ON THE BIFURCATION OF TRAPEZOID MAPS - AN EXAMPLE OF A CLASS OF ONE-DIMENSIONAL MAPS WITH A SUPERCONVERGENT PERIOD-DOUBLING CASCADE

We investigate the period doubling cascade in symmetric and asymmetric trapezoid maps. We prove that as the slope a of the trapezoid is incr eased, there occurs a cascade of period doubling bifurcations. Further we prove that the: symbolic sequence is the Metropolis-Stein-Stein se quence R(m) and that the ratio delta(m) of the interval of a, where t he 2(m)-cycle is stable, to the interval where the 2(m+l)-cycle is sta ble scales as delta(m) similar or equal to gamma((-1)m/3)(a(c gamma)(2 /3))(2m), where a(c) is the accumulation point of the period doubling cascade, and gamma is the ratio of the two slopes of the trapezoid. Th at is, convergence of the onset point of the 2m-cycle to a, is extreme ly rapid, and the Feigenbaum constant delta = lim(m-->infinity),delta( m) is infinite. Further, we give approximate expressions for the accum ulation point as functions of b, the length of the smaller side of the trapezoid. The agreement between theoretical results and numerically obtained data is excellent.