Limit cycles and their period detection via numeric and symbolic hybrid computations

•Computing attracting cycles near bifurcation parameters for logistic and cubic maps.•Realizing convergence to one limit point as opposed to numerical convergence to two.•Computing all the 209 superstable points of period length up to 11, and their degrees.•Calculating some singly or doubly superstable parameters for antisymmetric cubic map.•Relations to philosophy of computer-assisted mathematics, and quasicrystals discovery.