Bifurcation boundaries of three-frequency quasi-periodic oscillations in discrete-time dynamical system

• This paper analyzes quasi-periodic saddle–node bifurcations in a map.
• They exhibit complex bifurcation structures called Arnold resonance webs.
• An invariant two-torus is born when a stable and a saddle invariant one-torus merge.

This report presents an extensive investigation of bifurcations of quasi-periodic oscillations based on an analysis of a coupled delayed logistic map. This map generates an invariant two-torus (IT22) that corresponds to a three-torus in vector fields. We illustrate detailed Lyapunov diagrams and, by observing attractors, derive a quasi-periodic saddle–node (QSN) bifurcation boundary with a precision of 10−910−9. We derive a stable invariant one-torus (IT11) and a saddle IT11, which correspond to a stable two-torus and a saddle two-torus in vector fields, respectively. We confirmed that the QSN bifurcation boundary coincides with a saddle–node bifurcation point of a stable IT11 and a saddle IT11. Our major concern in this study is whether the qualitative transition from an IT11 to an IT22 via QSN bifurcations includes phase-locking. We prove with a precision of 10−910−9 that there is no resonance at the bifurcation point.