(r,o)~(r',o')= r-~-~-sin2~r0, r'+0,

Transcription

1 Physica 13D (1984) North-Holland, Amsterdam EXTENDED CHAOS AND DISAPPEARANCE OF KAM TRAJECTORIES David BENSIMON and Leo P. KADANOFF The James Franck Institute, University of Chicago, 564 S. Ellis, Chicago, IL 6637, USA Received 26 October 1983 In two-dimensional area-preserving maps, KAM trajectories serve as natural boundaries between stochastic regions. As they disappear, points may leak out from one region to the next. This escape rate is defined and related to the action-generating function. Analytic and numerical results are presented to describe the behavior of the escape rate near the disappearance of a KAM trajectory. I. Introduction The study of area-preserving maps in two dimensions is motivated by the fact that these maps exhibit the non-trivial dynamics of the simplest class of conservative systems (e.g., Hamiltonian systems with two degrees of freedom). In these mapping problems a sequence of points X/= (5, /) j =,1, 2... is generated by the mapping function T via Xj + 1 = T(Xj). We assume that the mapping is continuous and invertible so that we can study the behavior of the entire sequence Xj (j =, + 1, ). Depending upon the initial point, X, these sequences may show several different behaviors including: Cyclic. The sequence repeats with period q: Xj+q =Xj. Invariant (KAM) curve. The sequence never repeats but the points X/ fill out a curve F. That is, for each x on F, we can find a subsequence Xjo of Xj such that lim Xj =x. r/...~ O n It is also believed that for some starting points, {Xj} shows an area-filling chaotic behavior, namely: Chaos. There exists a region R c of finite area such that if x lies anywhere in R c there is a subsequence X A which approaches x. These different orbits are exhibited in fig. 1. Notice that if a KAM curve F encloses a region R r containing a fixed point X* of T (i.e., T(X*) = X*), then Vx~Rr, n~z: T"(x)~R r. In words: chaos cannot move across KAM curves. If the mapping T depends upon some parameter k, as for example, the standard map Tk, (r,o)~(r',o')= r-~-~-sin2~r, r'+, 1. L ' J ' I ' I ' J '.6 ~-- ~ _ + o + o o 2~- ~,.,..,~"... - R " " ~~..: ~ -.2 ;..~,- "c ~'~'~" ~" -.4 (1.1) -I., I, I, I, I, THETA Fig. 1. Three typical orbits of an area preserving map: the standard map at k = k c. a) Periodic orbit; b) KAM trajectory; c) bounded chaotic orbit /84/$3. Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

2 D. Bensimon and LP. Kadanoff / Extended chaos and disappearance of KAM trajectories 83 then as k is varied and reaches some critical initial value k c, a KAM curve may disappear, i.e., become discontinuous, Cantor set like. In that case a barrier to the extension of chaos vanishes and the chaotic region may thus suddenly grow in size. It is the purpose of this paper to understand what may happen in this sudden growth. Many of the results of the present paper were obtained in parallel by MacKay, Meiss and Percival with whom we had fruitful exchanges [1]. 2. The escape rate from a chaotic region Notice that a KAM curve bounds an invariant region: If R r contains a fixed point and if F is a KAM curve, the function T k maps R r into itself: Tk( Rr)= Rr. (2.1) Once the KAM curve has disappeared there exists no invariant region corresponding to R r. If we look for such a region Rv, bounded by a y which is a closed curve but not a KAM curve, then Tk(Rv) 4: R v. (2.2) If Rv contains a fixed point then the region Tg(Rv) will partially overlap with Rv, (2.3) Here ~2(Rv) is just the area of region Rv. As k approaches the critical value and R v approaches Rr, for F being the critical KAM curve, then the region of overlap will get closer and closer to the entire region R r and the left-hand inequality in eq. (2.3) will get closer and closer to an equality. In this limit we can say that the area escaping from Rr, L(T, R,) = n E- R,) (2.4) (where E - Rr is the complement of Rv), goes to zero. Eq. (2.4) represents the leakiness of the curve "t under the mapping T k. This leakiness is zero for -f being a KAM curve. In parallel, we can define an escape rate for any region as L(Tk,R) E(Tk, R) = ~2(R) (2.5) This escape rate serves as a bound on the rate at which points may leak out of R. Notice that if we apply q steps of T k we have L(Tq, R)<-qL(Tk,R), (2.6) so that the number of steps Q required to reduce the area in TkQ(R)nR to a proportion O of the original area I2(R) is bounded by 1-O (2.7) Q> e(rk,r)" Hence to get the best possible bounds on the permitted motion we should wish to minimize the escape rate (2.5). For this reason we consider, in this paper, the construction of the minimum escape rate from a given region [2]. Imagine that we wish to look for the escape from some region R 1 into another disjoint region R 2. Then consider all regions Rv containing R 1 (Rr ~ R1) and not overlapping R 2 (Rvc E - R2). Under these constraints we define theminimum escape rate from a neighborhood of R 1 into a neighborhood of R 2 to be e( Tk, R1, R2) =inf[e(tk,r,)], (2.8) with the minimum (inf) being taken over all Rv which satisfy the constraints. Since we are interested in the case in which L(Tk, R) may be very small, we also calculate the slightly simpler minimum escaping area l(t k, R1, R 2) = inf[ L(T k, R,)]. (2.9) We expect that the Rv's which give the minimum in (2.8) and (2.9) are essentially identical. Given these definitions, we set out to estimate e and l. To begin, we approximate a curve which is "almost" the KAM curve by constructing one

3 84 D. Bensimon and L.P. Kadanoff / Extended chaos and disappearance of KA M trajectories which goes through all the elements of cycles which approximate the KAM curve. This construction is meaningful even after the KAM curve has disappeared. We note that this curve gives an extremal of l: moving away from the unstable cycle elements increases / while moving away from the stable cycle elements decreases 1 (see appendix A). This extremum does give a bound on l, which can be estimated with the aid of a scaling theory for near critical KAM curves. Then we look at obtaining a better bound by considering curves which pass only through elements of an unstable cycle. 3. Bounding the escape rate by cycles For specificity, we concern ourselves with the mapping (1.1) defined on a cylinder, i.e., with = O + 1, and consider the KAM trajectory with winding number W = (~/5 + 1)/2, which disappears at k = k c = This curve is shown in fig. 1. The initial regions (R 1 and R2) are chosen to lie between two KAM curves by taking Rx to be a narrow strip around r = 1/2, while R 2 includes similar strips about r = + 3/2. We consider the test region Rv to be the area bounded by the curves Y and ~/1, where ~' is defined parametrically by writing O=O(t), Yo: r=r(t), _<t<l, with () = (1) +1, r() = r(1), and "tl is defined as the same curve displaced one step upward: O=O(t), Yt" r=r(t)+ l. A trivial estimate of the escape rate will be obtained if we choose Rv to include all r's and R ' I ' I i i [ i... r+ltl rltit] I]ITI]-I I I I i I, THETA Fig. 2. The escaped area (dashed region) for a trivial test region: the square ( _< r < 1; _< 8 < 1). O's in [,1]. Then the bounding curves are (see fig. 2) r(t) =, Y: O(t) = t; r(t) = 1, o(t)=t, while the images of these curves are (3.1) k r'(t) = 2~r sin2rrt, T(yo): k (3.2) O'(t) = t - ~ sinzlrt, and T(3q) is the same curve displaced one step upward. Since the area enclosed by Y and 7t is one, the escape rate and the escaping area are identical and given by E(Tk,Rv)= L(Tk,Rr) --~. Ikl (3.3) Notice that for k = k c this trivial estimate gives roughly 1% of the area escaping upon one iteration. It is instructive to obtain the result (3.3) in another way. Notice that the curves Y and its image T(Y) pass through the stable fixed point at (r, ) -- (,) and the unstable fixed point at (r, )

4 D. Bensimon and LP. Kadanoff / Extended chaos and disappearance of KAM trajectories 85 = (, ½). The escaped area is ½L(Tk,Rr)=Sol/2r(t)~dt e1/2, do'(t) -Jo r'(t)~dt. (3.4) Remark that the motion is generated by an action principle in which r'(t)= a A(O,O') o-or) ao' } ' '= (t) r(t)= O---~A(O,O') o=o(t), with O'=O'(t) (3.5) A(,') = -½(-') 2-- (2,r) 2 cos2rr. (3.6) Thus, L(T k, Rr) is given by L(Tk,RT)=2fol/2(dO dt do O) A(O,O')dt. dt' " Since the integrand is a total derivative, we find L(Tk, Rv)=2(Au-As). (3.7) Here A u =A(½, ½) is the action for the unstable cycle while A s = A(,) is the action for the stable one. Notice that in the derivation of eq. (3.7) we have used no properties of the paths 7 and 71 except the facts that they cross their image paths "/~ = T(7) and "/~ = T(71) at only the cycle points. Hence, the results (3.7) and (3.3) hold for all paths which pass through the fixed points and have no other crossing points. This result is thus extremal (indeed constant!) under variations of the paths. This result may be immediately generalized to higher order cycles. Let "/o pass through the 2q dements of a stable and unstable cycle of length q as in fig. 3. Let the image path cross the original o.,ok "f/ \ < ' t Lv - o.p \ /1 '\ / d.51, I, x-o'x-,~-'x,i THETA 1 ' I ' I ' I THETA Fig. 3. a) A curve "t (full line) and its image y'= Tk(-t ) (dashed line), passing through all stable () and unstable (x) elements of the p/q = 5/8 cycle; b) segment of the above curves lying in (.68 < r <.72;.2 < _<.55). The escaped area is the dashed region. path only at these 2q points. The stable cycle has -values 87 and the unstable one -values p. Then the escaped area is the union of the shaded regions shown. The net result is a form for L which is u u L(Tg'Rv)= 2E f i' rdo- fo; r'do'. JvO; By exactly the same calculation as before, we can reduce the result to L(Tk, Rv) = 2(Aqu - a q s), (3.8) where A q and A q are the total actions for the stable and unstable cycles of length q, namely q-l{ )2 } Aq= E -½(#j"-Oj~-i ---c s2*r~ ~ j=o (2Ir) 2 m ' (a = u,s). (3.9)

5 86 D. Bensimon and L.P. Kadanoff / Extended chaos and disappearance of KA M trajectories This result is, once again, path independent so long as the original path considered, To passes through the cycle elements and crosses its image path at these points only. This approach yields a sequence of estimates for L(T, R) and E(T, R), by choosing a succession of cycles converging to the KAM curve (or Cantor set for k > kc). For the Golden mean (W = (v~- + 1)/2) these cycles are of period [3] q. = F., where F. is the n th Fibonacci number, and have Oj+q, =~ +p,, with p, = Fn_ 1. For k < k c, the leakage rate Lq.(Tk, R) goes to zero faster than exponentially [4]. For k = k c, zq"(tk, R) "') algebraically in q~ [5, 6, 7]. In fact from the scaling and renormalization group analysis [5, 6, 7] of the action for the Golden mean KAM curve near its disappearance, we expect that Lq"(Tk, R) = qz(~ +Y )L*~ (q, lk- QI~), (3,1) with: d o = x o + Y = ; I, = and where L~(~) is a scaling function which applies respectively when k > k or k < k. Its exponential behavior (~ ~ oo) is k < kc: L* (~)- e -~ (T a constant), k>kc: L*(~)-~ do. Namely, in the supercritical case we expect (3.11) lim zq"(tk, R) = I~ - kcl "d - (3.12) qn"-) oo However, the region k > k is not directly accessible to this analysis since the "stable" cycles tend to bifurcate and increase in number as k passes above k c. Hence, we seek a formulation based only upon unstable cycles. There is another reason for looking to the unstable cycles. If we move our curve "t slightly from the unstable cycle elements, L(T k, Rv) increases at a rate proportional to the separation squared (appendix A). But a corresponding motion away from the stable cycle elements decreases L (T k, R v ). Since we are looking for a minimum of the escape rate we should seek to get away from the stable l I I I I I t.c?" / o.~ / / 1/ ~ x.,. f I I I I I I I I I \1-' Fig. 4. The escaped area (dashed region) for a curve V passing through unstable cycle elements only. A and C are neighboring points of the unstable cycle; B and B' are homoclinic points. elements, and use a curve which passes through unstable elements only. 4. Numerical results using unstable cycles only The curve T passing through all the elements of an unstable cycle is generated in the following way (see fig. 4). Choose two neighboring points A and C of an unstable cycle with winding number W = p/q ~ F,_I/F.. The curve y between these two points is composed of the unstable manifold of A and the stable manifold of C which intersect at the homoclinic point B*. Iterating that segment (ABC) of ~,, q - 1 times generates a continuous curve T passing through all the elements of the unstable cycle. Thus the curve v'= Tk(V) is identical with T except for the segment ABC. Due to the contraction along the stable manifold of C, homoclinic point B is mapped to homoclinic point B'. Since the mapping is area preserving T' must then intersect T in 2m + 1 points (for the standard map at the Golden mean: m = ). The escaped area, S, is the dashed area in fig. 4 between T and T'. The numerical results- concerning * The unstable (stable) manifolds at two neighboring points of the unstable cycle are determined by solving for the eigenvalues of the tangent map and applying the mapping on points along the relevant eigenvectors forward with T k (or backward with rk- 1).

6 D. Bensimon and L.P. Kadanoff / Extended chaos and disappearance of KA M trajectories 87 the dependence of S on k- validate the theoretical predictions: (a) k < kc-we observed the escaped area to scale as: S - q-d L* (ql k -- kcr ) _ q-do exp [ - yql k - kcl" ], taking d o = we could estimate ~/= 6. and ~, The observed value of v is in agreement with the R.G. value: v = The scaling function L* (x) is shown in fig. 5. (b) k = k c- We observed a power law decay of the leakage rate, S, as Q increases (fig. 6) s - q-do. The observed power is:.2. x o +Y = do = 3.5 X1() L' 1 I I 5 Symbol k '~\ ~.9o 4 "*~x.~ "~- --.,,, \ o x _A v ~ * ~-#xo "%~ -,,, I, x-qlk-kcl ~ Fig. 5. The scaling function for k< kc: L*(x). Notice the two branches which are generated by n even (odd) Fibonacci cycles (q, = F,,) ~.3 l Jlltiilll/.8 i ' ] ' I ' ' I ' ~ i~ I ~ 1 ~ I, I ~ I ' I ' I ' I Fig. 7. The escaped area (dashed region), S for a curve passing through unstable cycle elements only at k= 1.5 (> kc). a) p/q = 3/5, S = ; b) p/q = 5/8, S = X 1-3; C) p/q = 8/13, S = I I I I ] ~ I M I I I loglo(q) Fig. 6. The escaped area, S versus q (= F,) at k : kc. (c) k > k c- The results in that regime are particularly interesting since we expect the escaped area to converge to a constant non-zero value at large q. There exists, however, several numerical difficulties in converging to the correct unstable cycle for supercritical values of k. To overcome these difficulties we used an algorithm which consisted in predicting the initial point [ro(k), #(k)]- on the relevant symmetry line-based on the values of (r o, 8 o) for previous smaller values of k. An itera-

7 88 D. Bensimon and L.P. Kadanoff / Extended chaos and disappearance of KAM trajectories tive Newton method was then used to improve the determination of the initial point to an accuracy < 1-lo. To check for the correctness of the convergence, we verified that the order of the unstable cycle elements was the same as for the k = case. A typical example of the form the escaped area exhibits as higher cycles are used is shown in fig. 7 (at k = 1.5). Note the convergence of the leakage rate to four significant digits. In these numerical studies we could not study the behavior of the escaped area for arbitrarily large cycles, since the eigenvalue of the tangent map for large cycles is extremely high (for example: k = 2.28 X 15 for p/q = 55/89 at k = 1.1). Such a high eigenvalue means that after three iterations, with double precision arithmetic, one has lost all information about the initial point. The scaling function L~ (x) is shown in fig. 8. Its asymptotic behavior is given by eq. (3.11) with a critical exponent d o Note however the different form of the scaling function (L*(x) and L*(x)) for n-even (odd) Fibonacci cycles (q, = F,). We do not understand that result in the framework of the renormalization group for 2-D area-preserving maps [5, 6, 7]. t..i..i I I ' ' ' I ' ' - Symbol k o.985 x I.99-2, i.o I # ' I ' / / Acknowledgements We would like to acknowledge helpful discussions and correspondence on this work with R.S. MacKay, I. Percival and J.D. Meiss. This work is supported in parts by Grants NSF DMR MRL and NSF Appendix A We will prove that a curve passing through stable or unstable cycle dements extremizes the escaped area. Consider fig. 9a: In the vicinity of the q-cycle point one can use the tangent map to relate the curve 3' to its image 3" (after q iterations), The dashed area in fig. 9a is the escaped area in the vicinity of. Now shift a segment of 3' a distance e below 3', and consider the image 3" of F I, (/ IY fr' (b/ / ~ Y Y'.. YES// / # / I I IS i.. r, d-iiiiiiia C(I,-,) ~ X -8 I ~,, I -~ -2 2 log [x--q(k-k )v] Fig. 8. The scaling function for k>kc: L*(x). Notice the two branches which are generated by n even (odd) Fibonacci cycles ( q. = F n). Fig. 9. a) The escaped area for a curve y passing through cycle element O; b) the escaped area for the same curve "r moved away slightly (by e) in the neighborhood of cycle element.

8 D. Bensimon and LP. Kadanoff / Extended chaos and disappearance of KAM trajectories 89 that new line, which by now does not pass through (see fig. 9b). The escaped area, S', is the dashed one in fig. 9b. Clearly, - M22 and XA--, - M22 S' = S -- SOAC, D, + SABCD. (A.2) Points A and B are determined by finding the intersection of the line C'F' (the image of CF): Inserting (A.4) and (A.5) into (A.2): y = (M21x -- e)//m11, with the lines y = and y = -e: x^ = /M2x, (1-mxx) XB= M21 e, Then (A.3) /is = S'- S = (Mll + M22-2) e2" 21M2al (A.6) (We have introduced the absolute value of M21, since in fig. 9 M21 is implicitly assumed to be positive.) Therefore, unstable cycles (TrM> 2) minimize the escaped area, and stable cycles (ITr M I< 2) maximize it. References SABCD=e( 1 XA+XB)2 =el1 _(2-Mll (A.4) -To evaluate SOAC, o, we use the fact that the mapping is area conserving and thus So^c'D' = So^'co. A' is the pre-image of A, thus: YA = = M21" X^, [1] R.S. MacKay, J.D. Meiss and I.C. Percival, Physica 13D (1984) 55 (previous paper). [2] E. Wigner, J. Chem. Phys. 5 (1937) 72 and LC. Keck, Adv. Chem. Phys. 13 (1967) 85 consider analogous minimizations of flow rates in phase space. [3] J.M. Greene, J. Math. Phys. 2 (1979) [4] J.N. Mather, preprint, Princeton (1982). [5] S.J. Shenker and L.P. Kadanoff, J. Stat. Phys. 27 (1982) 631. [6] L.P. Kadanoff, Phys. Rev. Lett. 47 (1981) [7] R.S. MacKay, Proc. Conf. "Order in Chaos", Los Alamos (1982), published in Physica 7D (1983) 283. See also thesis (Princeton). The relation between the critical exponents (v, do) used here and those found by MacKay (& a,b) are: v = (logws)-l; do=logw(afl) with W=(vr5 + 1)/2.