Classical and quantum chaos for a kicked top

Reversing k-symmetries in dynamical systems

“…This verifies (11). Moreover (L} is a normal subgroup of Zr{x0 ), for every U ~'~r(x0) is a symmetry or reversing symmetry of L and hence…”

“…The angular momentum of the rotator precesses freely around the y-axis while feeling a periodic strain of impulses inducing a rotation around the z-axis. In case the period of the kicks is one fourth of the period of the free precession the equations of motion (of the components of the angular momentum of the rotator) can be written as [11] x' = z cos(kx) + y sin(kx), L: y' = y cos(kx) -z sin(kx), Hence, we find that L possesses a reversing 2-symmetry group. This group is identical to the group (139) that we discussed in relation to the Fibonacci trace map, but has a different orbit structure under thL.…”

“…3 one observes a dramatic change of the form of the web which can be interpreted as a bifurcation of the stochastic web. 11 The construction of this example is explained in Appendix B. x2 In fact, (98) is an example of map, having a so-called stochastic web possessing crystallographic symmetries, cf. [10].…”

“…This phenomenon can be observed in various well known dynamical systems that occur in the context of problems in physics, e.g. the standard map [14], web maps [10,12,15], a class of trace maps [13,16], and the kicked rotator [11,17]. These examples will be discussed in section 4.…”

Multiple return times in the quantum baker map

“…This map is a com- For small values of the parameters k and p, this system displays regular behaviour in large parts of the sphere, separated by chaotic regions. The regular islands shrink as the parameters grow, leading eventually to almost global chaos (See [4]). This is an interesting feature of this model: varying the control parameters k and p changes the behaviour of the classical model considerably.…”