Squared sine logistic map

• We propose a perturbated logistic map.
• We observe different routes to chaos.
• We observe an unlimited sequence of center-saddle bifurcations.
• The Lyapunov diagrams present qualitative differences for the routes to chaos.
• The bifurcations present robust scaling features.

A periodic time perturbation is introduced in the logistic map as an attempt to investigate new scenarios of bifurcations and new mechanisms toward the chaos. With a squared sine perturbation we observe that a point attractor reaches the chaotic attractor without following a cascade of bifurcations. One fixed point of the system presents a new scenario of bifurcations through an infinite sequence of alternating changes of stability. At the bifurcations, the perturbation does not modify the scaling features observed in the convergence toward the stationary state.