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

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 (IT 2 ) 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 1 0 − 9 . We derive a stable invariant one-torus (IT 1 ) and a saddle IT 1 , 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 IT 1 and a saddle IT 1 . Our major concern in this study is whether the qualitative transition from an IT 1 to an IT 2 via QSN bifurcations includes phase-locking. We prove with a precision of 1 0 − 9 that there is no resonance at the bifurcation point. [ABSTRACT FROM AUTHOR]