The bandcount increment scenario. II. Interior structures

Bifurcation structures in the two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In this second part, we investigate fine substructures nested into the basic structures reported and explained in part I. It is demonstrated that the overall structure of the chaotic domain is caused by a complex interaction of bandcount increment, bandcount adding and bandcount doubling structures, whereby some of them are nested into each other ad infinitum leading to self-similar structures in the parameter space.

[1]  M. Feigenbaum The universal metric properties of nonlinear transformations , 1979 .

[2]  Michael Schanz,et al.  Codimension-three bifurcations: explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  E. Ott Chaos in Dynamical Systems: Contents , 1993 .

[4]  Michael Schanz,et al.  On the fully developed bandcount adding scenario , 2008 .

[5]  E. Ott Chaos in Dynamical Systems: Contents , 2002 .

[6]  Michael Schanz,et al.  The bandcount increment scenario. I. Basic structures , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[7]  Soumitro Banerjee,et al.  Robust Chaos , 1998, chao-dyn/9803001.

[8]  J. Milnor On the concept of attractor , 1985 .

[9]  Michael Schanz,et al.  On detection of multi-band chaotic attractors , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  Michael A. Zaks,et al.  Universal scenarios of transitions to chaos via homoclinic bifurcations , 1989 .