On the Computation of Finite Invariant Sets of Mappings

This paper suggests a new computational method for determining closed curves that are invariant under a given mapping. Unlike other authors, we discretize not only the curve but also the mapping itself. This allows us to avoid completely the computational difficulties connected with the numerical solution of large linear systems. The method uses simple recurrence formulas, which greatly reduce the execution times. The problem of determining closed curves that are invariant under some given mapping arises in many applications, and various numerical techniques have been proposed for the calculation of such invariant cycles. We refer here only to Doedel [1], Iooss et al. [3], Kevrekidis et al. [4], van Veldhuizen [6], where further references may be found. As in [4], we consider a continuously differentiable mapping (1) (s,x)eS1 xRm^(^(s),F(s,x)), $:51^51, F:Sl xRm^Rm, where S1 is the unit circle in R2, parametrized by s. In other words, (1) is a mapping of the cylinder S1 x Rm into itself. Then (2) 1:S1^S1xRm, 7(8) = (s,r(s)), s 6 S1, is invariant under (1) exactly if (3) r($(s)) = F(s,r(8)), stS1. As usual, for the computation we approximate this curve by some polygon with vertices x¿ « r(s¿), t = 1,2,..., n. But, other than in the cited references, we also discretize the circle mapping $ by the following mapping from N = {1,2,..., n} into itself: (4) V.N^N, V(i) -y [/*(*) \ ieN. Here, [a] denotes the closest integer to a € R1 and {a} the nonnegative fractional part of a. Then, our discretization of (3) has the form (5) x*m =F(sí,Xí), i'eJV, Received March 7, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 14E99, 39-04, 39B30, 65D20. 'This work was supported in part by the Office of Naval Research under contract N-00014-80C-9455, the National Science Foundation under grant DCR-8309926, and the Air Force Office of Scientific Research under grant 84-0131. ©1989 American Mathematical Society 0025-5718/89 $1.00 + $.25 per page