From Mandelbrot-like sets to Arnold tongues

•We propose a coupling of two quadratic maps.•There are two coupling terms, one being linear and the other quadratic.•A two-dimensional parameter-space of the coupling is investigated.•A transition from a Mandelbrot-like set to Arnold tongues is detected.

A transition from Mandelbrot-like sets to Arnold tongues is characterized via a coupling of two non-identical quadratic maps proposed by us. A two-dimensional parameter-space considering the parameters of the individual quadratic maps was used to demonstrate numerically the event. The location of the parameter sets where Naimark–Sacker bifurcations occur, which is exactly the place where Arnold tongues of arbitrary periods are born, was computed analytically.