One-dimensional invariant measure in periodicity hubs: A tribute to Professor Jason A. C. Gallas

Chaotic behavior near a periodicity hub is characterized in five different three-dimensional systems, namely, the paradigmatic Rössler system, the Rosenzweig-MacArthur predator-prey model, a semiconductor laser model, the Gaspard-Nicolis chemical oscillator, and the Nishio-Inaba electronic circuit. Return maps of local maxima for a selected dynamical variable in each system were extracted from numerical solutions. By rescaling the data and assuming full ergodicity in the unit interval, we show that excellent fits to the ubiquitously U-shaped invariant densities are obtained with weighted combinations of the beta and Kumaraswamy distributions.