Physics LettersA 158 (1991) 51—56 PHYSICS LETTERS A North-Holland Symbolic kinetic analysis of two-dimensional maps A.B. Rechester’ and R.B. White Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA Received 25 April 1991; revised manuscript received 16 June 1991; accepted for publication 17 June 1991 Communicated by A.R. Bishop Symbolic dynamics is used to partition phase space for the Hénon attractor and the Chirikov—Taylor map. The method used for constructing the generating partitions appears to be applicable to many dynamical systems. The discretized time evolution equation is used to calculate sta.~tical properties such as the correlation function. In a previous work [1] symbolic dynamics has of phase space (coarse graining) we define as fol- been used to construct a kinetic equation for a one- lows: let D, be the set of all x in some initial domain dimensional chaotic attractor. Good agreement was D such that any orbit with x 0=x will evolve in time found between the results of the symbolic kinetic to produce exactly the same sequence (Se, Si, ..., equation and orbit averages for the invariant distri- or equivalently the associated value of 1. If the map bution function and correlation function. In this work has an inverse, it can be more convenient to define we apply the same method to two quite different sys- D, using the value of x at some other time than 1=0. tems, that of the chaotic attractor given by the Hénon D, defines a coarse grain element of phase space. For map, and the area-preserving Chirikov—Taylor map. our purposes the sequences define a symbolic dy- Although some differences are found, the method namics for the system if for a given 1 each set D, is described earlier appears to be readily applicable to simply connected. We introduce a rule for construct- these more complex cases. ing the symbol domains which appears to have wide Symbolic dynamics [21 provides a means of par- applicability. As will become apparent, the accuracy titioning phase space so that information concerning of the coarse grain approximations which we will in- the particle orbits is imbedded in the partitioning. troduce depends on the fact that the domains D, be- Consider a sequence of phase space points x0, x1, x2, come small for large n. given by the dynamics, which we call an orbit, the For the maps considered in this work we cannot subscript referring to time, Define a sequence of in- prove that the sets D, are simply connected for all n. tegerss0, s,, s2, ... associated with this orbit according Indeed, the remarkable result we wish to demon- to a prescribed rule. Normally the rule is given by the strate is that for fairly small n, where this property division of p-dimensional phase space into a small can be readily verified numerically, the symbolic ki- number of symbol domains separated by a few netic analysis resulting from this coarse grain par- (p— 1)-dimensional surfaces, with one integer la- titioning reproduces the essential statistical proper- beling each symbol domain, and s~ is determined by ties of the map. the symbol domain to which x, belongs. Now introduce a coarse grain approximation, i.e. Further consider a truncated sequence (So, 5~, ..., approximate the distribution functionf(x, 1) as con- s~1)of length n. It is convenient to introduce a sin- stant in each coarse grain domain D, through gle integer 1 which characterizes the sequence ~ — ft 1) ~ uniquely (see eqs. (7) and (14)). The partitioning “ — .3 MIT, Cambridge, MA, USA. Similarly, let D,, be that part of the set D, which 0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved. 51