How do the classical dynamics of a composite system relate to the entanglement characteristics of the corresponding quantum system? We show that entanglement in nonlinear bipartite systems can be associated with a fixed point bifurcation in the classical description. In a non dissipative system a fixed point corresponds to a quantum stationary state, usually a ground state. Using the example of coupled giant spins we show that, when the fixed point undergoes a supercritical pitchfork bifurcation, the corresponding quantum state achieves a maximum amount of entanglement. By way of contrast, we consider a molecular BEC system that experiences a different kind of bifurcation and does not exhibit a peak in the entanglement corresponding to the bifurcation parameter
