Doubly twisted Neimark–Sacker bifurcation and two coexisting two-dimensional tori

•This study investigates a three-coupled delayed logistic map.•It generates complex quasiperiodic bifurcations.•Two periodic solution coexist in a conventional Arnol'd tongue.•Two two-tori coexist in a high-dimensional Arnol'd tongue.

We discuss a complicated bifurcation structure involving several quasiperiodic bifurcations generated in a three-coupled delayed logistic map where a doubly twisted Neimark–Sacker bifurcation causes a transition from two coexisting periodic attractors to two coexisting invariant closed circles (ICCs) corresponding to two two-dimensional tori in a vector field. Such bifurcation structures are observed in Arnol'd tongues. Lyapunov and bifurcation analyses suggest that the two coexisting ICCs and the two coexisting periodic solutions almost overlap in the two-parameter bifurcation diagram.