### Abstract

In some examples in coupled dynamical systems, stability of attractors is lost with the increase of the number of coupled elements, where the threshold number for the instability is found to often lie at around 5-10. Numerical results on globally coupled maps, feed-forward neural networks, and genetic networks are described, where prevalance of Milnor attractors as a result of instability is noted. The origin of this critical number 5-10 for instability is discussed as the dominance of 'factorial' over 'exponential'. Following this general argument, we report some results on hierarchical cell differentiation, based on coupled cell dynamics with internal catalytic reactions. We show that Milnor attractors provide a state of a stem cell that can both proliferate and differentiate, i.e., have both stability for reproduction and instability to switch to other cell states.