On the Bifurcation of Trapezoid Maps : An Example of a Class of One-Dimensional Maps with a Superconvergent Period Doubling Cascade : Condensed Matter and Statistical Physics

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We investigate the period doubling cascade in symmetric and asymmetric trapezoid maps. We prove that as the slope a of the trapezoid is increased, there occurs a cascade of period doubling bifurcations. Further we prove that the symbolic sequence is the Metropolis-Stein-Stein sequence R^<*m> and that the ratio δ_m of the interval of a, where the 2^m-cycle is stable, to the interval where the 2^<m+1>-cycle is stable scales as δ_m ≃ γ^<(-1)^m/3>(a_cγ^<2/3>)^2^m, where a_c is the accumulation point of the period doubling cascade, and γ is the ratio of the two slopes of the trapezoid. That is, convergence of the onset point of the 2^m -cycle to a_c is extremely rapid, and the Feigenbaum constant δ ≡ lim_<m→∞>δ_m is infinite. Further, we give approximate expressions for the accumulation 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.
理論物理学刊行会の論文
1998-07-25