Bifurcating vector fields driven by time-scale separated motivational dynamics

Dynamical systems can be designed to exhibit a range of distinct behaviors, which all arise from the same set of continuous dynamics when the latter bifurcates, triggered by a switch in one of its scalar parameters. Building on recent advances that introduce motivation and value dynamics as an efficient way to design multi-behavioral systems, this paper lifts some of the existing restrictions on what kind of planar vector fields can be combined to produce bifurcations. This relaxation enriches the class of dynamical systems that such an approach applies, and gives rise to new behaviors. The paper identifies new analytical conditions under which this new set of planar vector fields can undergo Hopf bifurcations and result in a multi-behavioral system. Numerical simulations and experimental results confirm the theoretical predictions for the existence of the Hopf bifurcations and the applicability of the theory in real systems.