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.14)This is illustrated in Fig.11.1 for the simple case of a fixed point of order 1on a 2-dimensional surface of section (m = 1): the points z.z3 lie on anellipse, if the orbit is stable.They would lie on a hyperbola in the case of anunstable fixed point.Fig.11.1.An elliptic fixed pointr0 is located at the origin of a2-dimensional surface of section.Consecutive points of a neighbor-ing orbit, , lie on an ellipse.To prove this property, consider Eq.(11.13) for any given value of k.Writethe 2m values of »k as an array, x , and the 2m components of z as another» karray, yµ.The result has the form of a linear transformation:(11.15) 336 Chaos and Irreversibilityvwhere A is the µth component of c in the expansion of z0 in Eq.(11.12).µ» » »By inverting (11.15), we obtain , where B is the inverse matrix»of A.Recall now that the 2m values of » are paired, as and , so thatthe 2m values of x a" »k are also paired: x x = 1.It follows that» » »'(11.16)which is just another way of writing Eq.(11.14).The physical interpretation of this result is a conservation law: in the vicinityof the fixed point r0, there is some function of r (that is, of q and p) which isconstant, and Eq.(11.14) represents the first terms in a power expansion of thisfunction.However, some care is needed here, and various possibilities must bedistinguished.For a stable (also called elliptic) fixed point, the sequence of z may denselykcover a closed manifold C.This is an ellipse (or ellipsoid) if the zk are infini-tesimal, a more complicated manifold if they are finite.In particular, for thesimple case of a 2-dimensional surface of section, C is a closed curve whichdivides the surface of section into two parts.If an initial point r is inside thedomain bounded by C, all the consecutive mappings of r will also be inside,by continuity.(Proof: let a continuous curve R connect the fixed point r0 tor0 + z and pass through r.Consecutive mappings of R will connect r0 tor + z without ever leaving the domain bounded by C.)0 kIn the general case of a finite deviation z, the existence of such a manifold isguaranteed, under appropriate conditions, by the KAM theorems.2 4 Considera n integrable Hamiltonian system, namely one with n analytic constants ofmotion in involution.The orbit of r then lies on a n -dimensional torus in the2n-dimensional phase space.If that Hamiltonian is perturbed in a way whichmakes it nonintegrable, there will be fewer than n constants of integration(there may remain only one, the energy).The KAM theorems assert that if theperturbation is analytic, or at least differentiable sufficiently many times, thesetori will be distorted but they will not disappear as long as the perturbation issmall enough.[This property apparently holds under less restrictive conditions,such as for the nonanalytic law of force (11.20) discussed below, but there isno formal proof.] As the perturbation increases, the KAM manifolds graduallybreak down, and the motion of a point r in phase space can explore domainsof dimensionality higher than n (up to 2n 1 dimensions, if energy is the onlyremaining constant of motion of the perturbed Hamiltonian).The situation is more complicated in the case of an unstable (or hyperbolic)fixed point.In the simplest case of a 2-dimensional surface of section, it mayhappen that there exists a continuous curve which behaves as a hyperbola inthe vicinity of the unstable fixed point, and which is mapped onto itself by2A.N.Kolmogorov, Dokl.Akad.Nauk 98 (1954) 527.3V.I.Arnol d, Usp.Mat.Nauk 18 (1963) 13 [transl.Russ.Math.Surveys 18 (1963) 9].4J.Moser, Nachr.Akad.Wiss.Göttingen, Math.Phys.K1 (1962) 1. Discrete maps 337rthe nonlinear transformation r ’! F( ).However, in the generic case, thesequence of zk wanders over all accessible parts of the surface of section thatcan be reached without crossing a KAM line.Such a motion is called irregularor chaotic, because it becomes effectively unpredictable for long times: theslightest perturbation of the initial conditions is amplified, step after step, untilany computer with finite resources is unable to determine new points in a reliableway.The situation is illustrated by Fig.11.2, which displays both KAM linesand chaotic areas.That figure also shows chains of islands of stability, namely KAM lines whichconsist of several disjoint parts.The latter arise whenever a mapping such asthe one sketched in Fig.11.1 does not densely cover a closed curve, but containsonly a finite number of points, because it is itself periodic: z = z.Each one ofsthese points thus is a fixed point of order s.If this new periodic orbit is stable,the associated KAM line consists of s disjoint parts.Case study: area preserving map on a torusConsider a compact 2-dimensional phase space, 1 d" q,p d" 1.The end points 1 and 1 are identified.The discrete map consists of two steps, namely( mod 2 ), (11.17)q ’! q' = q + pfollowed by( mod 2 ).(11
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