Generic Hopf–Neĭmark–Sacker bifurcations in feed-forward systems

摘要

We show that generic Hopf–Neĭmark–Sacker bifurcations occur in the dynamics of a large class of feed-forward coupled cell networks. To this end we present a framework for studying such bifurcations in parametrized families of perturbed forced oscillators near weak resonance points. Our approach is based on fine-tuning existing normal form techniques. We then apply this framework to show that certain cells in the feed-forward networks exhibit this forced oscillator dynamics and, hence, undergo generic Hopf–Neĭmark–Sacker bifurcations. These bifurcations correspond to the occurrence of resonance tongues in parameter space with a ‘standard’ geometry.