Quasi-periodic resonances and the Landau–Hopf scenario

The effect of resonances on a cascade of quasi-periodic bifurcations, the sequence of which occur in accordance with the Landau–Hopf scenario, is examined using an ensemble of discrete van der Pol-Duffing oscillators. With small frequency detunings of the oscillators, tongues of quasi-periodic modes emerge, analogous to Arnold tongues, and in the region of the highest frequency oscillations. With a large frequency detuning, the general structure of regimes transformation in accordance with Landau–Hopf scenario remains, but the quasi-periodic Hopf bifurcation in the cascade can be replaced by a saddle-node bifurcation of torus. Narrow resonance regions based on tori of different dimensions are also observed. At high values of the Duffing oscillator-like nonlinear parameter, resonances can destroy high-dimensional tori in the Landau–Hopf cascade.