Hopf and quasi-periodic Hopf bifurcations and deterministic coherence in coupled Duffing-Holmes and Van der Pol oscillators: the Arnol’d resonance web

Abstract

A survey on coupled Van der Pol (VDP) and Duffing-Holmes (DH) oscillators, a model widely encountered in various branches of Physics and engineering is done. A bifurcation analysis of the model is presented when both unidirectional and mutual couplings are considered. It is found that stable, unstable and chaotic behaviors appear in the models. Using the method of charts of dynamics regimes in parameter planes, numerical study of the parameters space of the initial differential equations is done. Results from both approaches are compared and discussed. Features of the bifurcation picture are discussed when varying control parameters and analysis of slow-flow equations is presented. We show that the local bifurcation transition from an invariant one-torus (IT1) to an invariant two-torus (IT2) is caused by a Neimark-Saker (NS) bifurcation, also known as a one dimension-higher quasi-periodic Hopf (QH) bifurcation, these by analysing the graph of Lyapunov exponents. We observe that in the mutual coupling case, a complex structure generally-called Arnold’s resonance web phenomenon appears. Numerical simulations are compared to experimental measurements to illustrate the above behaviors. We also note the birth of coherence resonance in the slave oscillator for a certain coupling strength in the case of unidirectional coupling. We also analyze the effect of the coupling strength on the generalized synchronization between the master system and the slave system in the case of unidirectional coupling, through the concept of mutual false nearest neighbors. It allows us to understand when and how closeness in response space implies closeness in driving space.