Period doubling for trapezoid function iteration: Metric theory

Perturbed period-doubling bifurcation. I. Theory.

Superconvergence of period doubling cascade in trapezoid maps Its Rigorous proof and superconvergence of period doubling cascade starting from a period p solution

In the symmetric and the asymmetric trapezoid maps, as a slope a of the trapezoid is increased, the period doubling cascade occurs and the symbolic sequence of periodic points is the Metropolis-Stein-Stein sequence R∗m and the convergence of the onset point am of the period 2 m solution to the accumulation point ac is exponentially fast. In the previous paper, we proved these results. In this p...

Lacunarity and Period-doubling

We show that the deviation from power laws of the scaling of chaotic measures such as Lyapunov exponents and topological entropy is periodic in the logarithm of the distance from the accumulation of period doubling. Moreover, this periodic function is asymptotically universal for each measure (for functions in the appropriate universality class). This is related to the concept of lacunarity kno...

Period-doubling cascades galore

Period-doubling cascades for the quadratic family are familiar to every student of elementary dynamical systems. Cascades are also often observed in both numerical and scientific experiments. Yet in all but the simplest cases, very little is known about why they exist. When there is one cascade, there are almost always infinitely many. Further, there has been no systematic way of distinguishing...

Chaotic Period Doubling

The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser in the nineteen-seventieth to study the asymptotic small scale geometry of the attractor of one-dimensional systems which are at the transition from simple to chaotic dynamics. This geometry turns out to not depend on the choice of the map under rather mild smoothness conditions. The ex...