The Breakdown of KAM Trajectories

Transcription

1 The Breakdown of KAM Trajectories D. BENSIMONO ~t L. P. KADANOFF~ OAT& T Bell Laboraiories Murray Hill,New Jersey bthe James Franck Insrirure Chicago, Illinois INTRODUCl'ION Hamiltonian systems with two degrees of freedom, for example, two coupled oscillators, are the simplest class of conservative systems that may exhibit a nontrivial dynamics. In this introduction, we briefly review the phenomenology associated with that class of systems. If the coupling between the two oscillators is linear, the system is integrable and the motion decouples into normal modcs. These are best described in terms of action-angle variables:' I&) -IS I,(r) -I$ 0'0) - dg% + 0:; e2w - %(fit + 05 (1) because the energy is conserved, E - E(If, 13, the motion in phase-space is confined to a torus. If one considers a cross section of the torus (Poincarb section, see Raum I), then the dynamics defines an area-preserving mapping of that section into itself: en+, Wn+J 2~s In+, - I, (2) where the winding number Q(In+,) is the ratio of the two normal modes frequencies, Q - wi/*. Two types of motion are possible: (1) Cyclic motion: If Q is rational (Q- p/q), there exist an infinite number of different cyclic orbits. For any initial condition (8, lo), the sequence {Om) repeats itself with period q: On+, - 0, [mod 24. (2) KAM curve: If Q is irrational, the sequence {On} never repeats, but the points 0, fill out a curve r. If the coupling between the two oscillators is nonlinear, the system is in general nonintegrable. According to the Poincar&Birkhoff theorem? only two cyclic orbits remain for every rational winding number (Q -p/q): an elliptic orbit that nearby points tend to cycle around and a hyperbolic one that nearby points are repelled from. Some KAM curves (irrational Q) may still exist depending on the strength of the nonlinear coupling and their proximity to a rational orbit (a destabilizing factor). However, the most important qualitative and quantitative difference between the 110

2 BENSIMON & KADANOFF BREAKDOWN OF KAM TRAJECTORIES 111 dynamics of integrable and nonintegrable systems is the existence in the latter of regions of stochasticity. These are regions, R, of finite area such that if x lies in R, then there is a subsequence xjn in R that approaches x. Because the dynamics is reversible, stochastic regions are bounded by KAM curves; see FIGURE 2. However, as the strength of the nonlinear coupling is increased, more and more KAM curves disappear, that is, become discontinuous, cantor set-like. As they disappear, the barrier to the extension of chaos vanishes, becomes leaky, and the stochastic regions may thus grow in size. The purpose of this discussion will be to describe how this may happen. Because the phenomenology described previously is very generic and quite independent of the form of the Hamiltonian and its nonlinearities, we will choose to study some particular area-preserving map, namely, the standard map: /-h r K A M CURVE FIGURE 1. Phase-space dynamics for linearly coupled oscillators. The continuous curve is a KAM trajectory. The crosses are elements of an Q = y3 cycle. ESCAPE FROM A CHAOTIC REGION As previously noted, stochastic regions are bounded by KAM curves. Thus, the action of the mapping Tk on a region Rr bounded by a KAM curve r is to map Rr onto itself Tk (R,) = RP (4) In order to understand the breakdown of r, one would like to construct a good estimate to it; this requires one to define what we mean by a good estimate. If we bound a region not by a KAM curve, but by an approximant of r, say 7, then Tk(R7) Ry. (5)

3 112 ANNALS NEW YORK ACADEMY OF SCIENCES The leakiness of y can be measured by the area of Tk(R,) that has escaped, that is, that overlaps with the complement of R, : E - Ry This leakiness is designated LV,, R,) = W,W, n E - R,I; (6) for a KAM curve, L( T,, R,.) - 0. Thus, a good estimate y to the KAM curve r is one that minimizes L( T,, R,) - (6L - 0 and 6 L < 0). RELATING THE LEAKINESS TO THE ACTION For specificity, we concern ourselves with the mapping (equation 3) defined on a cylinder, that is, with 0 = B + 1, and consider the KAM trajectory with winding number Q - (6-1)/2 (the Golden Mean), which is apparently the last one to disappear at k - k, Consider a rational approximant, Q,, = pjq,,. to fl and construct a curve yo passing through all the elements of the elliptic {(0;, r;)} and hyperbolic {(e), r))} cycles of FIGURE 2. Phase-space dynamics for nonlinearly coupled oscillators. Dots: stable elliptic cycles. Crosses: unstable hyperbolic cycles. Continuous curves: KAM trajectories. Shaded area: bounded chaos. (Actually, the whole picture should be shaded because nonstochastic regions have measure zero.) This picture is repeated ad infiniturn in the neighborhood of every elliptic cycle element.

4 BENSIMON & KADANOFF BREAKDOWN OF KAM TRAJECTORIES 113 r FIGURE 3. The leakiness of an approximant, 7, of the KAM curve, r, passing through all the elements of the elliptic and hyperbolic cycles of winding number Q = %. The escaped area is the dashed region. winding number Q,. Let y, be the same curve as yo. but displaced by one unit (see FIGURE 3). R, is the region bounded by yo and y,. Notice that T,(y) intersects y at all of the elements of the cycles. Thus, the escaped area is 1 - L( T,, R,) = 18 r d0-4 e r d6. (7) 2 j *i Also, note that the motion is generated by an action principle in which with a a r = - A(e, e ), r = - A(e, e ), (8) ae ae Therefore, 1 - L( Tk, R,) = z: a A(& 0) d0 + 2 A(& 0 ) do = A; - A ;, (9) 2, e; ae ei ae j

5 114 ANNALS NEW YORK ACADEMY OF SCIENCES where A: and Aj are the total actions for the hyperbolic and elliptic cycles, respectively, of length q, namely, One can show3 that this result is path independent as long as the original path considered, yo, passes through the cycle elements. Thus, yo satisfies the first criterion for a good estimate of P (L is stationary, 61. = 0). However, it does not satisfy the second: yo does not minimize L (actually b2l = 0). Nevertheless, it yields the correct scaling form for the leakiness, that is, the way L scales with q,, (as one gets a better and better estimate of I') and with k - k,. Thus, in order to study the disappearance of a KAM curve and its becoming leaky, one needs to study how the action behaves near k = kr3*' This can be done by a Renormalization Group (RG) analysis, whose principal ideas we discuss next. RENORMALIZATION GROUP IDEAS FOR MAPPINGS Presenting the scaling and RG analysis of the action for the Golden Mean KAM curve near its disappearance is beyond the scope of the present discussion and can be found in references 5-7. However, for the benefit of the neophyte, we will present and exemplify here the essential ideas behind the RG for mappings: (1) Formulate an approximate problem and obtain a recursion relation between the structure of the problem, that is, some characteristic function at the level n + 1, to the structure at previous levels of the approximation. (2) Formulate a scaling ansatz for the characteristic function. (3) Let the approximation be exact, that is, let n - m. Then solve the resulting fixed point equation and study its stability. For example, consider the rational approximant to the Golden Mean KAM curve: where the F,'s are Fibonacci numbers: F,+, - F. = Fa-,, FO - FI = 1. Let6 k T,: r' - r -- sin 2r8, 2u 8' = r' + 0, R: r' = r, Notice that Tk o R = R o Tk [where "0" stands for the compositionfo g =f(g(x))]. Let zf - (rf, 0) be a point belonging to a pn/q,, cycle, ~f = C,,(Z:) s TP o R"'(Z;). (12)

6 BENSIMON & KADANOFF BREAKDOWN OF KAM TRAJECTORIES 115 because P~+I =Pn + Pn-l and qn+l = ; thus, by using the commutativity of R and T,, one obtains the desired relation between the structure of the problem at level n + 1, that is, G,,,, and its structure at levels nand n - 1: Next, make the scaling ansatz, with and s = r - r*. where r* is the value of the KAM curve at B = 0. Finally, let n - m (g,- g*) and write the fixed point equation: g* = &g* 0 & 0 g* 0 G-2 = &-lg* g* 0 & -1. (15) Notice that g* actually represents two functions. This then makes equation 15 harder to solve than a similar fixed point equation obtained for a scalar function such as the action.'" Indeed, to our knowledge, equation 15 has not yet been solved. However, similar RG ideas can be applied to the action for the Golden Mean KAM curve near its disappearance.'-' They result in the following prediction for the scaling of the leakiness L (equation 9): UTk, RqJ = qidol:(qn(k - k,i"). applies respectively when k > k, and k < k,. Its asymptotic behavior ([- m) is k < k, : I,,+(() - Pt (y is a constant), with do = , u = , and where L: is a scaling function that k > k,: L$ - Fd0. Namely, in the supercritical region, we expect (16) (17) This RG analysis using nonoptimal estimates of r was checked against numerical results with optimal estimates of r, that is, curves y, passing through all the elements of the hyperbolic cycle of length q,, which can be shown to minimize the leakiness. The predictions of the RG calculation were thus confirmed.' REFERENCES 1. LICHTENBERG, A. J. & M. A. LIEBERMAN Regular and Stochastic Motion. Springer- Verlag. New York/Berlin.

7 116 ANNALS NEW YORK ACADEMY OF SCIENCES 2. BIRKHOFF, G. D Dynamical Systems. Amer. Math. Soc. New York; see also reference BENSIMON, D. & L. P. UDANOFF Physica 13D MAcKAY, R. S., J. D. MEIS & 1. C. PERCIVAL Physica 13D KADANOFF. L. P Phys. Rev. Lett. 47: SHENKER, S. J. & L. P. KADANOFF J. Stat. Phys. 27: MAcKAY, R. S Physica 'ID283; thesis (Princeton).