  How does the classical phase space structure for a composite system relate to
the entanglement characteristics of the corresponding quantum system? We
demonstrate how the entanglement in nonlinear bipartite systems can be
associated with a fixed point bifurcation in the classical dynamics. Using the
example of coupled giant spins we show that when a fixed point undergoes a
supercritical pitchfork bifurcation, the corresponding quantum state - the
ground state - achieves its maximum amount of entanglement near the critical
point. We conjecture that this will be a generic feature of systems whose
classical limit exhibits such a bifurcation.
