By Hans Jürgen Korsch; Hansjörg Jodl; Timo Hartmann

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Princeton University Press, Princeton, 1963 14. D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys. 20 (1971) 167, reprinted in: Hao Bai-Lin, Chaos, World Scientiﬁc, Singapore, 1984 15. S. E. Newhouse, D. Ruelle, and F. Takens, Occurrence of strange axiom A attractors near quasiperiodic ﬂows on T m , m ≥ 3, Commun. Math. Phys. 64 (1978) 35 16. P. Manneville and Y. Pomeau, Intermittency and the Lorenz model, Phys. Lett. A 75 (1979) 1 17. Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Commun.

This section can be omitted at a ﬁrst reading. 1 The Poincar´ e-Birkhoﬀ Theorem According to the KAM theorem for Hamiltonian systems with two degrees of freedom (see Sect. 2) invariant tori with a suﬃciently irrational ratio of the two basic frequencies ω1 /ω2 remain invariant under a small perturbation of the Hamiltonian. Tori with a rational frequency ratio are excluded by the conditions of the KAM theorem and are, in most cases, destroyed. Let us consider this rational case in more detail. 37) or simply ρi+1 θi+1 =T ρi .

The frictionless motion of a particle on a plane billiard table bounded by a closed curve [2]– [7]. The limiting cases of strictly regular (‘integrable’ ) and strictly irregular (‘ergodic’ or ‘mixed’ ) systems can be illustrated, as well as the typical case, which shows a complicated mixture of regular and irregular behavior. , the consecutive destruction of invariant tori for increasing deviation from integrability as described by the KAM theory and the Poincar´e-Birkhoﬀ theorem discussed in Chap.