Numerical evidences of universality and self-similarity in the Forced Logistic Map

Details Numerical evidences of universality and self-similarity in the Forced Logistic Map Numerical evidences of universality and self-similarity in the Forced Logistic Map

2011-12-18

arxiv.org/abs/1112.4143v1

Mathematics

Dynamical Systems

Scientific paper

We explore different families of quasi-periodically Forced Logistic Maps for the existence of universality and self-similarity properties. In the bifurcation diagram of the Logistic Map it is well known that there exist parameter values $s_n$ where the $2^n$-periodic orbit is superattracting. Moreover these parameter values lay between one period doubling and the next. Under quasi-periodic forcing, the superattracting periodic orbits give birth to two reducibility-loss bifurcations in the two dimensional parameter space of the Forced Logistic Map, both around the points $s_n$. In the present work we study numerically the asymptotic behavior of the slopes of these bifurcations with respect to $n$. This study evidences the existence of universality properties and self-similarity of the bifurcation diagram in the parameter space.

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