are admissible. In principle, we can also locate such an orbit as accurately as the precision allows.
5. Conclusion To a good approximation, the attractor of the Lorenz model on the section z = r - 1 can be viewed as a onedimensional object. The family of maps proposed by Sparrow models most aspects of the Lorenz system behavior very well. Such maps can be extracted directly from numerical data. In the above we have discussed the symbolic dynamics of the Lorenz-Sparrow map, and proposed new procedures to generate admissible sequences for given kneading sequences. Based on the symbolic dynamics a detailed numerical study of the Lorenz model can be performed, including the construction of chaotic orbits. We have analyzed only the case of r = 125, but the procedures are valid for other r. When the attractor is a simple stable period, we may locate an unstable short orbit and then construct the unstable manifolds of the period to find K and H. A n alternative method to obtain a chaotic attractor remnant i s t o take several initial points and keep transient orbits. The procedures to generate admissible sequences for given kneading sequences are based on the argument of continuity. The same idea may be applied to maps such as the circle map and the generic cubic map. The map in a Poincar6 section of the Lorenz model is actually two-dimensional. A complete analysis must be based on two-dimensional symbolic dynamics. Even in a two-dimensional analysis the one-dimensional symbolic dynamics still plays an essential role in determining what we should search for, as we know f r o m the study of symbolic dynamics for the H r n o n map.
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Acknowledgements This work was partially supported by the National Natural Science Foundation of China. The author acknowledges fruitful discussions with Bai-lin Hao and Jun-xian Liu, and is grateful for the kind hospitality of C. Tsallis at the Centro Brasileiro de Pesquisas Ffsicas where this work was conducted.
References [1] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [2] C. Sparrow, The Lorenz Equation: Bifurcations, Chaos, and Strange Attractors (Springer, New York, 1982). [3] J.Mi•n•randW.Thurst•n•in:Dynamica•Systems•ed.J.C.A•exander•LectureN•tesinMathematics•V••. 342(Springer, Berlin, 1988); J. Guckenheimer, in: Dynamical Systems, eds. J. Guckenheimer, J. Moser and S.E. Newhouse, CTME Lectures 1978 (Birkh~iuser, Basel, 1980). [4] P. Collet and J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhaiiser, Basel, 1980). [5] N. Metropolis, M.L. Stein and P.R. Stein, J. Combin. Theory 15 (1973) 25. [6] W.M. Zheng and B.L. Hao, Intern. J. Mod. Phys. B 3 (1989) 1183, [7] W.M. Zheng and J.X. Liu, Commun. Theor. Phys. 27 (1997) 423.
Physica D 109 (1997) 191-198
Predicting orbits of the Lorenz equation from symbolic dynamics Wei-Mou Zheng 1 Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China
Abstract
To a good approximation the family of maps proposed by Sparrow (1982) for the Lorenz system characterizes the dynamical behavior of the system very well. Such maps are numerically constructed. Symbolic dynamics of the maps is discussed. The procedures to find admissible sequences at given kneading sequences are proposed. By means of the symbolic dynamics allowed orbits are predicted for the Lorenz equation at a typical combination of parameters, and numerically located. PACS:
05.45.+b
1. Introduction
The Lorenz system Jc = - - c r x + cry,
~ = - - x y + r x -- y ,
~ = x y -- b x ,
(1)
where parameters r, cr and b are, respectively, the Rayleigh number, Prandtl number and geometric ratio, has been attracting great attention since the 1960s after the model of fluid dynamics was prop0sedto show that dynamical systems with a few degrees of freedom can possess stochasticity [ 1]. The numerical simulation of the model suggests the existence of contracting foliations due to a strong dissipation in the system. By identifying each contracting foliation with a single point the dimension of the problem can be reduced by 1 at the price of the loss of unique histories for the trajectories. In this simplification one may view the intersection of a trajectory on the attractor with a transversal surface as a one-dimensional object, and then construct a one-dimensional map. An alternative onedimensional map used by Lorenz is extracted by recording the successive local maximum values of the z-coordinate, and plotting them one against the next. In the monograph by Sparrow [2] on the Lorenz equations a family of maps was proposed as 'an obvious choice' for modeling the behavior of the system. We shall call this family of maps the Lorenz-Sparrow (LS) map. In fact, the map may easily be obtained from the return map of the upwards Poincar~ section of a trajectory in the plane z = r - 1, which contains the stationary points. It is our purpose to develop symbolic dynamics theory for the LS map, and then apply it to study the Lorenz system. l Corresponding author. 0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved SO 167-2789(97)00169-3
PII
192
W.-M. Zheng/Physica D 109 (1997) 191-198
Symbolic dynamics of one-dimensional maps is well-understood [3,4]. After numerically constructing the LS map in Section 2; we describe the main ingredients of symbolic dynamics such as kneading sequences, ordering rules and admissibility conditions. When kneading sequences are known for a map, we may predict all the allowed orbits based on the admissibility conditions. However, except for simple maps with single kneading sequence such as the unimodal map and anti-symmetric cubic map, it has never been addressed in the literature how to construct admissible sequences for given kneading sequences. In Section 3 based on the continuity argument we give several propositions for determination of admissible sequences for a given map. Section 4 is devoted to the application of symbolic dynamics for the LS map to the Lorenz system. Finally, in Section 5 we make a few concluding remarks.
2. The LS map and its symbolic dynamics In the study of the Lorenz model the best known path in parameter space is the r-line of the varying Rayleigh number for fixed ~r = 10 and b = 8/3. Let us consider r = 125 as a specific example. The upward Poincar6 section of the attractor in the plane z = r - 1 is shown in Fig. 1. When viewing the attractor as a one-dimensional object every point can be identified with its x-coordinate. By making the transformation
=
x - 36.786 x+36.786
for'x > 0, forx < 0 ,
(2)
we interchange the left subinterval with the right subinterval. The return map for successive ~ is shown in Fig. 2 which belongs to the family of maps proposed by Sparrow for the Lorenz model. We now discuss symbolic dynamics of such maps. For the LS map on the interval shown in Fig. 2, which we denote by f , we divide the interval into four subintervals N, R, L and M at the two critical points C and D, and the middle discontinuous (breaking) point B. The mapping function is anti-symmetric, and is monotonic in each subinterval. The subsequent iterations of an initial point then generate a symbolic sequence that is obtained by registering which subinterval the successive points of an orbit fall in. Assigning the natural order of initial points to their symbolic sequences we may define the order of sequences. We then have the first ordering rule for sequences (3)
N
where, as shown in Fig. 2, letters N. R, L and M are the codes for the four subintervals from left to right, and C, B and D are dividing points. In subintervals R and L the mapping function increases, whereas in subintervals N and M it decreases. From the monotonicity of the mapping function branches we have the second ordering rule for sequences EN... ON...>
< EC... OC...
< ER... > OR...
< EB... > OB...
< EL... > OL...
< ED... > OD...
< EM.... > OM....
(4)
where E (or O) represents a finite string of N. M, R and L containing an even (or odd) total number of letters N and M. We call such strings E and O even and odd. respectively. Due to the symmetry the map is completely determined by two sequences: sequence K of the critical value f ( C ) and sequence H of the rightmost point f ( B _ ) . These may be called the kneading sequences of the map. From the symmetry the sequence/4 of the leftmosl point f ( B + ) is just the mirror image of H. i.e. sequence/1 can be obtained from H by interchanging letters R, N and C with L, M and D, respectively. Similarly, the sequence/( of the other critical value f ( D ) is the mirror image of K. The kneading sequences of the map determine all the
W.-M. Zheng/Physica D 109 (1997) 19l-.198
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R
-15
J3 0
L
DI 15
M
-4
30
Fig. 2. The iterated map of ~ extracted from the Poincar6 map of Fig. 1. Compared with Fig. 1, the intervals left and right Io Ihc origin zu'e interchanged here. admissible sequences which c o r r e s p o n d to real orbits o f the map. In terms o f K and H the admissibility conditions for an a l l o w e d s e q u e n c e I o f the map f can be derived to be
K < Ir <_ H,
fI < Nln < N K ,
(5a)
1:1 < It <_ K ,
Mf~ < Mlm <_ H,
(5b)
where lr stands for any element in the set R of all subsequenccs of I that h)llow a letter R, i.e. 1, ~ R = {YI1 = X R Y } .
(6)
The m e a n i n g o f It, lm and In is analogous. O f course, the k n e a d i n g sequences t h e m s e l v e s must also satisfy the admissibility conditions (5). We say that two such sequences which satisfy conditions (5) and then form a k n e a d i n g
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pair are compatible to each other. From conditions (5a) we have K = min{Kr, K n } , i,e, K is shift minimal with respect to R and N. A sequence which has such a property shall, for brevity, be called shift minimal later on.
When we examine the admissibility of a given sequence ! by means of conditions (5) an alternative way is to check only conditions (5a) for both I and its mirror image [ , i.e. we need to look at subsequences following R and N only. We may now define the LS map more precisely by a compatible kneading pair K and H associated with the admissibility conditions (5). While the LS map family with two parameters described by K and H is rather general~ the r-line of the Lorenz system only explores a special path of the parameter plane. Sometimes the Lorenz system may correspond to certain reduced cases. When r (< 30.1) is small enough the branches M and N will disappear. Points C and D reduce to the end points of the interval. Sequence K is then the minimal in R. Furthermore, K and H become dependent on each other, and satisfy H = L/~. When r is large enough, the subinterval N U R always maps to L U M, and vice versa. Sequences K and H are then related by K = L / t . In these two reduced cases conditions (5) are still the same. In this paper we shall consider only the typical case of r = 125.
3. Propositions for admissible sequences The main goal of symbolic dynamics is to determine all the possible motions of the system under study. That is, for a given kneading pair K and H of the LS map we determine all admissible sequences. In practice all the allowed short periodic sequences up to a certain period are very important. In principle, one can generate all possible sequences, and then check their admissibility. However, we may derive some propositions which will make the job much easier. All these propositions are based on the argument of continuity. Due to the symmetry we may consider only periodic sequences whose non-repeating fundamental strings end with R and N. (i) If 11 = U0Ul " • • U n / x " " " a n d 1 2 ~ - U o U 1 • • • U n V ' ' " a r e admissible for a given kneading pair where u i , / X , v {N, R, L, M} and/X ¢ v, then 11 = u o u l . • . u n r is also admissible where ~ 6 {C, B, D} is between/X and v. Of course, C must be followed by K and D by/~, and B should be interpreted as either R H or L/~. This proposition is rather obvious. For example, let u i - 1 = R . Both K <_%u i u i + l . . . U n / x . . . < H and K <_ u i u i + l . . , U n P . . . < H are satisfied. Thus, K < u i u i + l . . . U n T < H is also valid since u i u i + 1 " " U n ~ is between u i u i + 1 . . . U n / x a n d btibli+l
" • • UnP.
It is natural that, under the condition of this proposition, if U C ~ u o u l . - . u n C is admissible, so are U R K and U N K . In fact, the admissibility of U C always implies that of U t K , where t stands for R and N. Assume K = SOSl • • • Sn • "" a n d u i - 1 : R . From the admissibility of U C we have c~ =---s o s l • • • S n - i + l < u i " ' " U n t ~ Ott. If o t ¢ ~r, then U t K satisfies the condition due to K. If ot = ~r, then from the admissibility of the U C string ot must be odd. So, we have to prove S n _ i + 2 S n _ i + 3 • . • > K , w h i c h i s , of course, satisfied. Similarly, we can prove that U t K satisfies the condition due to K for u i - 1 = N . T h e condition due to H can be checked in a similar way to verify that the condition is also satisfied. (ii) If I1 = U O U l . • • u n B = U B and I2 = U C are admissible, so is ( U R ) ~ . As we know, U B a n d U C imply U R H a n d U R K , respectively. Since for 0 < k < n + 1 the kth shift of ( U R ) ~ is always between the kth shifts of U R H and U R K , the admissibility of ( U R ) ~ is then guaranteed. Similarly, one can prove that if U C a n d U N I S I a r e admissible, so is ( U N ) ~ . (iii) If UOUl • • • u n C =- U C a n d U / X . . . a r e admissible, and, in addition, U / x - . . > ( U t ) c~ > U t K , where t {R, N}, then ( U t ) ~ is also admissible. We regard this ordering of the three admissible sequences as 'right'. The proof of this proposition is rather trivial, and is thus omitted. When U / x . . . is just some shorter u o u l . . . u k r , where k < n and r 6 {C, B, D}, the wrong
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ordering will imply a non-admissible ( U t ) °c. For example, for r = C from UOUl .. • u k C > U C string UOUl . . . Uk must be odd, The ordering would lead to uk+l " • " u n t ( U t ) °~ < K , which forbids ( U t ) ec. Similarly, if U C and U v . . . are admissible, and U w K > ( U w ) ~ > U v . . . , where w 6 {R, N}, then ( u o u l . . . u ~ w ) ~ is also admissible. Here, when U v . . . is generated from U with some Un-i replaced by v {C, B, D}, the wrong ordering also means that ( U w ) °° is non-admissible. Furthermore, one can prove: (iv) If U l z . . . , U D and U v . . . are admissible, U t and U w are, respectively, the greater and smaller of U M and U L , then ( U t U w ) °c is admissible when U / z . . - > ( U t U w ) ec > UtB2, and ( U w U t ) ~ is admissible when Uwf~ > (Uwfrt) ~ > Uv....
In this case, we have also the similar statement that the wrong ordering implies the non-admissibility of the periodic sequence under examination. We have discussed admissible sequences associated with sequences ending with C or D. The next proposition is related to a sequence ending with B. (v) If I1 = u o u l " " u n B = U B and /2 = U R . . . are admissible, and I~ = u o u l ' . . u k v , where k < n and 6 {C, B, D}, then ( U R ) ~z is admissible when it is between 11 and 12, otherwise it is non-admissible. Similarly, if I1 = U B and I2 = U L . . . are admissible, ( U L ( J R ) ~z is admissible when it is in between, and non-admissible otherwise.
4. Application to the Lorenz equation We now determine allowed periodic orbits up to period 6 of the Lorenz system at parameters r = 125, cr = 10 and b = 8/3. It is not difficult to find numerically the kneading sequences, which are K = RLRLL
NLRLR...,
H = MRRLR
MLRLR....
(7)
To avoid counting the same period more than once, we regard the shift minimal of all its shifts as its representative, and consider only such sequences. Thus, the non-repeating fundamental string of a periodic sequence in this form ends with either R or N. Due to the symmetry, if a sequence is admissible, so is its mirror image. We may keep only one of the two. Furthermore, we may zonsider only sequences starting with L or R. A sequence starting with N will violate the conditions due to K. The general form of fundamental strings starting with M is M U l q M U z t 2 . . . M U n t n -- W , where ti ~ { R , N } a n d U i is a blank or a string of L and M. String Ua cannot start with M, otherwise its mirror image N N . . • would contradict with K. If the first letter of U1 is L, then the shift minimal of 1~, i.e. the mirror image of W, starts with R. If U1 is a blank, tl cannot be N, otherwise W = ( M N ) °z would contradict with H. In this case, tl = R implies that the shift minimal of !~ starts with L or R. We shall use the propositions given in the Section 3 to generate allowed shift minimal periodic sequences. The greatest sequence starting with L is L K , and the smallest shift minimal sequence is K. The interval J1 = [K, L K ] bounded by these two sequences is the image of the interval J0 = [ N D , R D ] which includes C at the center. Due to the discontinuity of the mapping function at B, after iterations the single-piece interval J0 or J1 may map to multiple pieces, i.e. every time a subinterval contains B its image usually becomes two pieces. Denoting the shift operator by a , then the end points of subintervals must be either a i c , o r c r i B , o r okD, where i, j and k are non-negative integers including zero. When a period n sequence exists, a subinterval in the nth image of J0 will overlap with J0, and contain a region inside which sequences take the non-repeating string of the period as their first n letters. We put L / ( at the top and K at the bottom. Comparing the two sequences, we see that there exists B, which is interpreted as either L/£ = L N L L R L N . • • or R H = R M R R L R M . . . , in between. Comparing further L/£ and L / 4 we find LLIYI = L L N L L R L . . . , LB, LRH = LRMRRLR... and L C between L/~ and L/~. Near L C , which is regarded as either L R K = L R R L R L L . . . or L N K = L N R L R L L . , ., both ( L R ) ~ and ( L N ) ~ are rightly ordered, so they are admissible. Up to this step, the result is shown in the first column of Table 1.
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Table 1 Generating admissible periodic sequences up to period 6 for K = R L R L L N L . LK : LLR~R
LLRLRRMR
LLIZl = LLNLLRLN LB L R H = LRMRRLRM
111111111111 LLCRLRLL LLNI~N
LRLRLR LC = LCRLRLLN LNLNLN
LLNLLCRL IllI~HIIIltHH
• • and H = M R R L R M L .
••
LLNLLRLN LI=I = L N L L R L N B RH = RMRRLRM RMLNRM RD = R D L R L R R M RLLRRL K = RLRLLNLR
There exists L L C between L / ( and L B . ( L L R ) °° results in a wrong ordering, but ( L L N ) c~ does not. So, °° is admissible, but ( L L R ) °° is not. Comparing L K with L L C , we see that there exist no shift minimal sequences up to period 6 in between. (In fact, we may find admissible sequences ( L L R L N ) °°, ( L L N L N ) °~ and (LLNLR) °° there, and all are not shift minimal.) Although L L N L L C exists between ( L L N ) °~ and L B , sequence (LLNLLR) °~ is wrongly ordered, hence forbidden. The above steps are shown in the second column of Table 1. We may continue the proceedures to find more allowed sequences until any nearby two sequences coincide in the first six letters. When both of them do not come from any periodic sequence we should check whether the periodic sequence made of the leading six letters is between the two. The period may be shorter than 6, e.g. when the leading string is R M L N R M , the periodic sequences to be checked are ( R M L N ) °° a n d ( R M L N R M ) °~. If such a periodic sequence is between the two non-periodic sequence, the sequence is allowed, otherwise it is forbidden. We keep only those periodic sequences which are shift minimal. After we examine all periodic sequences associated with allowed sequences ending with C, B and D, as well as those not associated with any sequences ending with C, B and D, we can find all the allowed periodic sequences. They are listed in the first column of Table 2, where only non-repeating strings are given, and for a sequence and its mirror image only one of the two is kept. All these orbits are numerically located [7], and the corresponding orbit points are also listed in the next two columns of the same table. From the table we see that some orbits, e.g. (RMLNLN) ~ a n d ( R M L N ) c~, are very close, and would be easily missed from a numerical search by using homogenously distributed initial points if the precision is not high enough. The knowledge of symbolic dynamics is very useful in locating orbits. We roughly determine the critical point C at x = 19.9029 from 2000 x-values on the attractor by sorting pairs ( x n , X n + l ) according to Xn. (The point C is the minimal point of Xn+l as a function of Xn.) We then assign letters L, M, N and R to intervals (--40.0, --19.9029), (--19.9029, 0.0), (0.0, 19.9029) and (19.9029, 40.0), repectively. From pairs (x, y) of several points on the attrctor sorted according to x, we pick up some points which are roughly equally spaced in x, and then generate sequences for each point. From the ordering rule we know where to look for a certain sequence. We may scan attractor points at a low precision, and generate sequences at a higher precision when the x-coordinate of a point falls in the expected subinterval. If a sequence is found coincidant with the target one in the first few, say 7, letters, we can use the point as an initial trial point for the Newton method to locate the target orbit. In this way we can determine orbits very efficiently. (LLN)
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Table 2 Periodic orbits up to period 6 at r = 125, b = 8/3 and a = 10 Sequence
x
y
LLN LR LN LNLR LNLRLN LNLRLR LNLRMR LNLLN LNLLR RMRLN RMRLR RMLNLR RMLNLN RMLN RLLR RLLN RLLNLN RLLNLR RLRMLN RLRMLR RLRLLR RLRLLN
-23.7940 -25.8026 -32.9691 --34.1507 -34.1665 -34.5169 -35.6238 -35.6728 -36.3245 35.5826 34.9728 32.8501 32.7998 32.7873 28.1808 27.4689 27.3435 27.3254 27.2031 27.1660 26.5101 26.4647
-41.8178 -48.7146 -76.6436 -81.7424 -81.8116 -83.3486 -88.2773 -88.4978 -91.4554 88.0914 85.3643 76.1318 75.9185 75.8651 57.0741 54.2912 53.8014 53.7205 53.2534 53.0988 50.5501 50.3789
:
Furthermore, from the admissibility conditions we may construct allowed chaotic sequences. For example, it can be verified that any sequences consisting of only the segment L R and L N L R are admissible. In principle, we can also locate such an orbit as accurately as the precision allows.
5. Conclusion To a good approximation, the attractor of the Lorenz model on the section z = r - 1 can be viewed as a onedimensional object. The family of maps proposed by Sparrow models most aspects of the Lorenz system behavior very well. Such maps can be extracted directly from numerical data. In the above we have discussed the symbolic dynamics of the Lorenz-Sparrow map, and proposed new procedures to generate admissible sequences for given kneading sequences. Based on the symbolic dynamics a detailed numerical study of the Lorenz model can be performed, including the construction of chaotic orbits. We have analyzed only the case of r = 125, but the procedures are valid for other r. When the attractor is a simple stable period, we may locate an unstable short orbit and then construct the unstable manifolds of the period to find K and H. A n alternative method to obtain a chaotic attractor remnant i s t o take several initial points and keep transient orbits. The procedures to generate admissible sequences for given kneading sequences are based on the argument of continuity. The same idea may be applied to maps such as the circle map and the generic cubic map. The map in a Poincar6 section of the Lorenz model is actually two-dimensional. A complete analysis must be based on two-dimensional symbolic dynamics. Even in a two-dimensional analysis the one-dimensional symbolic dynamics still plays an essential role in determining what we should search for, as we know f r o m the study of symbolic dynamics for the H r n o n map.
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Acknowledgements This work was partially supported by the National Natural Science Foundation of China. The author acknowledges fruitful discussions with Bai-lin Hao and Jun-xian Liu, and is grateful for the kind hospitality of C. Tsallis at the Centro Brasileiro de Pesquisas Ffsicas where this work was conducted.
References [1] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [2] C. Sparrow, The Lorenz Equation: Bifurcations, Chaos, and Strange Attractors (Springer, New York, 1982). [3] J.Mi•n•randW.Thurst•n•in:Dynamica•Systems•ed.J.C.A•exander•LectureN•tesinMathematics•V••. 342(Springer, Berlin, 1988); J. Guckenheimer, in: Dynamical Systems, eds. J. Guckenheimer, J. Moser and S.E. Newhouse, CTME Lectures 1978 (Birkh~iuser, Basel, 1980). [4] P. Collet and J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhaiiser, Basel, 1980). [5] N. Metropolis, M.L. Stein and P.R. Stein, J. Combin. Theory 15 (1973) 25. [6] W.M. Zheng and B.L. Hao, Intern. J. Mod. Phys. B 3 (1989) 1183, [7] W.M. Zheng and J.X. Liu, Commun. Theor. Phys. 27 (1997) 423.