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On the fully developed bandcount adding scenario. (English) Zbl 1158.37012

The paper investigates the dynamics of the following piecewise-linear discontinuous map \[ x_{n+1}=\left\{ \begin{aligned} f_l(x)&=ax_n +\mu+1\quad\text{if}\quad x_n<0\\ f_r(x)&=ax_n +\mu-1\quad\text{if}\quad x_n>0 \end{aligned} \right. \eqno(1) \] where \(a>0\) and \(\mu\in [-1,1]\) are parameters. In particular, the structure of the chaotic region in the 2D parameter space is described. It is shown that this region has a complex and presumably self-similar structure caused by interior crises of one- and multi-band chaotic attractors. The overall 2D structure is formed by two specific 1D bifurcation scenarios, namely, bandcount adding and bandcount doubling, nested into each other. For the bandcount doubling scenario the scaling properties are determined. For both scenarios it is shown which unstable periodic orbit is responsible for their formation. Some open questions that remain for further investigation are posed.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G10 Bifurcations of singular points in dynamical systems
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