By H.J. Korsch, H.-J. Jodl, T. Hartmann
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30), various limit cycles may coexist. This is explored in more detail in Chap. 8. In addition, the period of the limit cycle may be entirely determined by intrinsic properties of the system, and not by any external driving function. Other simple attractors are two-dimensional tori embedded in phase space and supporting a limiting quasiperiodic oscillation characterized by two frequencies. When a parameter of the systems is varied slowly, the attracting limit points and limit cycles also change, while remaining structurally similar.
This is discussed below. 93): both eigenvalues λ± can be real or complex conjugate. 94) (Tr L)2 = 4 det L in the (det L , Tr L) -plane. If both eigenvalues are inside the unit circle |λ± | < 1 the magnitude of the diﬀerence vector contracts, the iterates of the linearized map converge to ξ = 0, and the ﬁxed point is stable. The stability region in the (det L , Tr L) plane is a triangle bounded by the three straight-lines Tr L − det L = 1 , Tr L + det L = −1 , and det L = 1 . 95) When parameters of the system are varied so that one or two eigenvalues cross the stability boundary λ = 1, characteristic bifurcations occur.
3. One observes that the width of the destroyed zones decreases rapidly with increasing s. But, nevertheless, there are inﬁnitely many of them. 0 1 1 6 5 1 7 1 4 1 3 2 7 2 5 1 2 3 7 3 5 4 7 2 3 3 4 5 7 1 4 1 5 6 ω1 / ω2 6 7 Fig. 4. 5 at rational frequency ratio r/s. 24) −5/2 −3/2 = K(ε) s . 23) goes to zero for ε → 0, so that for a suﬃciently weak perturbation the phase space volume not ﬁlled with invariant curves can be made arbitrarily small. The invariant tori in the zones excluded by the KAM condition are in most cases destroyed.