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By Hans Jürgen Korsch; Hansjörg Jodl; Timo Hartmann

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When the parameter g is decreased to negative values, a limit cycle √ of radius −g appears, which attracts all solutions from inside or outside (see Fig. 19). 115) in Cartesian coordinates, yielding d dt x y = −g −ω ω −g x y . 117) The eigenvalues of the matrix are λ± = −g ± iω. At the bifurcation point g = 0, a pair of complex conjugate eigenvalues crosses the imaginary axis, which is characteristic for a Hopf bifurcation. A numerical study of a Hopf bifurcation can be found in the computer experiment in Sect.

R. Meyer, Generic Bifurcations of Periodic Points, Trans. AMS 149 (1970) 95 21. J. M. Greene, R. S. MacKay, F. Vivaldi, and M. J. Feigenbaum, Universal behaviour in families of area–preserving maps, Physica D 3 (1981) 468, reprinted in: R. S. MacKay and J. D. Meiss, Hamiltonian Dynamical Systems, Adam Hilger, Bristol, 1987 22. J. E. Marsden and M. , 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.

14. Pitchfork bifurcation: A stable period-one fixed point x∗ loses stability at a critical value of the parameter r = r1 and a pair of period-two fixed ∗ and x∗ is born. The dashed points x− + curve marks the position of the unstable fixed point x∗ . Another bifurcation of fixed points of one-dimensional maps can be observed when the mapping function f (x, r) becomes tangential to the bisector at a critical parameter value rc , as illustrated in Fig. 15. , two fixed points of the map. At the critical parameter, the slope of f is unity and therefore one of the fixed points is stable (slope smaller than unity) and one is unstable (slope larger than unity).

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