By Professor Dr. H. J. Korsch, Professor Dr. H.-J. Jodl (auth.)
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8) and the Chaos generator (Chap. 10). Px x Fig. 3. A family of invariant tori in phase space appears as invariant curves in a Poincare section. A subset of these invariant curves is filied with periodic orbits. 18 2. 3 The KAM Theorem For an integrable system, the entire phase space is filled with invariant tori and any trajectory will remain on the particular torus selected by the initial conditions. e. the existence of N integrals of motion. It is now of considerable interest to investigate the behavior of the system when the integrals of motion are destroyed.
It is investigated numerically in Chap. 9 . For increasing values of the parameter r, the fixed points of P can also lose their stability at r2 and bifurcate again into period-four orbits (fixed points of j4), and so on. 36 2. 14. Pitchfork bifurcation: A stable period-one fixed point x* loses stability at a critical value of the parameter r = rl and a pair of period-two fixed points x~ and x~ is born. The dashed curve marks the position of the unstable fixed point x*. x* x*. 15. e. two fixed points of the map.
The future fate and past history of such a homoclinic point is quite interesting. The continuity of the map implies that the iterates Tn(x) of a neighborhood X of a hornoclinie point must resemble the original set X, which contains an interval of the manifolds H+ and H-. Therefore H+ and H- must also cross in each of the iterates m(X), n = 1,2, ... , and we have aseries of infinitely many homoclinic points, which are mapped onto each other and converge to the hyperbolic fixed point. Moreover, the map is area-preserving and the area enclosed by the branches of H+ and H- between two such hornoclinie crossings is equal to the area enclosed by the branches between the subsequent crossings.