TRANSITION SPECTRA OF DYNAMICAL-SYSTEMS

The spectra of 'stretching numbers' (or 'local Lyapunov characteristic numbers') are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima c orrespond to maxima or minima in the curve a(theta), where a is the st retching number, and theta the azimuthal angle. We explain the appeara nce of new maxima in the spectra of ordered orbits. The orbits just ou tside the last KAM curve are confined close to this curve for a long t ime (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered or bits just inside the last KAM curve, but later these spectra tend to t he invariant spectrum of the chaotic domain. The sticky spectra are in variant during the stickiness time. The stickiness time increases expo nentially as we approach an island of stability, but very close to an island the increase is superexponential The stickiness time varies sub stantially for nearby orbits; thus we define a probability of escape P -n(x) at time n for every point x. Only the average escape time in a n ot very small interval Delta x around each x is reliable. Then we stud y the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spect rum within a given accuracy. In the regular domain N is small, while i n the chaotic domain it is large. In some ordered cases the convergenc e is anomalously slow. In these cases the maximum value of a(k) in the continued fraction expansion of the rotation number a = [a(0), a(1),. .. a(k),...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calc ulating orbits starting at various paints along a line parallel to the q-axis. A monotonic variation of the sup {q} as a function of the ini tial condition q(0) indicates ordered motions, a jump indicates the cr ossing of a localized chaotic domain, and a V-shaped structure indicat es the crossing of an island. But sometimes the V-shaped structure dis appears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands.