Ordered and isomorphic mapping of periodic structures in the parametrically forced logistic map

• A direct description of the internal structure of a periodic window in terms of winding numbers is proposed.
• Periodic structures in parameter spaces are mapped in a recurrent and isomorphic way.
• Sequences of winding numbers show global and local organization of periodic domains.

We investigate the periodic domains found in the parametrically forced logistic map, the classical logistic map when its control parameter changes dynamically. Phase diagrams in two-parameter spaces reveal intricate periodic structures composed of patterns of intersecting superstable orbits curves, defining the cell of a periodic window. Cells appear multifoliated and ordered, and they are isomorphically mapped when one changes the map parameters. Also, we identify the characteristics of simplest cell and apply them to other more complex, discussing how the topography on parameter space is affected. By use of the winding number as defined in periodically forced oscillators, we show that the hierarchical organization of the periodic domains is manifested in global and local scales.