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

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

“…These bifurcations are generated when a stable invariant two-torus and a saddle invariant two-torus merge and disappear. Discussions in the literature [18,19,25] and the results of the present study strongly suggest that a qualitative transition from a stable n-dimensional torus to a stable (n + 1)-dimensional torus is generated by the saddle-node (SN) bifurcations of a stable n-dimensional torus and a saddle n-dimensional torus.…”

“…In the present study, we investigate complex quasi-periodic bifurcations generated in a higher-dimensional map than those discussed previously [9,14,25]. Our model uses a three-coupled delayed logistic map that can generate an invariant three-torus, and is represented as follows:…”

“…furcation, quasi-periodic saddle-node (QSN) bifurcation, and quasiperiodic period-doubling bifurcation. QH bifurcations are also called quasi-periodic Neimark-Sacker (QNS) bifurcations [9,17,25]. In particular, Broer et al revealed that QSN bifurcations generate extremely complex partial entrainment bifurcation structures called Arnol'd resonance webs where the partial synchronization regions of the lower dimensional quasi-periodic oscillations extend like a web in many directions in the parameter space.…”

“…This transition was identified as a QSN bifurcation in continuous-time dynamical systems. Kamiyama et al attempted to fill the gap between the analyses of Broer et al [1] and Anishchenko et al [18,19] by illustrating Arnol'd resonance webs in a coupled delayed logistic map that generates an invariant two-torus [25]. They showed that QSN bifurcations occur when a stable invariant one-torus and a saddle invariant one-torus merge and disappear.…”

Numerical and experimental observation of Arnol’d resonance webs in an electrical circuit

“…They showed that a stable three-dimensional torus occurs because of the saddle-node bifurcation of a stable two-dimensional torus and a saddle twodimensional torus. Anishchenkos' and our previous results [14,15] strongly suggest that a stable (n + 1)-dimensional torus emerges because of the saddle-node bifurcation of the stable and saddle n-dimensional tori, which is identified as a QSN bifurcation. Such bifurcation structures are noteworthy.…”