By Professor Dr. H. J. Korsch, Professor Dr. H.-J. Jodl (auth.)
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In this case, there is apriori no confinement to a two-dimensional submanifold of phase space, and the trajectory may explore all those parts of the phase space which are energetically accessible. This is, however, not the case. The transition from an integrable to a nonintegrable system is most clearly analyzed in one of the most fundamental results in the theory of Hamiltonian (conservative) systems: the celebrated theorem of Kolmogoroff, Arnold, and Moser, which describes the infiuence of perturbations on an integrable system.
Finally, after an infinite number of such bifurcations, the system becomes chaotic. 1O. e. the system approaches a stable equilibrium. For a critical parameter value, this point attractor turns into a limit cyde (a Hopf bifurcation, as discussed in Sect. 6) and the system oscillates periodically. 1O). The motion on the attractor is quasiperiodic, characterized by two frequencies related to the two different rotations on the torus. When this torus-attractor is destabilized by a furt her parameter change, it turns into a strange attractor.
90) with L = (8 f j8xjl* 8f j8yjl* ) = 8(J, g) I 8gj8xjl* 8gj8yjl* The eigenvalues of the matrix L are given by 8(x,y) *. 91) 38 2. 16. Stability of a twodimensional map L. Crossing the boundary of the stability triangle leads to characteristic bifurcations. In the convex region bounded by the dashed line the eigenvalues are complex conjugate. 92) (with Tr L = Ln - L 22 and det L = L n L 22 - L 12 L 21 ), which are interrelated by A+ + A_ = Tr L , A+A_ = det L . 93) The eigenvalues therefore depend on two real valued parameters det Land Tr L, and the character of the fixed point can be conveniently related to different regions in the (detL, TrL)-plane.