Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus

Abstract

We study synchronization of a resonant limit cycle on a two-dimensional torus with an external harmonic signal. The regime of the resonant limit cycle is realized in a system of two coupled Van der Pol oscillators; we consider the resonances 1:1 and 1:3. We analyze the influence of coupling strength between the oscillators. We show that the resonant limit cycle can be generally synchronized on the torus through the resonance destruction followed by the locking of one and then another one of the basic frequencies. We consider the bifurcational mechanism of the synchronization effect.

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.…”

“…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.…”

“…Independently, and approximately over the same period, a significant understanding of quasi-periodic bifurcations was provided by the powerful analyses of Anishchenko et al [18,19]. By investigating coupled van der Pol oscillators containing a forcing term, they demonstrated that the transition from a stable twodimensional torus to a stable three-dimensional torus is triggered by the SN bifurcations of a stable two-dimensional torus and a saddle two-dimensional torus.…”

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

“…Moreover, Anishchenko et al investigated the geometric structure of a QSN bifurcation [12,13]. 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.…”

Multistability in nonlinearly coupled ring of Duffing systems

“…The first one is a global coupling, where all-to-all coupling is present, hence the oscillators are connected with each other by direct links or by the mean field [1][2][3][4]. The second type is a local coupling, where the single oscillator is connected to nodes in its nearest neighborhood [5][6][7][8][9][10]. In this work we consider the local coupling.…”