Bifurcation analysis of four-frequency quasi-periodic oscillations in a three-coupled delayed logistic map

• This study analyzes Arnol'd resonance web in a map generating invariant three-torus.
• The quasi-periodic saddle-node bifurcations generate complex bifurcations.
• They occur when a stable and saddle invariant two-torus merge and disappear.
• Just after the quasi-periodic saddle-node bifurcation, intermittent torus emerges.

We present herein an extensive analysis of the bifurcation structures of quasi-periodic oscillations generated by a three-coupled delayed logistic map. Oscillations generate an invariant three-torus, which corresponds to a four-dimensional torus in vector fields. We illustrate detailed two-parameter Lyapunov diagrams, which reveal a complex bifurcation structure called an Arnol'd resonance web. Our major concern in this study is to demonstrate that quasi-periodic saddle-node bifurcations from an invariant two-torus to an intermittent invariant three-torus occur because of a saddle-node bifurcation of a stable invariant two-torus and a saddle invariant two-torus. In addition, with some assumptions, we derive a bifurcation boundary between a stable invariant two-torus and a stable invariant three-torus due to a quasi-periodic Hopf bifurcation with a precision of 10−510−5.