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R. Jauslin / Physics Reports 365 (2002) 1 – 64 Fig. 15. Values of r as function of Fig. 16. Family of hyperbolas − 2 1 + 1 2 + 2 2 29 = ( 1 ; 2 ). = ±r for r = 1; 4; 5; 9; 11. 0 = ( −1 ; −1) and ∗ = (1; −1 ). Fig. 15 represents the values of r for 1=2 1 all such that maxi | i | 6 34. The value of r is invariant with respect to the map N since r N = | · (NQN ) | = | · Q | = r . The invariant r has the following geometric interpretation: Eq. 63) can also be written as with Q = − 2 1 + 1 2 + 2 2 = ±r : This equation gives a family of hyperbolas parametrized by r (see Fig.
The 50 C. R. l ) given by Eq. 7). ) such that H has a smooth invariant torus with frequency !. The dynamics on this attractor is determined by the Gauss map. e. ) such that H has a critical torus with frequency !. 4, we represent a section of this surface, and we analyze the critical function of a one-parameter family of Hamiltonians. ) on S are attracted. On this attractor, we have ÿxed periodic cycles of all the periods; these cycles correspond to quadratic frequencies. For √ instance, we have two ÿxed points: one corresponding to the golden mean and the other one to 2 − 1.
The ÿrst step of the transformation shifts the modes C. R. Jauslin / Physics Reports 365 (2002) 1 – 64 35 into k −1 for k = 1; 2. In particular, the mode 1 is shifted to 0 = (0; 1) which is a non-resonant mode according to the deÿnition of the non-resonant modes given in Section 1. After a rescaling procedure (rescaling of time and of the actions), we deÿne a transformation that reduces the size of the new non-resonant mode 0 from j to j2 (where j denotes the order of the perturbation h(A; ’)).